HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Introduction", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": [""], "variable_overrides": [[]], "questions": [{"name": "1.2.0. Introduction", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Indices, or powers, provide a convenient notation when we need to multiply a number by itself several times. In this Section we explain how indices are written, and state the rules which are used for manipulating them.
\nExpressions built up using non-negative whole number powers of a variable − known as polynomials − occur frequently in engineering mathematics. We introduce some common polynomials in this Section.
\nFinally, scientific notation is used to express very large or very small numbers concisely. This requires use of indices. We explain how to use scientific notation towards the end of the Section.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "The number $4 \\times 4 \\times 4$ is written, for short, as $4^3$ and read ‘$4$ raised to the power $3$’ or ‘$4$ cubed’. Note that the number of times ‘$4$’ occurs in the product is written as a superscript. In this context we call the superscript $3$ an index or power. Similarly we could write
\n$5 \\times 5 = 5^2,\\quad$ read ‘$5$ to the power $2$’ or ‘$5$ squared’
\nand
\n$7 \\times 7 \\times 7 \\times 7 \\times 7 = 7^5\\;, \\quad a \\times a \\times a = a^3\\;,\\quad m\\times m\\times m \\times m = m^4$
\nMore generally, in the expression $x^y$ , $x$ is called the base and $y$ is called the index or power. The plural of index is indices. The process of raising to a power is also known as exponentiation because yet another name for a power is an exponent. When dealing with numbers your calculator is able to evaluate expressions involving powers, probably using the $x^y$ button.
\nUse a calculator to evaluate $3^{12}$
\nUsing the $x^y$ button on the calculator check that you obtain $3^{12}=531441$.
\nIdentify the index and base in the following expressions. (a) $8^{11}$,$\\quad$ (b) $(-2)^5$, $\\quad$ (c) $p^{-q}$
\n(a) In the expression $8^{11}$, $8$ is the base and $11$ is the index.
\n(b) In the expression $(-2)^5$, $-2$ is the base and $5$ is the index.
\n(c) In the expression $p^{-q}$, $p$ is the base and $-q$ is the index. The interpretation of a negative index will be given in sub-section 4 of this book.
\nRecall from Section 1.1 that when several operations are involved we can make use of the BODMAS rule for deciding the order in which operations must be carried out. The BODMAS rule makes no mention of exponentiation. Exponentiation should be carried out immediately after any brackets have been dealt with and before multiplication and division. Consider the following examples.
\nEvaluate $7\\times 3^2$.
\nThere are two operations involved here, exponentiation and multiplication. The exponentiation should be carried out before the multiplication. So $7\\times 3^2 = 7 \\times 9 = 63$.
\nWrite out fully (a) $3m^4\\;,\\quad$ (b) $(3m)^4$.
\n(a) In the expression $3m^4$ the exponentiation is carried out before the multiplication by $3$. So
\n\\(3m^4=3\\times(m\\times m\\times m\\times m) = 3\\times m\\times m \\times m \\times m\\)
\n(b) Here the bracketed expression is raised to the power $4$ and so should be multiplied by itself four times:
\n\\((3m)^4 = (3m)\\times (3m)\\times (3m)\\times (3m)\\)
\nBecause of the associativity of multiplication we can write this as
\n\\( 3\\times 3\\times 3\\times 3 \\times m \\times m \\times m \\times m = 81m^4. \\)
\nNote the important distinction between $(3m)^4$ and $3m^4$.
", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base1": {"name": "base1", "group": "Q1", "definition": "random(2,3)", "description": "", "templateType": "anything", "can_override": false}, "index1": {"name": "index1", "group": "Q1", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "base2": {"name": "base2", "group": "Q1", "definition": "decimal(random(2..15)/10)", "description": "", "templateType": "anything", "can_override": false}, "base3": {"name": "base3", "group": "Q2", "definition": "random(4..29)", "description": "", "templateType": "anything", "can_override": false}, "base4": {"name": "base4", "group": "Q2", "definition": "4", "description": "Should be random(2..6)
\nSet to 4 to demonstrate problem
", "templateType": "anything", "can_override": false}, "index3": {"name": "index3", "group": "Q2", "definition": "random(3..8)", "description": "", "templateType": "anything", "can_override": false}, "index4": {"name": "index4", "group": "Q2", "definition": "7.9", "description": "decimal(random(1..49)/10)
\nSet to 7.9 to demonstrate problem
", "templateType": "anything", "can_override": false}, "ans4": {"name": "ans4", "group": "Q2", "definition": "precround(base4^index4,1)", "description": "", "templateType": "anything", "can_override": false}, "Q3aBase": {"name": "Q3aBase", "group": "Q3", "definition": "random(random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z']),string(random(2..29)))", "description": "", "templateType": "anything", "can_override": false}, "Q3ablockbit": {"name": "Q3ablockbit", "group": "Q3", "definition": "latex(Q3aBase + ' ' + '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "Q3aIdx": {"name": "Q3aIdx", "group": "Q3", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "Q3aExpansion": {"name": "Q3aExpansion", "group": "Q3", "definition": "latex(join(iterate(x,x,Q3aBlockBit,q3aIdx-2),\" \")+' '+Q3aBase)", "description": "", "templateType": "anything", "can_override": false}, "Q3bAlphabet": {"name": "Q3bAlphabet", "group": "Q3", "definition": "shuffle(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "Q3bBaseDeal": {"name": "Q3bBaseDeal", "group": "Q3", "definition": "deal(21)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "Q3bBase1": {"name": "Q3bBase1", "group": "Q3", "definition": "q3bAlphabet[Q3bBaseDeal[0]]", "description": "", "templateType": "anything", "can_override": false}, "Q3bBase2": {"name": "Q3bBase2", "group": "Q3", "definition": "q3bAlphabet[Q3bBaseDeal[1]]", "description": "", "templateType": "anything", "can_override": false}, "Q3bIdx1": {"name": "Q3bIdx1", "group": "Q3", "definition": "Q3bIdxDeal[0]", "description": "", "templateType": "anything", "can_override": false}, "Q3bIdx2": {"name": "Q3bIdx2", "group": "Q3", "definition": "Q3bIdxDeal[1]", "description": "", "templateType": "anything", "can_override": false}, "Q3bBlockBit1": {"name": "Q3bBlockBit1", "group": "Q3", "definition": "latex(Q3bBase1 + ' '+ '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "Q3bBlockBit2": {"name": "Q3bBlockBit2", "group": "Q3", "definition": "latex(Q3bBase2 + ' ' + '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "Q3bExpansion": {"name": "Q3bExpansion", "group": "Q3", "definition": "latex(\nlatex(join(iterate(x,x,Q3bBlockBit1,q3bIdx1-2),\" \"))+\" \"+q3bbase1\n+ latex(\"\\\\times\") + \" \" +\nlatex(join(iterate(x,x,Q3bBlockBit2,q3bIdx2-2),\" \"))+\" \"+q3bbase2\n)", "description": "", "templateType": "anything", "can_override": false}, "Q3bIdxDeal": {"name": "Q3bIdxDeal", "group": "Q3", "definition": "shuffle(2..8)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "q4ans": {"name": "q4ans", "group": "Q4", "definition": "(Q4num/Q4den)^Q4idx", "description": "", "templateType": "anything", "can_override": false}, "Q4num": {"name": "Q4num", "group": "Q4", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "q4den": {"name": "q4den", "group": "Q4", "definition": "random(2..5 except q4num)", "description": "", "templateType": "anything", "can_override": false}, "q4idx": {"name": "q4idx", "group": "Q4", "definition": "random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "Q3dBaseNum": {"name": "Q3dBaseNum", "group": "Q3", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "Q3dBaseDen": {"name": "Q3dBaseDen", "group": "Q3", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "Q3dIdx": {"name": "Q3dIdx", "group": "Q3", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "Q3dBlockBit": {"name": "Q3dBlockBit", "group": "Q3", "definition": "latex('\\\\\\frac{'+Q3DBaseNum+'}{'+Q3dBaseDen + '} '+ '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "Q3dExpansion": {"name": "Q3dExpansion", "group": "Q3", "definition": "latex(join(iterate(x,x,Q3dBlockBit,q3dIdx-2),\" \")+' '+'\\\\\\frac{'+Q3DBaseNum+'}{'+Q3dBaseDen + '} ')", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "Q1", "variables": ["base1", "index1", "base2"]}, {"name": "Q2", "variables": ["base3", "index3", "base4", "index4", "ans4"]}, {"name": "Q3", "variables": ["Q3aBase", "Q3aIdx", "Q3ablockbit", "Q3aExpansion", "Q3bAlphabet", "Q3bBaseDeal", "Q3bIdxDeal", "Q3bBase1", "Q3bBase2", "Q3bIdx1", "Q3bIdx2", "Q3bBlockBit1", "Q3bBlockBit2", "Q3bExpansion", "Q3dBaseNum", "Q3dBaseDen", "Q3dIdx", "Q3dBlockBit", "Q3dExpansion"]}, {"name": "Q4", "variables": ["q4ans", "Q4num", "q4den", "q4idx"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.1.1 Non-calculator index notation evaluation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Asks students to compute (base)^index without a calculator for two simple questions. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Evaluate, without using a calculator,
", "advice": "(a) $\\var{base1}^\\var{index1} =\\var{base1^index1}$
\n(b) $\\var{base2}^2 =\\var{base2^2}$
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base1": {"name": "base1", "group": "Ungrouped variables", "definition": "random(2,3)", "description": "", "templateType": "anything", "can_override": false}, "index1": {"name": "index1", "group": "Ungrouped variables", "definition": "random(2..5)", "description": "", "templateType": "anything", "can_override": false}, "base2": {"name": "base2", "group": "Ungrouped variables", "definition": "decimal(random(2..15)/10)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["base1", "index1", "base2"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "(a) $\\var{base1}^\\var{index1} = $ [[0]]
\n(b) $\\var{base2}^2 = $ [[1]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "part a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{base1}^{index1}", "maxValue": "{base1}^{index1}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "part b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{base2}^2", "maxValue": "{base2}^2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.1.2 Calculator index notation evaluation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Use a calculator to compute (base)^index. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Evaluate using a calculator
", "advice": "Using a calculator,
\n(a) $\\var{base3}^\\var{index3} = \\var{base3^index3}$
\n(b) $(\\var{base4})^\\var{index4} = \\var{ans4}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base3": {"name": "base3", "group": "Ungrouped variables", "definition": "random(4..29)", "description": "", "templateType": "anything", "can_override": false}, "index3": {"name": "index3", "group": "Ungrouped variables", "definition": "random(3..8)", "description": "", "templateType": "anything", "can_override": false}, "base4": {"name": "base4", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}, "index4": {"name": "index4", "group": "Ungrouped variables", "definition": "decimal(random(1..49 except [10,20,30,40])/10)", "description": "", "templateType": "anything", "can_override": false}, "ans4": {"name": "ans4", "group": "Ungrouped variables", "definition": "precround(base4^index4,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["base3", "index3", "base4", "index4", "ans4"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "(a) $\\var{base3}^\\var{index3} = $ [[0]]
\n(b) $(\\var{base4})^\\var{index4} = $[[1]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "part a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{base3}^{index3}", "maxValue": "{base3}^{index3}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "part b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{ans4}", "maxValue": "{ans4}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "dp", "precision": "1", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": false, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.1.3 Write values using index notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Students are given three expanded products and asked to write them in index notation. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write each of the following using index notation.
\nTo enter $3\\times\\left(\\frac{2}{7}\\right)^4$, type 3*(2/7)^4.
