Fourth part of the Core Foundation Maths pre-arrival self-assessment material:
\nQuestion 1: Algebra IV: Properties of indices (1) - Multiplication/Division
\nQuestion 2: Algebra IV: Properties of indices (2) - Fractions
\nQuestion 3: Algebra IV - Properties of Indices (4) - Further
\nQuestion 4: Numbers V: standard index form (conversions and operations)
\n"}, "timing": {"allowPause": true, "timeout": {"message": "", "action": "none"}, "timedwarning": {"message": "", "action": "none"}}, "question_groups": [{"questions": [{"name": "Algebra IV: Properties of indices (1) - Multiplication/Division", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j"], "tags": [], "preamble": {"css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "advice": "Recall the laws of indices to help solve the problems:
\n$x^a \\times x^b = x^{a+b}$
\n$x^a \\div x^b = x^{a-b}$
\n$x^{-a} = \\frac{1}{x^a}$
\n$(x^a)^b = x^{ab}$
\n$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$
\n$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$
\n$x^0 = 1$
\n\nWorked Solutions:
\nPart a) $x^{(\\var{a}+\\var{b})}=\\simplify{x^{({a}+{b})}}$
\nPart b) $p^{(\\var{c}+\\var{d})}=\\simplify{p^{({c}+{d})}}$
\nPart c) $\\var{a}^\\var{f}\\times{k^{(\\var{b}\\times\\var{f})}}=\\simplify{{a}^{f}*k^{({b}*{f})}}$
\nPart d) $y^{((\\var{a}+\\var{b})/(\\var{a}\\times\\var{b}))}=y^{\\frac{\\simplify{{a}+{b}}}{\\simplify{{a}*{b}}}}$
\nPart e) $c^{(\\var{a}-\\var{b})}=c^\\simplify{({a}-{b})}$
\nPart f) $\\frac{\\var{a}}{\\var{b}}h^{\\var{c}-\\var{d}}=\\frac{\\var{a}}{\\var{b}}{\\simplify{h^{{c}-{d}}}}$
\nPart g) $\\frac{4^\\var{g}}{2^\\var{h}}\\times{d^{\\var{g}-\\var{h}}}=\\simplify{(4^{g})/(2^{h})*d^{g-h}}$
\nPart h) $\\frac{6^\\var{g}}{9^\\var{h}}\\times{p^{\\var{h}\\var{j}-\\var{g}\\var{f}}}=\\simplify{(6^{g})/(9^{h})*p^{h*j-g*f}}$
", "rulesets": {}, "parts": [{"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "$x^\\var{a} \\times x^\\var{b}$
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Use the following indices law to help answer this question:
\n$x^a \\times x^b = x^{a+b}$
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "scripts": {}, "answer": "x^({a}+{b})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "$p^\\var{c} \\times p^\\var{d}$
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Use the following law to help answer this question:
\n$x^a \\times x^b = x^{a+b}$
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "p^({c}+{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "$(\\var{a}k^\\var{b})^\\var{f}$
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Use the following law to answer this question:
\n$(ax^b)^c = a^cx^{bc}$
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{a}^{f}*k^({b}*{f})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "$y^{1/\\var{a}} \\times y^{1/\\var{b}}$
", "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Use the following law:
\n$x^a \\times x^b = x^{a+b}$
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "y^(({a}+{b})/({a}*{b}))", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "Use the following law:
\n$x^a \\div x^b = x^{a-b}$
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "c^({a}-{b})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "Use the following law to answer this question:
\n$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "{a}/{b}*h^({c}-{d})", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "This question differs from part f due to the brackets. Using principles of BODMAS, the brackets need to be expanded first.
\n$(4d)^\\var{g}$ expands to $4^\\var{g}d^\\var{g}$ and $(2d)^\\var{h}$ expands to $2^\\var{h}d^\\var{h}$
\nNow you are left with a simple division question as follows:
\n$\\frac{4^{\\var{g}}d^{\\var{g}}}{2^{\\var{h}}d^{\\var{h}}}$
\n\nUse the principle:
\n$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$ to answer the question.
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(4^{g})/(2^{h})*d^{g-h}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}, {"stepsPenalty": 0, "vsetrangepoints": 5, "prompt": "Using principles of BODMAS, the brackets need to be expanded first.
\n$(6p^{-\\var{f}})^{\\var{g}}$ expands to $6^{\\var{g}}p^{({-\\var{f}}\\times{\\var{g}})}$ and $(9p^{-\\var{j}})^{\\var{h}}$ expands to $9^{\\var{h}}p^{({-\\var{j}}\\times{\\var{h}})}$
\nNow you are left with a dividend and divisor,
\n$\\frac{6^{\\var{g}}p^{({-\\var{f}}\\times{\\var{g}})}}{9^{\\var{h}}p^{({-\\var{j}}\\times{\\var{h}})}}$
\nwhich can be simplfied according to the known patterns of indices.
\ni.e.
\n$\\frac{ax^c}{bx^d}= \\frac{a}{b}x^{(c-d)}$
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "showCorrectAnswer": true, "answersimplification": "all", "scripts": {}, "answer": "(6^{g})/(9^{h})*p^{h*j-g*f}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "statement": "Simplify each of the following expressions, giving your answer in its simplest form.
\nClick 'Show steps' for guidance on which index law is applicable.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(2..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "random(0..9)", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..9 except a)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "d": {"definition": "random(-9..-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2..3)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "random(2..5 except g)", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "j": {"definition": "random(2..3 except f)", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}}, "metadata": {"description": "Simplifying indices.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Algebra IV: Properties of indices (2) - Fractions", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}], "functions": {}, "ungrouped_variables": ["a", "b", "c", "d", "f", "g", "h", "j", "k", "l", "f1", "f2", "f3", "f4", "t1", "t2"], "tags": [], "advice": "Recall the laws of indices:
\n$x^a \\times x^b = x^{a+b}$
$x^a \\div x^b = x^{a-b}$
$x^{-a} = \\frac{1}{x^a}$
$(x^a)^b = x^{ab}$
$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$
$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$
$x^0 = 1$
$\\var{c}^{\\frac{1}{\\var{b}}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "2", "minValue": "2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{1/z}$ is the same as $\\sqrt[z]{x}$
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$\\var{d}^{\\frac{1}{\\var{a}}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "3", "minValue": "3", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{1/z}$ is the same as $\\sqrt[z]{x}$
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$\\var{h}^{\\frac{\\var{f}}{\\var{g}}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "3^{f}", "minValue": "3^{f}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$\\var{f3}^{\\frac{\\var{f1}}{