(a) $\\var{Q3aExpansion}=\\var{expression(Q3aBase)}^\\var{Q3aIdx}$
\n(b) $\\var{Q3dExpansion}=\\left(\\frac{\\var{Q3dBaseNum}}{\\var{Q3dBaseDen}}\\right)^{\\var{Q3dIdx}}$
\n(c) $\\var{Q3bExpansion}=\\var{expression(Q3bBase1)}^\\var{Q3bIdx1}\\var{expression(Q3bBase2)}^\\var{Q3bIdx2}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"Q3aBase": {"name": "Q3aBase", "group": "q3a", "definition": "random(random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z']),string(random(2..29)))", "description": "", "templateType": "anything", "can_override": false}, "Q3aIdx": {"name": "Q3aIdx", "group": "q3a", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "q3aBlockBit": {"name": "q3aBlockBit", "group": "q3a", "definition": "latex(Q3aBase + ' ' + '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "q3aExpansion": {"name": "q3aExpansion", "group": "q3a", "definition": "latex(join(iterate(x,x,Q3aBlockBit,q3aIdx-2),\" \")+' '+Q3aBase)", "description": "", "templateType": "anything", "can_override": false}, "q3bAlphabet": {"name": "q3bAlphabet", "group": "q3b", "definition": "shuffle(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "q3bIdxDeal": {"name": "q3bIdxDeal", "group": "q3b", "definition": "shuffle(2..8)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "q3bBase1": {"name": "q3bBase1", "group": "q3b", "definition": "q3bAlphabet[0]", "description": "", "templateType": "anything", "can_override": false}, "q3bBase2": {"name": "q3bBase2", "group": "q3b", "definition": "q3bAlphabet[1]", "description": "", "templateType": "anything", "can_override": false}, "q3bIdx1": {"name": "q3bIdx1", "group": "q3b", "definition": "Q3bIdxDeal[0]", "description": "", "templateType": "anything", "can_override": false}, "q3bIdx2": {"name": "q3bIdx2", "group": "q3b", "definition": "Q3bIdxDeal[1]", "description": "", "templateType": "anything", "can_override": false}, "q3bBlockBit1": {"name": "q3bBlockBit1", "group": "q3b", "definition": "latex(Q3bBase1 + ' '+ '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "q3bBlockBit2": {"name": "q3bBlockBit2", "group": "q3b", "definition": "latex(Q3bBase2 + ' ' + '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "q3bExpansion": {"name": "q3bExpansion", "group": "q3b", "definition": "latex(\nlatex(join(iterate(x,x,Q3bBlockBit1,q3bIdx1-2),\" \"))+\" \"+q3bbase1\n+ latex(\"\\\\times\") + \" \" +\nlatex(join(iterate(x,x,Q3bBlockBit2,q3bIdx2-2),\" \"))+\" \"+q3bbase2\n)", "description": "", "templateType": "anything", "can_override": false}, "q3dBaseNum": {"name": "q3dBaseNum", "group": "q3d", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "q3dBaseDen": {"name": "q3dBaseDen", "group": "q3d", "definition": "random(2..10)", "description": "", "templateType": "anything", "can_override": false}, "q3dIdx": {"name": "q3dIdx", "group": "q3d", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "q3dBlockBit": {"name": "q3dBlockBit", "group": "q3d", "definition": "latex('\\\\\\frac{'+Q3DBaseNum+'}{'+Q3dBaseDen + '} '+ '\\\\\\times')", "description": "", "templateType": "anything", "can_override": false}, "q3dExpansion": {"name": "q3dExpansion", "group": "q3d", "definition": "latex(join(iterate(x,x,Q3dBlockBit,q3dIdx-2),\" \")+' '+'\\\\\\frac{'+Q3DBaseNum+'}{'+Q3dBaseDen + '} ')", "description": "", "templateType": "anything", "can_override": false}, "q3bBase1e": {"name": "q3bBase1e", "group": "q3b", "definition": "expression(q3bBase1)", "description": "For display
", "templateType": "anything", "can_override": false}, "q3bBase2e": {"name": "q3bBase2e", "group": "q3b", "definition": "expression(q3bBase2)", "description": "For display
", "templateType": "anything", "can_override": false}, "q3aBasee": {"name": "q3aBasee", "group": "q3a", "definition": "expression(q3abase)", "description": "for display in the advice.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "q3a", "variables": ["Q3aBase", "Q3aIdx", "q3aBlockBit", "q3aExpansion", "q3aBasee"]}, {"name": "q3b", "variables": ["q3bAlphabet", "q3bIdxDeal", "q3bBase1", "q3bBase2", "q3bIdx1", "q3bIdx2", "q3bBlockBit1", "q3bBlockBit2", "q3bExpansion", "q3bBase1e", "q3bBase2e"]}, {"name": "q3d", "variables": ["q3dBaseNum", "q3dBaseDen", "q3dIdx", "q3dBlockBit", "q3dExpansion"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "(a) $\\var{Q3aExpansion}$ = [[0]]
\n(b) $\\var{Q3dExpansion}$ = [[1]]
\n(c) $\\var{Q3bExpansion} =$ [[2]]
", "gaps": [{"type": "jme", "useCustomName": true, "customName": "a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expression(Q3aBase)}^{Q3aIdx}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "You must write your answer using an index"}, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{(Q3dBaseNum/Q3dBaseDen)^Q3dIdx}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "You must write your answer using an index"}, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "c", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expression(Q3bBase1)}^{Q3bIdx1}*{expression(Q3bBase2)}^{Q3bIdx2}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "You must write your answer using indices."}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.1.4 Evaluate a fraction to a power (non-calculator)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Evaluate a simple fraction squared or cubed. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Evaluate without using a calculator. Leave any fractions in frational form.
\nNote: to type $(\\frac{2}{3})^6$, type (2/3)^6.
\\[\\left(\\frac{\\var{Q4num}}{\\var{Q4den}}\\right)^{\\var{Q4idx}}=\\frac{\\var{Q4num}^{\\var{Q4idx}}}{\\var{Q4den}^{\\var{Q4idx}}}=\\frac{\\var{Q4num^Q4idx} }{\\var{Q4den^Q4idx} }\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q4num": {"name": "q4num", "group": "Ungrouped variables", "definition": "random(1..5)", "description": "", "templateType": "anything", "can_override": false}, "q4den": {"name": "q4den", "group": "Ungrouped variables", "definition": "random(2..5 except q4num)", "description": "", "templateType": "anything", "can_override": false}, "q4idx": {"name": "q4idx", "group": "Ungrouped variables", "definition": "random(2..3)", "description": "", "templateType": "anything", "can_override": false}, "q4ans": {"name": "q4ans", "group": "Ungrouped variables", "definition": "(Q4num/Q4den)^Q4idx", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q4num", "q4den", "q4idx", "q4ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\left(\\frac{\\var{Q4num}}{\\var{Q4den}}\\right)^{\\var{Q4idx}}=$[[0]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "answer", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q4ans}", "maxValue": "{q4ans}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": false, "mustBeReducedPC": "50", "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Laws of indices", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "1.2.2. Laws of Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Add, subtract and multiply indices. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "There is a set of rules that enables us to manipulate expressions involving indices. These rules are known as the laws of indices, and they occur so commonly that it is worthwhile to memorise them.
\nThe laws of indices state:
\n| First law: | \n$a^m\\times a^n = a^{m+n}$ | \nadd indices when multiplying numbers with the same base | \n
| Second law: | \n$\\frac{a^m}{a^n} = a^{m-n}$ | \nsubtract indices when dividing numbers with the same base | \n
| Third law: | \n$(a^m)^n = a^{mn}$ | \nmultiply indices together when raising a number to a power | \n
\n
Example 16
\nSimplify (a) $a^5\\times a^4$, $\\qquad$ (b) $2x^5(x^3)$.
\nSolution
\nIn each case we are required to multiply expressions involving indices. The bases are the same and we use the first law of indices.
(a) The indices must be added, thus $a^5\\times a^4 = a^{5+4} = a^9$.
(b) Because of the associativity of multiplication we can write
\\(2x^5(x^3)=2(x^5x^3)=2x^{(5+3)}=2x^8\\)
\nThe first law of indices (Key Point 5) extends in an obvious way when more terms are involved:
\nExample 17
\nSimplify $b^5\\times b^4 \\times b^7$.
\nSolution
\nThe indices are added. Thus $b^5\\times b^4 \\times b^7=b^{5+4+7}=b^{16}$.
\n", "advice": "Task 1:
\n$\\var{a_name}^{\\var{i1_expr}}\\times \\var{a_name}^{\\var{i2_expr}}\\times \\var{a_name}^{\\var{i3_expr}} = \\var{a_name}^{\\var{i1_expr}+\\var{i2_expr}+\\var{i3_expr}} = \\var{a_ans}$
\nTask 2:
\n$\\var{b_name}^\\var{b_i1_expr}\\div\\var{b_name}^\\var{b_i2_expr} = \\var{b_ans}$
\nTask 3:
\n$(\\var{c_name}^\\var{c_i1_expr})^\\var{c_i2_expr} = \\var{c_name}^{\\var{c_i1_expr}\\times\\var{c_i2_expr}}=\\var{c_ans}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Question a", "definition": "random(random(['a','b','c','d','e','f','g','h','k','m','n','p','q','r','s','t','w','x','y','z']),string(random(2..9)))", "description": "", "templateType": "anything", "can_override": false}, "a_name": {"name": "a_name", "group": "Question a", "definition": "expression(a)", "description": "", "templateType": "anything", "can_override": false}, "i1": {"name": "i1", "group": "Question a", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "i2": {"name": "i2", "group": "Question a", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "i3": {"name": "i3", "group": "Question a", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "question b", "definition": "random(random(['a','b','c','d','e','f','g','h','k','m','n','p','q','r','s','t','w','x','y','z']),string(random(2..9)))", "description": "", "templateType": "anything", "can_override": false}, "b_name": {"name": "b_name", "group": "question b", "definition": "expression(b)", "description": "", "templateType": "anything", "can_override": false}, "b_i1": {"name": "b_i1", "group": "question b", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "b_i2": {"name": "b_i2", "group": "question b", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "question c", "definition": "random(random(['a','b','c','d','e','f','g','h','k','m','n','p','q','r','s','t','w','x','y','z']),string(random(2..9)))", "description": "", "templateType": "anything", "can_override": false}, "c_name": {"name": "c_name", "group": "question c", "definition": "expression(c)", "description": "", "templateType": "anything", "can_override": false}, "c_i1": {"name": "c_i1", "group": "question c", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "c_i2": {"name": "c_i2", "group": "question c", "definition": "random(string(random(2..10)),random(alphabet))", "description": "", "templateType": "anything", "can_override": false}, "ex1_base": {"name": "ex1_base", "group": "Exercises", "definition": "[random(string(random(2..29)),random(alphabet)),\n random(string(random(2..29)),random(alphabet)),\n random(string(random(2..29)),random(alphabet))]", "description": "", "templateType": "anything", "can_override": false}, "ex1_idx": {"name": "ex1_idx", "group": "Exercises", "definition": "[random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10))]", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','k','m','n','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "ex1_base_expr": {"name": "ex1_base_expr", "group": "Exercises", "definition": "[expression(ex1_base[0]),expression(ex1_base[1]),expression(ex1_base[2])]", "description": "", "templateType": "anything", "can_override": false}, "c_i1_expr": {"name": "c_i1_expr", "group": "question c", "definition": "expression(c_i1)", "description": "", "templateType": "anything", "can_override": false}, "c_i2_expr": {"name": "c_i2_expr", "group": "question c", "definition": "expression(c_i2)", "description": "", "templateType": "anything", "can_override": false}, "c_ans": {"name": "c_ans", "group": "question c", "definition": "simplify(expression(c+\"^\"+\"(\"+c_i1 + \"*\" + c_i2+\")\"),[\"basic\",\"collectNumbers\"])", "description": "", "templateType": "anything", "can_override": false}, "i1_expr": {"name": "i1_expr", "group": "Question a", "definition": "expression(i1)", "description": "", "templateType": "anything", "can_override": false}, "i2_expr": {"name": "i2_expr", "group": "Question a", "definition": "expression(i2)", "description": "", "templateType": "anything", "can_override": false}, "i3_expr": {"name": "i3_expr", "group": "Question a", "definition": "expression(i3)", "description": "", "templateType": "anything", "can_override": false}, "b_i1_expr": {"name": "b_i1_expr", "group": "question b", "definition": "expression(b_i1)", "description": "", "templateType": "anything", "can_override": false}, "b_i2_expr": {"name": "b_i2_expr", "group": "question b", "definition": "expression(b_i2)", "description": "", "templateType": "anything", "can_override": false}, "b_ans": {"name": "b_ans", "group": "question b", "definition": "simplify(expression(b+\"^\"+\"(\"+b_i1 + \"-\" + b_i2+\")\"),[\"basic\",\"collectNumbers\"])", "description": "", "templateType": "anything", "can_override": false}, "a_ans": {"name": "a_ans", "group": "Question a", "definition": "simplify(expression(a+\"^\"+\"(\"+i1 + \"+\" + i2 + \"+\" + i3 +\")\"),[\"basic\",\"collectNumbers\"])", "description": "", "templateType": "anything", "can_override": false}, "ans1c": {"name": "ans1c", "group": "Exercises", "definition": "expression(ex1_base[2] + \"^\" + string(ex1_idx[4]*ex1_idx[5]))", "description": "", "templateType": "anything", "can_override": false}, "ans1b": {"name": "ans1b", "group": "Exercises", "definition": "expression(ex1_base[1] + \"^\" + string(ex1_idx[2]-ex1_idx[3]))", "description": "", "templateType": "anything", "can_override": false}, "ans1a": {"name": "ans1a", "group": "Exercises", "definition": "expression(ex1_base[0] + \"^\" + string(ex1_idx[0]+ex1_idx[1]))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet"], "variable_groups": [{"name": "question c", "variables": ["c", "c_ans", "c_i1", "c_i1_expr", "c_i2", "c_i2_expr", "c_name"]}, {"name": "Exercises", "variables": ["ex1_base", "ex1_idx", "ex1_base_expr", "ans1c", "ans1b", "ans1a"]}, {"name": "Question a", "variables": ["a", "a_name", "i1", "i2", "i3", "i1_expr", "i2_expr", "i3_expr", "a_ans"]}, {"name": "question b", "variables": ["b", "b_name", "b_i1", "b_i2", "b_i1_expr", "b_i2_expr", "b_ans"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": true, "customName": "Task 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Simplify $\\var{a_name}^{\\var{i1_expr}}\\times \\var{a_name}^{\\var{i2_expr}}\\times \\var{a_name}^{\\var{i3_expr}}$
\nNote: to enter $x^{(y + z)}$, type x^(y+z)
All quantities have the same base. To multiply the quantities together, the indices are added.
"}], "answer": "{a_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Example 18
\nSimplify (a) $\\displaystyle{\\frac{8^4}{8^2}}$,$\\qquad$(b)$x^{18}\\div x^7$.
\nSolution
\nIn each case we are required to divide expressions involving indices. The bases are the same and we use the second law of indices (Key Point 5).(a) The indices must be subtracted, thus $\\displaystyle{\\frac{8^4}{8^2}=8^{4-2}=8^2=64}$.
(b) Again the indices are subtracted, and so $x^{18}\\div x^7 = x^{18-7}=x^{11}$
Simplify $\\var{b_name}^\\var{b_i1_expr}\\div\\var{b_name}^\\var{b_i2_expr}$.
\nNote: to enter $x^{(y + z)}$, type x^(y+z)
The bases are the same, and the division is carried out by subtracting the indices.
"}], "answer": "{b_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Example 19
\nSimplify (a) $(8^2)^3$,$\\qquad$(b)$(z^3)^4$.
\nSolution
\nWe use the third law of indices (Key Point 5).
\n(a) $(8^2)^3 = 8^{2\\times 3} = 8^6$
\n(b) $(z^3)^4 = z^{3\\times 4} = z^{12}$
\n"}, {"type": "jme", "useCustomName": true, "customName": "Task 3", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Simplify $(\\var{c_name}^\\var{c_i1_expr})^\\var{c_i2_expr}$
\nNote: to enter $x^{yz}$, type x^(yz) or x^(y*z)
Using the third law of indices, the two powers are multiplied.
"}], "answer": "{c_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Two important results which can be derived from the laws of indices state:
\nAny non-zero number raised to the power $0$ has the value $1$, that is $a^0 = 1$.
\nAny number raised to power $1$ is itself, that is $a^1 = a$.
\nA generalisation of the third law states:
\n\\( (a^m b^n)^k = a^{mk}b^{nk} \\)
\n\n
Example 20
\nRemove the brackets from (a) $(3x)^2$,$\\quad$ (b) $(x^3y^7)^4$.
\nSolution
\n(a) Noting that $3=3^1$ and $x=x^1$ then $(3x)^2=(3^1x^1)^2=3^2x^2=9x^2$
\nor, alternatively, $(3x)^2=(3x)\\times(3x)=9x^2$
\n(b) $(x^3y^7)^4=x^{3\\times 4}y^{7\\times 4} = x^{12}y^{28}$
\nShow that $(-xy)^2$ is equivalent to $x^2y^2$ whereas $(-xy)^3$ is equivalent to $-x^3y^3$.
\n"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.2.1 Write expressions with a single index", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Use the index laws to simplify 3 simple expressions;
\nn^a*n^b, n^a/n^b, (n^a)^b, where n is a randomised variable or number, and a and b are randomised nonzero integers.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Rewrite each of the following expressions with a single index. In each case, write the value of the index in the box.
", "advice": "(a)
\n\\[\\var{abase}^\\var{ex1_idx[0]}\\var{abase}^\\var{ex1_idx[1]}=\\var{abase}^{(\\var{ex1_idx[0]})+(\\var{ex1_idx[1]})}=\\var{abase}^{\\var{ex1_idx[0]+ex1_idx[1]}}\\]
\n(b)
\n\\[\\frac{\\var{bbase}^\\var{ex1_idx[2]}}{\\var{bbase}^\\var{ex1_idx[3]}}=\\var{bbase}^{(\\var{ex1_idx[2]})-(\\var{ex1_idx[3]})}= \\var{bbase}^{\\var{ex1_idx[2]-ex1_idx[3]} } \\]
\n(c)
\n\\[\\left(\\var{cbase}^\\var{ex1_idx[4]}\\right)^\\var{ex1_idx[5]}=\\var{cbase}^{(\\var{ex1_idx[4]})\\times (\\var{ex1_idx[5]}) }=\\var{cbase}^{\\var{ex1_idx[4]*ex1_idx[5]} } \\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','k','m','n','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "ex1_base": {"name": "ex1_base", "group": "Ungrouped variables", "definition": "[random(string(random(2..29)),random(alphabet)),\n random(string(random(2..29)),random(alphabet)),\n random(string(random(2..29)),random(alphabet))]", "description": "", "templateType": "anything", "can_override": false}, "ex1_idx": {"name": "ex1_idx", "group": "Ungrouped variables", "definition": "[random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10)),\n random(random(-10..-2),random(2..10))\n]", "description": "", "templateType": "anything", "can_override": false}, "ex1_base_expr": {"name": "ex1_base_expr", "group": "Ungrouped variables", "definition": "[expression(ex1_base[0]),expression(ex1_base[1]),expression(ex1_base[2])]", "description": "", "templateType": "anything", "can_override": false}, "cbase": {"name": "cbase", "group": "Ungrouped variables", "definition": "expression(ex1_base[2])", "description": "(m^7)^-5 is not accepting 1/m^35 as an answer, but it does accept m^-35
", "templateType": "anything", "can_override": false}, "ans1b": {"name": "ans1b", "group": "Ungrouped variables", "definition": "expression(ex1_base[1] + \"^\" + string(ex1_idx[2]-ex1_idx[3]))", "description": "", "templateType": "anything", "can_override": false}, "ans1a": {"name": "ans1a", "group": "Ungrouped variables", "definition": "expression(ex1_base[0] + \"^\" + string(ex1_idx[0]+ex1_idx[1]))", "description": "", "templateType": "anything", "can_override": false}, "abase": {"name": "abase", "group": "Ungrouped variables", "definition": "expression(ex1_base[0])", "description": "", "templateType": "anything", "can_override": false}, "bbase": {"name": "bbase", "group": "Ungrouped variables", "definition": "expression(ex1_base[1])", "description": "", "templateType": "anything", "can_override": false}, "ans1c": {"name": "ans1c", "group": "Ungrouped variables", "definition": "expression(ex1_base[0] + \"^\" + string(ex1_idx[4]*ex1_idx[5]))", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet", "ex1_base", "ex1_idx", "ex1_base_expr", "ans1a", "ans1b", "ans1c", "abase", "bbase", "cbase"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "(a) $\\var{abase}^\\var{ex1_idx[0]}$ $\\var{abase}^\\var{ex1_idx[1]} = \\var{abase}^u$, where $u=$[[0]]
\n(b) $\\frac{\\var{bbase}^\\var{ex1_idx[2]}}{\\var{bbase}^\\var{ex1_idx[3]}} = \\var{bbase}^u$, where $u=$[[1]]
\n(c) $\\left(\\var{cbase}^\\var{ex1_idx[4]}\\right)^\\var{ex1_idx[5]} = \\var{cbase}^u$, where $u=$[[2]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "a index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex1_idx[0]+ex1_idx[1]", "maxValue": "ex1_idx[0]+ex1_idx[1]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "b index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex1_idx[2]-ex1_idx[3]", "maxValue": "ex1_idx[2]-ex1_idx[3]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "c index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ex1_idx[4]*ex1_idx[5]", "maxValue": "ex1_idx[4]*ex1_idx[5]", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.2.2 Remove the brackets index question", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Remove the brackets from (na)^k, or from n(a)^kwhere n is a number and a is a variable.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Remove the brackets from $\\var{expr}$.
", "advice": "\\[\\var{expr}=\\var{c}\\var{ve[0]}^{\\var{idx}}\\var{ve[1]}^{\\var{idx}}\\]
\n\\[\\var{expr}=\\var{c}^{\\var{idx}}\\var{ve[0]}^{\\var{idx}}\\]
\n\\[\\var{expr}=\\var{c}^{\\var{idx}}\\var{ve[0]}^{\\var{idx}}\\var{ve[1]}^{\\var{idx}}\\]
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "shuffle(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "nv": {"name": "nv", "group": "Ungrouped variables", "definition": "random(0,1,2)", "description": "number of variables to include.
\nIf nv=0 then 2 variables are used and the constant is not powered.
", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random(2..9)", "description": "", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "if(nv=0,\n expression(c+\"*(\"+v[0]+\"*\"+v[1]+\")^\"+idx)\n ,\n expression(\n \"(\"+c+\"*\"+v[0]+ [\"\",\"*\"+v[1]][nv-1] +\")^(\"+idx+\")\"\n )\n) ", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "if(nv=0,\n simplify(expression(c+\"*\"+v[0]+\"^\"+idx+\"*\"+v[1]+\"^\"+idx),\"all\"),\n simplify(expression(\n c+\"^\"+idx+\"*\"+v[0]+\"^\"+idx + [\"\",\"*\"+v[1]+\"^\"+idx][nv-1]\n ),\"all\")\n)", "description": "", "templateType": "anything", "can_override": false}, "ve": {"name": "ve", "group": "Ungrouped variables", "definition": "[expression(v[0]),expression(v[1])]", "description": "for display in the advice
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v", "nv", "c", "idx", "expr", "ans", "ve"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": true, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "notallowed": {"strings": ["(", ")", "You must remove the brackets."], "showStrings": false, "partialCredit": 0, "message": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.2.3 Simplify expression with indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Simplify n1.v^k1.(n2.v^k2), where n1, n2 are positive integers, v is a random letter variable, and k1 and k2 are nonzero integers.
\nThe answer should be expressed as n.v^k
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Simplify $\\var{c[0]}\\var{v}^\\var{idx[0]}(\\var{c[1]}\\var{v}^\\var{idx[1]})$
", "advice": "\\[\\begin{align*}\\var{c[0]}\\var{v}^\\var{idx[0]}(\\var{c[1]}\\var{v}^\\var{idx[1]})&= \\var{c[0]}\\times\\var{c[1]}\\times\\var{ve}^{\\var{idx[0]}}\\times\\var{ve}^{\\var{idx[1]}}\\\\ &=\\var{c[0]*c[1]}\\var{ve}^{(\\var{idx[0]})+(\\var{idx[1]})}\\\\ &=\\var{expr} \\end{align*}\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"v": {"name": "v", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "the variable to use
", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "repeat(random(-12..12 except [0,1]),2)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "repeat(random(2..9),2)", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "simplify(expression(c[0]+\"*\"+v+\"^\"+idx[0]+\"(\"+c[1]+\"*\"+v+\"^\"+idx[1]+\")\"),\"all\")", "description": "", "templateType": "anything", "can_override": false}, "ve": {"name": "ve", "group": "Ungrouped variables", "definition": "expression(v)", "description": "for display purposes in the advice.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["v", "idx", "c", "expr", "ve"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{expr}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": ["0.5", "0.6"], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$n*$v`?^?", "partialCredit": 0, "message": "You have not fully factorised the expression.Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "An important group of mathematical expressions which use indices are known as polynomials. Examples of polynomials are:
\n\\(4x^3 +2x^2 +3x−7,\\quad x^2 +x,\\quad 17 −2t+7t^4, \\quad z −z^3 \\)
\nNotice that they are all constructed using non-negative whole number powers of the variable. Recall that $x^0 = 1$ and so the number $−7$ appearing in the first expression can be thought of as $−7x^0$. Similarly the $17$ appearing in the third expression can be read as $17t^0$.
\nA polynomial expression takes the form
\n\\( a_0 + a_1x + a_2x^2 + a_3x^3 + : : : + a_nx^n \\)
\nwhere $a_0$, $a_1$, $a_2$, $a_3$, $\\dots$ $a_n$ are all constants called the coeffcients of the polynomial. The number $a_0$ is also called the constant term. The highest power in a polynomial is called the degree of the polynomial.
\nPolynomials with low degrees have special names and subscript notation is often not needed:
\n| Polynomial | \nDegree | \nName | \n
|---|---|---|
| $ax^3+bx^2+cx+d$ | \n3 | \ncubic | \n
| $ax^2+bx+c$ | \n2 | \nquadratic | \n
| $ax+b$ | \n1 | \nlinear | \n
| $a$ | \n0 | \nconstant | \n
The reason for the convoluted double simplification is because without this, you occasionally got an expression with a +- in front of the second term. Why? I have no idea. Multiple iterations of \"simplify\" did not change it. But simplifying the expression, converting to a string and then repeating seems to fix it!!!
", "templateType": "anything", "can_override": false}, "q3_deg": {"name": "q3_deg", "group": "question 3", "definition": "if(q3_coeffs[0]<>0,9,\n if(q3_coeffs[1]<>0,8,\n if(q3_coeffs[2]<>0,7,\n if(q3_coeffs[3]<>0,6,\n if(q3_coeffs[4]<>0,5,\n if(q3_coeffs[5]<>0,4,\n if(q3_coeffs[6]<>0,3,\n if(q3_coeffs[7]<>0,2,\n if(q3_coeffs[8]<>0,1,0\n )))))))))", "description": "", "templateType": "anything", "can_override": false}, "q3_coeffs": {"name": "q3_coeffs", "group": "question 3", "definition": "[weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n random(-9..9)\n ]", "description": "", "templateType": "anything", "can_override": false}, "q3_expr_bits": {"name": "q3_expr_bits", "group": "question 3", "definition": "shuffle([q3_coeffs[0] + letter + \"^9\",\n q3_coeffs[1] + letter + \"^8\" , \n q3_coeffs[2] + letter + \"^7\" , \n q3_coeffs[3] + letter + \"^6\" , \n q3_coeffs[4] + letter + \"^5\" , \n q3_coeffs[5] + letter + \"^4\" , \n q3_coeffs[6] + letter + \"^3\" , \n q3_coeffs[7] + letter + \"^2\" , \n q3_coeffs[8] + letter , \n q3_coeffs[9] \n ])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet", "letter"], "variable_groups": [{"name": "question 1", "variables": ["q1_expr_array", "q1_expr", "q1_idx", "q1_ans_array", "q1_ans"]}, {"name": "question 2", "variables": ["weighted_random_list", "q2_quadratic", "q2_linear", "q2_constant", "q2_expr", "q2_sgn1", "q2_sgn2", "q2_expr_parts"]}, {"name": "question 3", "variables": ["q3_expr", "q3_deg", "q3_coeffs", "q3_expr_bits"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.3.1 Is this a polynomial?", "extensions": [], "custom_part_types": [{"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.
", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["Yes", "No"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"Yes\"?", "help_url": "", "hint": "An expression which evaluates totrue or false.", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Identify whether or not an expression is a polynomial. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Is $\\var{q1_expr}$ a polynomial?
", "advice": "$\\var{q1_expr}$
\nThis expression contains a fraction term with the variable $\\var{expression(letter)}$ on the denominator, so it is not a polynomial.
\nThis expression contains a term with the square root of the variable $\\var{expression(letter)}$, so it is not a polynomial.
\nThis is a linear polynomial.
\nThis is a quadratic polynomial.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1_expr_array": {"name": "q1_expr_array", "group": "Ungrouped variables", "definition": "[\n string(random(-9,-8,-7,-6,-5,-4,-3-2,2,3,4,5,6,7,8,9)) + letter + \"^2 + \\\\frac\\{\" + string(random(1..9)) + \"\\}\\{\"+ letter +\"\\} +2\",\n \"\\\\frac\\{\"+ string(random(-9,-8,-7,-6,-5,-4,-3-2,2,3,4,5,6,7,8,9)) + \"\\}\\{\" + letter + \"+\" + string(random(1..9)) + \"\\}\",\n \"\\\\sqrt\\{\"+string(random(1..9))+letter+\"\\}\",\n string(random(-9,-8,-7,-6,-5,-4,-3-2,2,3,4,5,6,7,8,9)) + letter + \"+\" + string(random(1..9)),\n string(random(-9,-8,-7,-6,-5,-4,-3-2,2,3,4,5,6,7,8,9)) + letter + \"^2+\"+ string(random(1..9)) + letter +\"+\"+string(random(1..9))\n ]", "description": "[containing an x^(-1) term, hyperbola, surd, linear, quadratic]
", "templateType": "anything", "can_override": false}, "letter": {"name": "letter", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','m','n','p','q','r','s','t','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "q1_ans_array": {"name": "q1_ans_array", "group": "Ungrouped variables", "definition": "[false, false, false, true, true]", "description": "", "templateType": "anything", "can_override": false}, "q1_expr": {"name": "q1_expr", "group": "Ungrouped variables", "definition": "latex(q1_expr_array[q1_idx])", "description": "", "templateType": "anything", "can_override": false}, "q1_idx": {"name": "q1_idx", "group": "Ungrouped variables", "definition": "random(0..4)", "description": "", "templateType": "anything", "can_override": false}, "q1_ans": {"name": "q1_ans", "group": "Ungrouped variables", "definition": "q1_ans_array[q1_idx]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q1_expr_array", "letter", "q1_ans_array", "q1_expr", "q1_idx", "q1_ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "yes-no", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "settings": {"correct_answer_expr": "{q1_ans}"}}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.3.2 What type of polynomial is this?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Is this polynomial a quadratic, linear or constant? Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "What type of polynomial is $\\var{q2_expr}$?
", "advice": "$\\var{q2_expr}$
\nThere are no variables in this expression, so this is a constant.
\nThe highest degree is 1, so this is a linear expression.
\nThe highest degree is 2, so this is a quadratic expression.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"weighted_random_list": {"name": "weighted_random_list", "group": "Ungrouped variables", "definition": "[[0,10],[1,1],[2,1],[3,1],[4,1],[5,1],[6,1],[7,1],[8,1],[9,1]]", "description": "", "templateType": "anything", "can_override": false}, "q2_quadratic": {"name": "q2_quadratic", "group": "Ungrouped variables", "definition": "weighted_random(weighted_random_list)", "description": "", "templateType": "anything", "can_override": false}, "q2_linear": {"name": "q2_linear", "group": "Ungrouped variables", "definition": "weighted_random(weighted_random_list)", "description": "", "templateType": "anything", "can_override": false}, "q2_constant": {"name": "q2_constant", "group": "Ungrouped variables", "definition": "random(0..9 except q2_quadratic)", "description": "", "templateType": "anything", "can_override": false}, "q2_expr": {"name": "q2_expr", "group": "Ungrouped variables", "definition": "simplify(expression(\n q2_expr_parts[0] + q2_sgn1 + q2_expr_parts[1] + q2_sgn2 + q2_expr_parts[2]),\n [\"basic\",\"unitFactor\",\"unitPower\",\"zeroFactor\",\"zeroTerm\"])", "description": "", "templateType": "anything", "can_override": false}, "q2_expr_parts": {"name": "q2_expr_parts", "group": "Ungrouped variables", "definition": "shuffle([q2_quadratic + letter + \"^2\",\n q2_linear + letter,\n q2_constant\n ])", "description": "", "templateType": "anything", "can_override": false}, "q2_sgn1": {"name": "q2_sgn1", "group": "Ungrouped variables", "definition": "random(\"+\",\"-\")", "description": "", "templateType": "anything", "can_override": false}, "q2_sgn2": {"name": "q2_sgn2", "group": "Ungrouped variables", "definition": "random(\"+\",\"-\")", "description": "", "templateType": "anything", "can_override": false}, "letter": {"name": "letter", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','m','n','p','q','r','s','t','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["weighted_random_list", "q2_quadratic", "q2_linear", "q2_constant", "q2_expr", "q2_expr_parts", "q2_sgn1", "q2_sgn2", "letter"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "1_n_2", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minMarks": 0, "maxMarks": 0, "shuffleChoices": false, "displayType": "radiogroup", "displayColumns": 0, "showCellAnswerState": true, "choices": ["constant", "linear", "quadratic"], "matrix": ["if(q2_quadratic=0 and q2_linear=0,1,0)", "if(q2_quadratic=0 and q2_linear<>0,1,0)", "if(q2_quadratic<>0,1,0)"], "distractors": ["", "", ""]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.3.3 What is the degree of this polynomial?", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given an arbitrary polynomial, identify its degree. Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "What is the degree of the polynomial $\\var{q3_expr}$?
", "advice": "Look for the largest index on the variable. This is the degree.
\nIf there are no indices, look for a variable. If there is a variable with no index, then this could be rewritten with an index of 1, and this is the degree.
\nIf there is no variable, and only a constant, then the degree is $0$.
\nSo the degree of $\\var{q3_expr}$ is $\\var{q3_deg}$.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q3_expr": {"name": "q3_expr", "group": "Ungrouped variables", "definition": "simplify(expression(string(simplify(\nexpression( \n q3_expr_bits[0] + \"+\" + q3_expr_bits[1] + \"+\" + q3_expr_bits[2] + \"+\" +\n q3_expr_bits[3] + \"+\" + q3_expr_bits[4] + \"+\" + q3_expr_bits[5] + \"+\" +\n q3_expr_bits[6] + \"+\" + q3_expr_bits[7] + \"+\" + q3_expr_bits[8] + \"+\" +\n q3_expr_bits[9] ),\n[\"all\"]))),[\"all\"])", "description": "", "templateType": "anything", "can_override": false}, "q3_expr_bits": {"name": "q3_expr_bits", "group": "Ungrouped variables", "definition": "shuffle([q3_coeffs[0] + letter + \"^9\",\n q3_coeffs[1] + letter + \"^8\" , \n q3_coeffs[2] + letter + \"^7\" , \n q3_coeffs[3] + letter + \"^6\" , \n q3_coeffs[4] + letter + \"^5\" , \n q3_coeffs[5] + letter + \"^4\" , \n q3_coeffs[6] + letter + \"^3\" , \n q3_coeffs[7] + letter + \"^2\" , \n q3_coeffs[8] + letter , \n q3_coeffs[9] \n ])", "description": "", "templateType": "anything", "can_override": false}, "q3_deg": {"name": "q3_deg", "group": "Ungrouped variables", "definition": "if(q3_coeffs[0]<>0,9,\n if(q3_coeffs[1]<>0,8,\n if(q3_coeffs[2]<>0,7,\n if(q3_coeffs[3]<>0,6,\n if(q3_coeffs[4]<>0,5,\n if(q3_coeffs[5]<>0,4,\n if(q3_coeffs[6]<>0,3,\n if(q3_coeffs[7]<>0,2,\n if(q3_coeffs[8]<>0,1,0\n )))))))))", "description": "", "templateType": "anything", "can_override": false}, "q3_coeffs": {"name": "q3_coeffs", "group": "Ungrouped variables", "definition": "[weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,4],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,3],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n weighted_random([[0,2],[1,1]]) * random(-9..9),\n random(-9..9)\n ]", "description": "", "templateType": "anything", "can_override": false}, "letter": {"name": "letter", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','m','n','p','q','r','s','t','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q3_expr", "q3_expr_bits", "q3_deg", "q3_coeffs", "letter"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "q3_deg", "maxValue": "q3_deg", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Negative indices", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "1.2.4. Negative Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Sometimes a number is raised to a negative power. This is interpreted as follows:
\n\\( a^{-m}=\\frac{1}{a^m}\\qquad a^m = \\frac{1}{a^{-m}} \\)
\nThus a negative index can be used to indicate a reciprocal.
\nExample 21
\nWrite each of the following expressions using a positive index and simplify if possible.
\n(a) $2^{-3}$,$\\quad$ (b)$\\frac{1}{4^{-3}}$,$\\quad$ (c) $x^{-1}$, $\\quad$ (d) $x^{-2}$, $\\quad$ (e) $10^{-1}$
\nSolution
\n(a) $2^{-3}=\\frac{1}{2^3}=\\frac{1}{8}$,
\n(b) $\\frac{1}{4^{-3}} = 4^3 = 64$
\n(c) $x^{-1} = \\frac{1}{x^1} = \\frac{1}{x}$,
\n(d) $x^{-2}=\\frac{1}{x^2}$,
\n(e) $10^{-1}=\\frac{1}{10^1}=\\frac{1}{10}$ or $0.1$.
", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1_expr": {"name": "q1_expr", "group": "question 1", "definition": "if(q1_1on,\n expression(q1_letter + \"^-\" + q1_idx),\n expression(\"1/\" + q1_letter + \"^-\" + q1_idx)\n)", "description": "", "templateType": "anything", "can_override": false}, "q1_idx": {"name": "q1_idx", "group": "question 1", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "q1_letter": {"name": "q1_letter", "group": "question 1", "definition": "random(random(alphabet),random(2..20))", "description": "", "templateType": "anything", "can_override": false}, "q1_1on": {"name": "q1_1on", "group": "question 1", "definition": "random(0,1)", "description": "", "templateType": "anything", "can_override": false}, "q1_ans": {"name": "q1_ans", "group": "question 1", "definition": "if(q1_1on,\n expression(\"1/\" + q1_letter + \"^\" + q1_idx),\n expression(q1_letter + \"^\" + q1_idx)\n)", "description": "", "templateType": "anything", "can_override": false}, "q2_expr": {"name": "q2_expr", "group": "question 2", "definition": "expression( \n \"(\" + q2_letter + \"^\" + q2_idx1 + \"*\" + q2_letter + \"^\" + q2_idx2 + \")\" +\n \"/\" + q2_letter + \"^\" + q2_idx3 )", "description": "", "templateType": "anything", "can_override": false}, "q2_letter": {"name": "q2_letter", "group": "question 2", "definition": "random(random(alphabet),random(2..20))", "description": "", "templateType": "anything", "can_override": false}, "q2_idx1": {"name": "q2_idx1", "group": "question 2", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx2": {"name": "q2_idx2", "group": "question 2", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx3": {"name": "q2_idx3", "group": "question 2", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx": {"name": "q2_idx", "group": "question 2", "definition": "q2_idx1+q2_idx2-q2_idx3", "description": "", "templateType": "anything", "can_override": false}, "q2_ans": {"name": "q2_ans", "group": "question 2", "definition": "simplify(if(q2_idx > 0,\n expression(q2_letter+\"^\"+q2_idx),\n expression(\"1/\"+q2_letter+\"^\"+q2_negidx)\n),[\"basic\",\"unitPower\",\"zeroPower\",\"simplifyFractions\"]) ", "description": "", "templateType": "anything", "can_override": false}, "q2_negidx": {"name": "q2_negidx", "group": "question 2", "definition": "-q2_idx", "description": "", "templateType": "anything", "can_override": false}, "q3_expr": {"name": "q3_expr", "group": "Question 3", "definition": "expression(\"10^-\"+q3_idx)", "description": "", "templateType": "anything", "can_override": false}, "q3_ans": {"name": "q3_ans", "group": "Question 3", "definition": "rational(1/(10^(q3_idx)))", "description": "", "templateType": "anything", "can_override": false}, "q3_idx": {"name": "q3_idx", "group": "Question 3", "definition": "random(0..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet"], "variable_groups": [{"name": "question 1", "variables": ["q1_idx", "q1_letter", "q1_1on", "q1_ans", "q1_expr"]}, {"name": "question 2", "variables": ["q2_letter", "q2_idx1", "q2_idx2", "q2_idx3", "q2_idx", "q2_negidx", "q2_expr", "q2_ans"]}, {"name": "Question 3", "variables": ["q3_expr", "q3_ans", "q3_idx"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.4.1 Rewrite an expression using a positive index", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given an expression (either a^-k or 1/a^-k) with a negative index, rewrite it with a positive index.
\nThe variable a and the index k are randomised.
\nPart of HELM Book 1.2
\n", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write $\\displaystyle{\\var{expr}}$ using a positive index.
", "advice": "\\[\\var{expr}=\\frac{1}{\\var{v}^{\\var{idx}}}\\]
\n\\[\\var{expr}=\\var{v}^{\\var{idx}}\\]
\n\\[=\\var{v}\\]
\n\\[=\\frac{1}{\\var{v}}\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"expr": {"name": "expr", "group": "Ungrouped variables", "definition": "if(isFrac,\n expression(letter + \"^-\" + idx),\n expression(\"1/\" + letter + \"^-\" + idx)\n)", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "if(isFrac,\n expression(\"1/\" + letter + \"^\" + idx),\n expression(letter + \"^\" + idx)\n)", "description": "", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "random(1..9)", "description": "", "templateType": "anything", "can_override": false}, "letter": {"name": "letter", "group": "Ungrouped variables", "definition": "random(random(alphabet),random(2..20))", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "isFrac": {"name": "isFrac", "group": "Ungrouped variables", "definition": "random(0,1)", "description": "Is the answer a fraction?
", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(type(letter)=\"string\",\n expression(letter),\n expression(string(letter))\n)", "description": "The expression version of the variable so that it displays correctly in the advice.
", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet", "letter", "idx", "isFrac", "expr", "ans", "v"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": ["0.4", "0.8"], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.4.2 Simplify an expression using only positive indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Write an expression (a^k1*a^k2)/a^k3 using a single positive index. Variable a is randomised and can be a number or a letter. k1,k2 and k3 are randomised and can be positive or negative numbers.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Simplify $\\displaystyle{\\var{q2_expr}}$. Do not use any negative indices in your answer.
", "advice": "First simplify the numerator using the first law of indices.
\n\\[\\var{q2_expr}=\\frac{\\var{v}^{(\\var{q2_idx1})+(\\var{q2_idx2})}}{\\var{v}^{(\\var{q2_idx3})}}=\\frac{\\var{v}^{(\\var{q2_idx1+q2_idx2})}}{\\var{v}^{(\\var{q2_idx3})}}\\]
\nThen use the second law.
\n\\[=\\var{v}^{(\\var{q2_idx1+q2_idx2})-(\\var{q2_idx3})}=\\var{v}^{\\var{q2_idx1+q2_idx2-q2_idx3}}\\]
\n\\[=1\\]
\n\\[=\\var{v}\\]
\nFinally, convert to a positive index.
\n\\[=\\frac{1}{\\var{v}^{\\var{-1*(q2_idx1+q2_idx2-q2_idx3)}}}\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q2_expr": {"name": "q2_expr", "group": "Ungrouped variables", "definition": "expression( \n \"(\" + q2_letter + \"^\" + q2_idx1 + \"*\" + q2_letter + \"^\" + q2_idx2 + \")\" +\n \"/\" + q2_letter + \"^\" + q2_idx3 )", "description": "", "templateType": "anything", "can_override": false}, "q2_ans": {"name": "q2_ans", "group": "Ungrouped variables", "definition": "simplify(if(q2_idx > 0,\n expression(q2_letter+\"^\"+q2_idx),\n expression(\"1/\"+q2_letter+\"^\"+q2_negidx)\n),[\"basic\",\"unitPower\",\"zeroPower\",\"simplifyFractions\"])", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "q2_letter": {"name": "q2_letter", "group": "Ungrouped variables", "definition": "random(random(alphabet),random(2..20))", "description": "", "templateType": "anything", "can_override": false}, "q2_idx": {"name": "q2_idx", "group": "Ungrouped variables", "definition": "q2_idx1+q2_idx2-q2_idx3", "description": "", "templateType": "anything", "can_override": false}, "q2_negidx": {"name": "q2_negidx", "group": "Ungrouped variables", "definition": "-q2_idx", "description": "", "templateType": "anything", "can_override": false}, "q2_idx1": {"name": "q2_idx1", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx2": {"name": "q2_idx2", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx3": {"name": "q2_idx3", "group": "Ungrouped variables", "definition": "random(-9..9 except 0)", "description": "", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(type(q2_letter)=\"string\",\n expression(q2_letter),\n expression(string(q2_letter))\n)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet", "q2_letter", "q2_expr", "q2_ans", "q2_idx", "q2_negidx", "q2_idx1", "q2_idx2", "q2_idx3", "v"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "stepsPenalty": 0, "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "First simplify the numerator using the first law of indices.
"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Then use the second law.
"}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Finally, convert to a positive index, if needed.
"}], "answer": "{q2_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": ["0.7", "0.8"], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": true, "caseSensitive": false, "notallowed": {"strings": ["-"], "showStrings": false, "partialCredit": 0, "message": ""}, "mustmatchpattern": {"pattern": "$v^$n`? `| 1/$v^$n`? `| $n^$n`? `| 1/$n^$n`? `| 1", "partialCredit": 0, "message": "You need to give the answer as a single variable to a positive power.", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.4.3 Rewrite expression without using indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Given an expression 10^-k, rewrite it as a fraction with no index. k is a random positive integer from 1 to 6.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write $\\var{q3_expr}$ as a fraction with no indices.
", "advice": "\\[\\var{q3_expr}=\\frac{1}{10^{\\var{q3_idx}}}=\\frac{1}{\\var{10^q3_idx}}\\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q3_expr": {"name": "q3_expr", "group": "Ungrouped variables", "definition": "expression(\"10^-\"+q3_idx)", "description": "", "templateType": "anything", "can_override": false}, "q3_ans": {"name": "q3_ans", "group": "Ungrouped variables", "definition": "rational(1/(10^(q3_idx)))", "description": "", "templateType": "anything", "can_override": false}, "q3_idx": {"name": "q3_idx", "group": "Ungrouped variables", "definition": "random(1..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q3_expr", "q3_ans", "q3_idx"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q3_ans}", "maxValue": "{q3_ans}", "correctAnswerFraction": true, "allowFractions": true, "mustBeReduced": true, "mustBeReducedPC": "50", "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Fractional indices", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", "", ""], "variable_overrides": [[], [], [], []], "questions": [{"name": "1.2.5. Fractional Indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "So far we have used indices that are whole numbers. We now consider fractional powers. Consider the expression $\\displaystyle{\\left(16^{\\frac{1}{2}}\\right)^2}$. Using the third law of indices, $(a^m)^n=a^{mn}$, we can write
\n\\( \\left( 16^{\\frac{1}{2}} \\right)^2 = 16^{ \\frac{1}{2} \\times 2 } = 16^1 = 16. \\)
\nSo $\\displaystyle{16^{\\frac{1}{2}}}$ is a number which when squared equals $16$, that is $4$ or $−4$. In other words $\\displaystyle{16^{\\frac{1}{2}}}$ is a square root of $16$. There are always two square roots of a non-zero positive number, and we write $\\displaystyle{16^{\\frac{1}{2}}=\\pm 4}$
\nIn general $\\displaystyle{a^{\\frac{1}{2}}}$ is a square root of $a$, $a\\geq 0$
\nSimilarly
\n\\( \\left(8^{\\frac{1}{3}}\\right)^3=8^{\\frac{1}{3}\\times 3}=8^1=8 \\)
\nso that $\\displaystyle{8^{\\frac{1}{3}}}$ is a number which when cubed equals $8$. Thus $\\displaystyle{8^{\\frac{1}{3}}}$ is the cube root of $8$, that is $\\sqrt[3]{8}$, namely $2$. Each number has only one cube root, and so \\( 8^{\\frac{1}{3}} = 2 \\)
\nIn general
\nIn general $\\displaystyle{a^{\\frac{1}{3}}}$ is the cube root of $a$
\nMore generally we have
\nThe $n$th root of $a$ is denoted by $\\displaystyle{a^{\\frac{1}{n}}}$
\nWhen $a<0$ the $n$th root only exists if $n$ is odd.
\nWhen $a>0$ the positive $n$th root is denoted by $\\sqrt[n]{a}$.
\nIf $a<0$ the negative $n$th root is $-\\sqrt[n]{|a|}$.
\nYour calculator will be able to evaluate fractional powers, and roots of numbers. Check that you can obtain the results of the following Examples on your calculator, but be aware that calculators normally give only one root when there may be others.
\nExample 22
\nEvaluate (a) $\\displaystyle{144^{\\frac{1}{2}}}$,$\\quad$(b)$\\displaystyle{125^{\\frac{1}{3}}}$
\nSolution
\n(a) $\\displaystyle{144^{\\frac{1}{2}}}$ is a square root of $144$, that is $\\pm 12$.
\n(b) Noting that $5^3=125$, we see that $\\displaystyle{125^{\\frac{1}{3}}}=\\sqrt[3]{125}=5$
\nExample 23
\nEvaluate (a) $\\displaystyle{32^{\\frac{1}{5}}}$,$\\quad$ (b) $\\displaystyle{32^{\\frac{2}{5}}}$,$\\quad$ (c)$\\displaystyle{8^{\\frac{2}{3}}}$
\nSolution
\n(a) $\\displaystyle{32^{\\frac{1}{5}}}$ is the $5$th root of $32$, that is $\\sqrt[5]{32}$. Now $2^5=32$ and so $\\sqrt[5]{32}=2$.
\n(b) Using the third law of indices we can write $\\displaystyle{32^{\\frac{2}{5}} = 32^{\\left(2\\times\\frac{1}{5}\\right)}=(32^{\\frac{1}{5}})^2}$.Thus
\n\\( 32^{\\frac{2}{5}} = ((32)^{\\frac{1}{5}})^2=2^2=4 \\)
\n(c) Note that $\\displaystyle{8^{\\frac{1}{3}}=2}$. Then
\n\\( 8^{\\frac{2}{3}} = 8^{\\left( 2\\times\\frac{1}{3}\\right)}= (8^{\\frac{1}{3}})^2 = 2^2=4 \\)
\nNote the following alternatives:
\n\\( 8^{\\frac{2}{3}} = (8^{\\frac{1}{3}})^2 = (8^2)^{\\frac{1}{3}} \\)
\nExample 24
\nWrite the following as a simple power with a single index:
\n(a) $\\displaystyle{\\sqrt{x^5}}$,$\\quad$ (b) $\\displaystyle{\\sqrt[4]{x^3}}$
\nSolution
\n(a) $\\displaystyle{\\sqrt{x^5}=(x^5)^{\\frac{1}{2}}}$. Then using the third law of indices we can write this as $\\displaystyle{x^{\\left(5\\times\\frac{1}{2}\\right)}=x^{\\frac{5}{2}}}$.
\n(b) $\\displaystyle{\\sqrt[4]{x^3} = (x^3)^{\\frac{1}{4}}}$. Using the third law we can write this as $\\displaystyle{ x^{\\left( 3\\times\\frac{1}{4} \\right)} = x^{ \\frac{3}{4} } }$.
\nExample 25
\nShow that $\\displaystyle{ z^{-\\frac{1}{2}}=\\frac{1}{\\sqrt{z}} }$.
\nSolution
\n\\( z^{-\\frac{1}{2}} = \\frac{1}{z^\\frac{1}{2} } = \\frac{1}{\\sqrt{z}} \\)
", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1_expr": {"name": "q1_expr", "group": "question 1", "definition": "latex(\"\\\\frac\\{\" + q1_num+\"\\}\\{\"+q1_den + \"\\}\")", "description": "", "templateType": "anything", "can_override": false}, "q1_num": {"name": "q1_num", "group": "question 1", "definition": "'\\\\sqrt' + q1r1 + '\\{'+q1_letter+'\\}'", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "q1_letter": {"name": "q1_letter", "group": "question 1", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q1v1": {"name": "q1v1", "group": "question 1", "definition": "random(2..4)", "description": "", "templateType": "anything", "can_override": false}, "q1r1": {"name": "q1r1", "group": "question 1", "definition": "if(q1v1=2,'',\n if(q1v1=3,'[3]','[4]')\n)", "description": "", "templateType": "anything", "can_override": false}, "q1v2": {"name": "q1v2", "group": "question 1", "definition": "rational(random(1,-1)*random(5,4,3,2,1,1/4,1/3,1/2))", "description": "", "templateType": "anything", "can_override": false}, "q1v3": {"name": "q1v3", "group": "question 1", "definition": "rational(random(1,-1)*random([5,4,3,2,1,1/4,1/3,1/2] except q1v2))", "description": "", "templateType": "anything", "can_override": false}, "q1_den": {"name": "q1_den", "group": "question 1", "definition": "\"(\"+q1_letter + \"^\\{\" + q1v2 + \"\\}\" + q1_letter + \"^\\{\" + q1v3 + \"\\})\"", "description": "", "templateType": "anything", "can_override": false}, "q1_ans": {"name": "q1_ans", "group": "question 1", "definition": "expression(q1_letter+\"^\"+ \"(\" + (1/q1v1 - q1v2 - q1v3) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q2_expr": {"name": "q2_expr", "group": "question 2", "definition": "expression(q2_base+\"^(\"+q2_idx+\")\")", "description": "", "templateType": "anything", "can_override": false}, "q2_base": {"name": "q2_base", "group": "question 2", "definition": "random(2..99)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx": {"name": "q2_idx", "group": "question 2", "definition": "random(-1,1)*(q2_idxnum/q2_idxden)", "description": "", "templateType": "anything", "can_override": false}, "q2_idxden": {"name": "q2_idxden", "group": "question 2", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "q2_idxnum": {"name": "q2_idxnum", "group": "question 2", "definition": "random(1..q2_idxden-1)", "description": "", "templateType": "anything", "can_override": false}, "q3Letter": {"name": "q3Letter", "group": "question 3", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q3T1idx": {"name": "q3T1idx", "group": "question 3", "definition": "weighted_random([[0,0.5],[random(q3IdxSurds),0.25],[random(q3IdxFracs),0.25]])", "description": "", "templateType": "anything", "can_override": false}, "q3idxSurds": {"name": "q3idxSurds", "group": "question 3", "definition": "[1/2,1/3,1/4]", "description": "", "templateType": "anything", "can_override": false}, "q3idxFracs": {"name": "q3idxFracs", "group": "question 3", "definition": "eval(expression(random(-1,1)+\"*\"+(random(1..5)/random(1..9))))", "description": "", "templateType": "anything", "can_override": false}, "q3T2idx": {"name": "q3T2idx", "group": "question 3", "definition": "random(random(q3IdxSurds),random(q3IdxFracs))", "description": "", "templateType": "anything", "can_override": false}, "q3T3idx": {"name": "q3T3idx", "group": "question 3", "definition": "weighted_random([[0,0.5],[random(q3idxSurds),0.25],[random(q3idxFracs),0.25]])", "description": "", "templateType": "anything", "can_override": false}, "q3T4Idx": {"name": "q3T4Idx", "group": "question 3", "definition": "random(random(q3idxSurds),random(q3idxFracs))", "description": "", "templateType": "anything", "can_override": false}, "q3T1Latex": {"name": "q3T1Latex", "group": "question 3", "definition": "switch(q3T1idx=0,\"\",\n q3T1idx=1,q3letter,\n q3T1idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n q3T1idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n q3T1idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T1idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "q3T1idxStr": {"name": "q3T1idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T1idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T2idxStr": {"name": "q3T2idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T2idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T3idxStr": {"name": "q3T3idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T3idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T4idxStr": {"name": "q3T4idxStr", "group": "question 3", "definition": "string(simplify(expression(string(q3T4idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3T2Latex": {"name": "q3T2Latex", "group": "question 3", "definition": "switch(q3T2idx=0,\"\",\n q3T2idx=1,q3letter,\n q3T2idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n q3T2idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n q3T2idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T2idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "q3T3Latex": {"name": "q3T3Latex", "group": "question 3", "definition": "switch(q3T3idx=0,\"\",\n q3T3idx=1,q3letter,\n q3T3idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n q3T3idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n q3T3idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T3idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "q3T4Latex": {"name": "q3T4Latex", "group": "question 3", "definition": "switch(q3T4idx=0,\"\",\n q3T4idx=1,q3letter,\n q3T4idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n q3T4idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n q3T4idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T4idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "q3_num": {"name": "q3_num", "group": "question 3", "definition": "if(q3T1Latex=\"\",q3T2Latex, q3T1Latex + \"\\\\,\" + q3T2Latex)", "description": "", "templateType": "anything", "can_override": false}, "q3_den": {"name": "q3_den", "group": "question 3", "definition": "if(q3T3Latex=\"\",q3T4Latex, q3T3Latex + \"\\\\,\"+ q3T4Latex)", "description": "", "templateType": "anything", "can_override": false}, "q3_expr": {"name": "q3_expr", "group": "question 3", "definition": "latex(\"\\\\frac\\{\"+q3_num+\"\\}\\{\"+q3_den+\"\\}\")", "description": "", "templateType": "anything", "can_override": false}, "q3_ans": {"name": "q3_ans", "group": "question 3", "definition": "expression(q3letter+\"^(\"+ (q3T1idx+q3t2idx-q3t3idx-q3t4idx) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4a_expr": {"name": "q4a_expr", "group": "question 4", "definition": "expression(\"(\" + q4_letter + \"^(\" + q4indices[0] + \"))^(\" + q4indices[1]+\")\")", "description": "", "templateType": "anything", "can_override": false}, "q4b_expr": {"name": "q4b_expr", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + q4indices[2] + \")*\" + q4_letter + \"^(\" + q4indices[3] + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4c_expr": {"name": "q4c_expr", "group": "question 4", "definition": "expression( q4_letter + \"^(\" + q4indices[4] + \")/\" + q4_letter + \"^(\" + q4indices[5] +\")\")", "description": "", "templateType": "anything", "can_override": false}, "q4a_ans": {"name": "q4a_ans", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + (q4indices[0]*q4indices[1]) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4b_ans": {"name": "q4b_ans", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + (q4indices[2]+q4indices[3]) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4c_ans": {"name": "q4c_ans", "group": "question 4", "definition": "expression(q4_letter+\"^(\" + (q4indices[4]-q4indices[5]) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "q4_letter": {"name": "q4_letter", "group": "question 4", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "q4i_arr": {"name": "q4i_arr", "group": "question 4", "definition": "[1/5,1/4,1/3,1/2,2,3,4,5]", "description": "", "templateType": "anything", "can_override": false}, "q4indices": {"name": "q4indices", "group": "question 4", "definition": "[random(-1,1)*random(q4i_arr),\n random(-1,1)*random(q4i_arr),\n random(-1,1)*random(q4i_arr),\n random(-1,1)*random(q4i_arr),\n random(-1,1)*random(q4i_arr),\n random(-1,1)*random(q4i_arr)\n ]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["alphabet"], "variable_groups": [{"name": "question 1", "variables": ["q1_expr", "q1_letter", "q1v1", "q1r1", "q1_num", "q1v2", "q1v3", "q1_den", "q1_ans"]}, {"name": "question 2", "variables": ["q2_expr", "q2_base", "q2_idx", "q2_idxden", "q2_idxnum"]}, {"name": "question 3", "variables": ["q3Letter", "q3idxSurds", "q3idxFracs", "q3T1idx", "q3T1idxStr", "q3T1Latex", "q3T2idx", "q3T2idxStr", "q3T2Latex", "q3T3idx", "q3T3idxStr", "q3T3Latex", "q3T4Idx", "q3T4idxStr", "q3T4Latex", "q3_num", "q3_den", "q3_expr", "q3_ans"]}, {"name": "question 4", "variables": ["q4_letter", "q4i_arr", "q4indices", "q4a_expr", "q4b_expr", "q4c_expr", "q4a_ans", "q4b_ans", "q4c_ans"]}], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": true, "customName": "Task 1", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Simplify $\\displaystyle{\\var{q1_expr}}$
", "stepsPenalty": 0, "steps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "First, rewrite $\\var{latex(q1_num)}$ using an index and simplify the denominator using the first law of indices.
", "answer": "{expression(q1_letter+\"^(1/\"+q1v1+\")/\"+q1_letter+\"^(\"+(q1v2+q1v3)+\")\")}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Finally, use the second law to simplify the result.
"}], "answer": "{q1_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "?^? `| ?/?^?", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Example 26
\nThe generalisation of the third law of indices states that $(a^mb^n)^k = a^{mk}b^{nk}$. By
taking $m = 1$, $n = 1$ and $k = \\frac{1}{2}$ show that $\\sqrt{ab}=\\sqrt{a}\\sqrt{b}$.
Solution
Taking $m = 1$, $n = 1$ and $k = \\frac{1}{2}$ gives $(ab)^{\\frac{1}{2}} = a^{\\frac{1}{2}}b^{\\frac{1}{2}}$.
\nTaking the case when all these roots are positive we have $\\sqrt{ab}=\\sqrt{a}\\sqrt{b}$.
\n\\( \\sqrt{ab}=\\sqrt{a}\\sqrt{b},\\qquad a\\geq0,\\; b\\geq0 \\)
\nThis result often allows answers to be written in alternative forms. For example, we may write $\\sqrt{48}$ as $\\sqrt{3\\times 16} = \\sqrt{3}\\sqrt{16}=4\\sqrt{3}$
\nAlthough this rule works for multiplication we should be aware that it does not work for addition or subtraction so that
\n\\( \\sqrt{a\\pm b} \\neq \\sqrt{a}\\pm \\sqrt{b} \\)
"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.5.1 Evaluate a fractional power", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Use a calculator to evaluate a number to the power of a fractional index. Both the number (a positive integer) and the index (a rational) are randomised.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Using a calculator, evaluate $\\var{q2_expr}$ to 4 significant figures.
", "advice": "Using a calculator, $\\var{q2_expr}=\\var{eval(q2_expr)}$.
\nRounding this to 4 significant figures means that we round the answer to exactly 4 digits, beginning with the left hand most non-zero digit, even if this is to the right of the decimal point.
\nRounded to 4 significant figures, the answer is: $\\var{sigformat(eval(q2_expr),4)}$
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q2_expr": {"name": "q2_expr", "group": "Ungrouped variables", "definition": "expression(q2_base+\"^(\"+q2_idx+\")\")", "description": "", "templateType": "anything", "can_override": false}, "q2_base": {"name": "q2_base", "group": "Ungrouped variables", "definition": "random(2..99)", "description": "", "templateType": "anything", "can_override": false}, "q2_idx": {"name": "q2_idx", "group": "Ungrouped variables", "definition": "random(-1,1)*(q2_idxnum/q2_idxden)", "description": "", "templateType": "anything", "can_override": false}, "q2_idxden": {"name": "q2_idxden", "group": "Ungrouped variables", "definition": "random(2..7)", "description": "", "templateType": "anything", "can_override": false}, "q2_idxnum": {"name": "q2_idxnum", "group": "Ungrouped variables", "definition": "random(1..q2_idxden-1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q2_expr", "q2_base", "q2_idx", "q2_idxden", "q2_idxnum"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{eval(q2_expr)}", "maxValue": "{eval(q2_expr)}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "precisionType": "sigfig", "precision": "4", "precisionPartialCredit": 0, "precisionMessage": "You have not given your answer to the correct precision.", "strictPrecision": true, "showPrecisionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.5.2 Simplify expression with fractional indices", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Simplify (a^k1*a^k2)/(a^k3*a^k4) where a is a randomised variable and k1,k2,k3 and k4 are randomised fractions (k2 and/or k4 may be 0). They may be written in index form or in surd form, or even a combination of the two.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Simplify $\\displaystyle{ \\var{expr} }$
", "advice": "The basic approach is to:
\n(a) convert surds to indices
\n(b) add the indices on the numerator, and subtract indices on the denominator.
\n\\[ \\var{expr} = \\frac{\\var{adviceNum[0]}}{\\var{adviceDen}}= \\var{adviceNum[1]}=\\var{ans} \\]
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"expr": {"name": "expr", "group": "Ungrouped variables", "definition": "latex(\"\\\\frac\\{\"+num+\"\\}\\{\"+den+\"\\}\")", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "expression(q3letter+\"^(\"+ (T1idx+t2idx-t3idx-t4idx) + \")\")", "description": "", "templateType": "anything", "can_override": false}, "alphabet": {"name": "alphabet", "group": "Ungrouped variables", "definition": "['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z']", "description": "", "templateType": "anything", "can_override": false}, "q3letter": {"name": "q3letter", "group": "Ungrouped variables", "definition": "random(alphabet)", "description": "", "templateType": "anything", "can_override": false}, "t1idx": {"name": "t1idx", "group": "Ungrouped variables", "definition": "weighted_random([[0,0.5],[random(IdxSurds),0.25],[random(IdxFracs),0.25]])", "description": "", "templateType": "anything", "can_override": false}, "t2idx": {"name": "t2idx", "group": "Ungrouped variables", "definition": "random(random(IdxSurds),random(IdxFracs))", "description": "", "templateType": "anything", "can_override": false}, "t3idx": {"name": "t3idx", "group": "Ungrouped variables", "definition": "weighted_random([[0,0.5],[random(idxSurds),0.25],[random(idxFracs),0.25]])", "description": "", "templateType": "anything", "can_override": false}, "t4idx": {"name": "t4idx", "group": "Ungrouped variables", "definition": "random(random(idxSurds),random(idxFracs))", "description": "", "templateType": "anything", "can_override": false}, "den": {"name": "den", "group": "Ungrouped variables", "definition": "if(T3Latex=\"\",T4Latex, T3Latex + \"\\\\,\"+ T4Latex)", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "if(T1Latex=\"\",T2Latex, T1Latex + \"\\\\,\" + T2Latex)", "description": "", "templateType": "anything", "can_override": false}, "t3latex": {"name": "t3latex", "group": "Ungrouped variables", "definition": "switch(T3idx=0,\"\",\n T3idx=1,q3letter,\n T3idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n T3idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n T3idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T3idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "t4latex": {"name": "t4latex", "group": "Ungrouped variables", "definition": "switch(T4idx=0,\"\",\n T4idx=1,q3letter,\n T4idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n T4idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n T4idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T4idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "t1latex": {"name": "t1latex", "group": "Ungrouped variables", "definition": "switch(T1idx=0,\"\",\n T1idx=1,q3letter,\n T1idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n T1idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n T1idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T1idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "t2latex": {"name": "t2latex", "group": "Ungrouped variables", "definition": "switch(T2idx=0,\"\",\n T2idx=1,q3letter,\n T2idx=1/2,\"\\\\sqrt\\{\" + q3letter + \"\\}\",\n T2idx=1/3,\"\\\\sqrt\\[3]{\" + q3letter + \"\\}\",\n T2idx=1/4,\"\\\\sqrt\\[4]{\" + q3letter + \"\\}\",\n q3letter + \"^\\{\" + q3T2idxStr + \"\\}\"\n )", "description": "", "templateType": "anything", "can_override": false}, "q3t4idxstr": {"name": "q3t4idxstr", "group": "Ungrouped variables", "definition": "string(simplify(expression(string(T4idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "idxfracs": {"name": "idxfracs", "group": "Ungrouped variables", "definition": "eval(expression(random(-1,1)+\"*\"+(random(1..5)/random(1..9))))", "description": "", "templateType": "anything", "can_override": false}, "idxsurds": {"name": "idxsurds", "group": "Ungrouped variables", "definition": "[1/2,1/3,1/4]", "description": "", "templateType": "anything", "can_override": false}, "q3t3idxstr": {"name": "q3t3idxstr", "group": "Ungrouped variables", "definition": "string(simplify(expression(string(T3idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3t2idxstr": {"name": "q3t2idxstr", "group": "Ungrouped variables", "definition": "string(simplify(expression(string(T2idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "q3t1idxstr": {"name": "q3t1idxstr", "group": "Ungrouped variables", "definition": "string(simplify(expression(string(T1idx)),[\"simplifyFractions\",\"basic\",\"unitFactor\",\"unitDenominator\"]))", "description": "", "templateType": "anything", "can_override": false}, "adviceNum": {"name": "adviceNum", "group": "Ungrouped variables", "definition": "[if(T1idx=0,\n expression(q3letter+\"^(\"+T2idx+\")\"),\n expression(q3letter+\"^(\"+t1idx + \")*\" + q3letter+\"^(\"+T2idx+\")\")\n),\n if(T1idx=0 and T3idx=0,\n expression(q3letter+\"^(\"+T2idx+\"-\"+t4idx+\")\"),\n if(t1Idx=0 and t3idx<>0,\n expression(q3letter+\"^(\"+T2idx+\"-\"+t3Idx+\"-\"+t4idx+\")\"),\n if(t1Idx<>0 and t3Idx=0,\n expression(q3letter+\"^(\"+t1idx+\"+\"+T2idx+\"-\"+t4Idx+\")\"),\n expression(q3letter+\"^(\"+t1Idx+\"+\"+t2idx+\"-\"+t3idx+\"-\"+t4idx+\")\") \n ) \n )\n) \n]", "description": "", "templateType": "anything", "can_override": false}, "adviceDen": {"name": "adviceDen", "group": "Ungrouped variables", "definition": "if(T3idx=0,\n expression(q3letter+\"^\"+\"(\"+t4idx+\")\"),\n expression(q3letter+\"^(\"+t3idx + \")*\" + q3letter+\"^(\"+t4idx+\")\")\n)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["expr", "ans", "alphabet", "q3letter", "t1idx", "t2idx", "t3idx", "t4idx", "den", "num", "t3latex", "t4latex", "t1latex", "t2latex", "idxfracs", "idxsurds", "q3t1idxstr", "q3t2idxstr", "q3t3idxstr", "q3t4idxstr", "adviceNum", "adviceDen"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$v^`+-($n/$n`?)`? `| 1/$v^($n/$n`?)`? `| 1", "partialCredit": 0, "message": "You need to give the answer as a single variable with a single index.", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.5.3 Rewrite expressions using a single index", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Simplify three expressions: (a^b)^c, a^b * a^c, a^b/a^c where a, b and c are randomised. a is a letter, and b and c are rational numbers.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Write each of the following expressions with a single index:
", "advice": "(a) Multiply the indices:
\n\\[\\var{a_expr}=\\var{v}^{(\\var{iexp[0]})\\times(\\var{iexp[1]})}=\\var{a_ans}\\]
\n(b) Add the indices:
\n\\[\\var{b_expr}=\\var{v}^{(\\var{iexp[2]})+(\\var{iexp[3]})}=\\var{b_ans}\\]
\n(c) Subtract the indices:
\n\\[\\var{c_expr}=\\var{v}^{(\\var{iexp[4]})-(\\var{iexp[5]})}=\\var{c_ans}\\]
\nNote: if the final index is negative, you can also write the answer as a fraction with a positive index on the denominator.
", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"c_ans": {"name": "c_ans", "group": "Ungrouped variables", "definition": "simplify(expression(letter+\"^(\" + (indices[4]-indices[5]) + \")\"),\n [\"unitPower\",\"zeroPower\"])", "description": "", "templateType": "anything", "can_override": false}, "b_ans": {"name": "b_ans", "group": "Ungrouped variables", "definition": "simplify(expression(letter+\"^(\" + (indices[2]+indices[3]) + \")\"),\n [\"unitPower\",\"zeroPower\"])", "description": "", "templateType": "anything", "can_override": false}, "a_ans": {"name": "a_ans", "group": "Ungrouped variables", "definition": "simplify(expression(letter+\"^(\" + (indices[0]*indices[1]) + \")\"),\n [\"unitPower\",\"zeroPower\"])", "description": "", "templateType": "anything", "can_override": false}, "a_expr": {"name": "a_expr", "group": "Ungrouped variables", "definition": "expression(\"(\" + letter + \"^(\" + indices[0] +\n \"))^(\" + indices[1]+\")\")", "description": "", "templateType": "anything", "can_override": false}, "b_expr": {"name": "b_expr", "group": "Ungrouped variables", "definition": "expression(letter+\"^(\" + indices[2] + \")*\" + letter +\n \"^(\" + indices[3] + \")\")", "description": "", "templateType": "anything", "can_override": false}, "c_expr": {"name": "c_expr", "group": "Ungrouped variables", "definition": "expression( letter + \"^(\" + indices[4] +\n \")/\" + letter + \"^(\" + indices[5] +\")\")", "description": "", "templateType": "anything", "can_override": false}, "letter": {"name": "letter", "group": "Ungrouped variables", "definition": "random(['a','b','c','d','f','g','h','k','m','n','p','q','r','s','t','u','v','w','x','y','z'])", "description": "", "templateType": "anything", "can_override": false}, "indices": {"name": "indices", "group": "Ungrouped variables", "definition": "[random(-1,1)*random(i_arr),\n random(-1,1)*random(i_arr),\n random(-1,1)*random(i_arr),\n random(-1,1)*random(i_arr),\n random(-1,1)*random(i_arr),\n random(-1,1)*random(i_arr)\n ]", "description": "", "templateType": "anything", "can_override": false}, "i_arr": {"name": "i_arr", "group": "Ungrouped variables", "definition": "[1/5,1/4,1/3,1/2,2,3,4,5]", "description": "", "templateType": "anything", "can_override": false}, "iexp": {"name": "iexp", "group": "Ungrouped variables", "definition": "[expression(string(indices[0])),expression(string(indices[1])),\n expression(string(indices[2])),expression(string(indices[3])),\n expression(string(indices[4])),expression(string(indices[5]))]", "description": "The indices written as expressions. This allows them to be rendered correctly in the advice.
", "templateType": "anything", "can_override": false}, "v": {"name": "v", "group": "Ungrouped variables", "definition": "if(type(letter)=\"string\",\n expression(letter),\n expression(string(letter))\n)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["letter", "a_expr", "a_ans", "b_expr", "b_ans", "c_expr", "c_ans", "indices", "i_arr", "iexp", "v"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "(a) $\\var{a_expr}$ = [[0]]
\n\n(b) $\\var{b_expr}$ = [[1]]
\n\n(c) $\\var{c_expr}$ = [[2]]
", "gaps": [{"type": "jme", "useCustomName": true, "customName": "part a", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{a_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$v^`+-($n/$n`?)`? `| 1/$v^($n/$n`?)`? `| 1", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "part b", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{b_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$v^`+-($n/$n`?)`? `| 1/$v^($n/$n`?)`? `| 1", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}, {"type": "jme", "useCustomName": true, "customName": "part c", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "{c_ans}", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "mustmatchpattern": {"pattern": "$v^`+-($n/$n`?)`? `| 1/$v^($n/$n`?)`? `| 1", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}, {"name": "Scientific notation", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", ""], "variable_overrides": [[], []], "questions": [{"name": "1.2.6. Scientific Notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Part of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "It is often necessary to use very large or very small numbers such as $78000000$ and $0.00000034$.
Scientific notation can be used to express such numbers in a more concise form. Each number is written in the form
\\( a \\times 10^n \\)
\nwhere $a$ is a number between $1$ and $10$. We can make use of the following facts:
\n$10 = 10^1$; $\\quad 100 = 10^2$; $\\quad 1000 = 10^3\\quad$ and so on
\nand
\n$0.1 = 10^{-1}$; $\\quad 0.01 = 10^{-2}$; $\\quad 0.001 = 10^{-3}\\quad$ and so on.
\nFor example,
\nFurthermore, to multiply a number by $10^n$ the decimal point is moved $n$ places to the right if $n$ is a positive integer, and $n$ places to the left if $n$ is a negative integer. (If necessary additional zeros are inserted to make up the required number of digits before the decimal point.)
", "advice": "", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"q1base": {"name": "q1base", "group": "Ungrouped variables", "definition": "decimal(random(10001..99999)/10000)", "description": "", "templateType": "anything", "can_override": false}, "q1expr": {"name": "q1expr", "group": "Ungrouped variables", "definition": "q1base*10^q1Idx", "description": "", "templateType": "anything", "can_override": false}, "q1idx": {"name": "q1idx", "group": "Ungrouped variables", "definition": "random(-7..7 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "q2base": {"name": "q2base", "group": "Ungrouped variables", "definition": "eval(expression(split_regex(q2ans,\"e\")[0]))", "description": "", "templateType": "anything", "can_override": false}, "q2idx": {"name": "q2idx", "group": "Ungrouped variables", "definition": "eval(expression(split_regex(q2ans,\"e\")[1]))", "description": "", "templateType": "anything", "can_override": false}, "q2b1": {"name": "q2b1", "group": "Ungrouped variables", "definition": "decimal(random(11..99)/10)", "description": "", "templateType": "anything", "can_override": false}, "q2b2": {"name": "q2b2", "group": "Ungrouped variables", "definition": "decimal(random(11..99)/10)", "description": "", "templateType": "anything", "can_override": false}, "q2i1": {"name": "q2i1", "group": "Ungrouped variables", "definition": "random(-39..39 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "q2i2": {"name": "q2i2", "group": "Ungrouped variables", "definition": "random(-39..39 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "q2ans": {"name": "q2ans", "group": "Ungrouped variables", "definition": "formatnumber(eval(q2expr),\"scientific\")", "description": "", "templateType": "anything", "can_override": false}, "q2expr": {"name": "q2expr", "group": "Ungrouped variables", "definition": "expression(+q2b1 + \"* 10^(\" + q2i1 + \")*\" + q2b2 + \"* 10^(\" + q2i2 + \")\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["q1base", "q1idx", "q1expr", "q2b1", "q2b2", "q2i1", "q2i2", "q2expr", "q2base", "q2idx", "q2ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": true, "customName": "Task 1", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Write $\\var{q1Expr}$ in scientific notation:
\n[[0]]$\\times 10$[[1]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "q1Base", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q1Base}", "maxValue": "{q1Base}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "q1Idx", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{q1Idx}", "maxValue": "{q1Idx}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}, {"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "Many constants appearing in engineering calculations are expressed in scientific notation. For example the charge on an electron equals $1.6 \\times 10^{-19}$ coulomb and the speed of light is $3 \\times 10^8$ m s$^{-1}$. Avogadro's constant is equal to $6.02 \\times 10^{23}$ and is the number of atoms in one mole of an element. Clearly the use of scientific notation avoids writing lengthy strings of zeros.
\nYour scientific calculator will be able to accept numbers in scientific notation. Often the $E$ button (sometimes the $EXP$ button) is used and a number like $4.2\\times10^7$ will be entered as $4.2E7$. Note that $10E4$ means $10\\times 10^4$, that is, $10^5$. To enter the number $10^3$, say, you would key in $1E3$. Entering powers of $10$ incorrectly is a common cause of error. You must check how your particular calculator accepts numbers in scientific notation.
\nThe following task is designed to check that you can enter numbers given in scientific notation into your calculator.
"}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "1.2.6.1 Calculate product of numbers in scientific notation", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "tags": [], "metadata": {"description": "Calculate the product of two randomised scientific notation numbers and give the answer in scientific notation.
\nPart of HELM Book 1.2
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "Use your calculator to find
", "advice": "Multiply the two numbers together:
\n\\[\\begin{align*} \\var{expr}&= (\\var{b1}\\times\\var{b2}) \\times(10^{\\var{i1}}\\times 10^{\\var{i2}})\\\\ &= (\\var{b1*b2})\\times 10^{(\\var{i1}+\\var{i2})}\\\\ &=\\var{b1*b2}\\times 10^{\\var{i1+i2}}\\\\ \\textrm{Now adjust if necessary so that}&\\textrm{ there is exactly one digit in front of the decimal point:}\\\\ &=\\var{base}\\times 10^{\\var{idx}}\\end{align*}\\]
\n", "rulesets": {}, "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"base": {"name": "base", "group": "Ungrouped variables", "definition": "eval(expression(split_regex(ans,\"e\")[0]))", "description": "", "templateType": "anything", "can_override": false}, "idx": {"name": "idx", "group": "Ungrouped variables", "definition": "eval(expression(split_regex(ans,\"e\")[1]))", "description": "", "templateType": "anything", "can_override": false}, "expr": {"name": "expr", "group": "Ungrouped variables", "definition": "expression(+b1 + \"* 10^(\" + i1 + \")*\" + b2 + \"* 10^(\" + i2 + \")\")", "description": "", "templateType": "anything", "can_override": false}, "b1": {"name": "b1", "group": "Ungrouped variables", "definition": "decimal(random(11..99)/10)", "description": "", "templateType": "anything", "can_override": false}, "i1": {"name": "i1", "group": "Ungrouped variables", "definition": "random(-39..39 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "decimal(random(11..99)/10)", "description": "", "templateType": "anything", "can_override": false}, "i2": {"name": "i2", "group": "Ungrouped variables", "definition": "random(-39..39 except [-1,0,1])", "description": "", "templateType": "anything", "can_override": false}, "ans": {"name": "ans", "group": "Ungrouped variables", "definition": "formatnumber(eval(expr),\"scientific\")", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["base", "idx", "expr", "b1", "i1", "b2", "i2", "ans"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "$\\var{Expr}$ = [[0]] $\\times 10$[[1]]
", "gaps": [{"type": "numberentry", "useCustomName": true, "customName": "base", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{Base}", "maxValue": "{Base}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": true, "customName": "index", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "{Idx}", "maxValue": "{Idx}", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": false, "showresultspage": "oncompletion", "navigatemode": "menu", "onleave": {"action": "none", "message": ""}, "preventleave": true, "startpassword": "", "allowAttemptDownload": false, "downloadEncryptionKey": ""}, "timing": {"allowPause": true, "timeout": {"action": "none", "message": ""}, "timedwarning": {"action": "none", "message": ""}}, "feedback": {"showactualmark": true, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": true, "advicethreshold": 0, "intro": "This quiz is a Numbas implementation of the Helping Engineers Learn Maths (HELM) booklet 1.2, Indices. An html version can be found here.
\nThe first section of the quiz, \"Booklet\", comprises the original booklet. Questions generally have multiple versions, clicking the \"Try another question like this one\" button will generate a new version.
\nStarting a new instance of this quiz will create different questions in all sections after the first one.
", "end_message": "", "reviewshowscore": true, "reviewshowfeedback": true, "reviewshowexpectedanswer": true, "reviewshowadvice": true, "feedbackmessages": []}, "diagnostic": {"knowledge_graph": {"topics": [], "learning_objectives": []}, "script": "diagnosys", "customScript": ""}, "contributors": [{"name": "Merryn Horrocks", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4052/"}], "extensions": [], "custom_part_types": [{"source": {"pk": 1, "author": {"name": "Christian Lawson-Perfect", "pk": 7}, "edit_page": "/part_type/1/edit"}, "name": "Yes/no", "short_name": "yes-no", "description": "The student is shown two radio choices: \"Yes\" and \"No\". One of them is correct.
", "help_url": "", "input_widget": "radios", "input_options": {"correctAnswer": "if(eval(settings[\"correct_answer_expr\"]), 0, 1)", "hint": {"static": true, "value": ""}, "choices": {"static": true, "value": ["Yes", "No"]}}, "can_be_gap": true, "can_be_step": true, "marking_script": "mark:\nif(studentanswer=correct_answer,\n correct(),\n incorrect()\n)\n\ninterpreted_answer:\nstudentAnswer=0\n\ncorrect_answer:\nif(eval(settings[\"correct_answer_expr\"]),0,1)", "marking_notes": [{"name": "mark", "description": "This is the main marking note. It should award credit and provide feedback based on the student's answer.", "definition": "if(studentanswer=correct_answer,\n correct(),\n incorrect()\n)"}, {"name": "interpreted_answer", "description": "A value representing the student's answer to this part.", "definition": "studentAnswer=0"}, {"name": "correct_answer", "description": "", "definition": "if(eval(settings[\"correct_answer_expr\"]),0,1)"}], "settings": [{"name": "correct_answer_expr", "label": "Is the answer \"Yes\"?", "help_url": "", "hint": "An expression which evaluates totrue or false.", "input_type": "mathematical_expression", "default_value": "true", "subvars": false}], "public_availability": "always", "published": true, "extensions": []}], "resources": []}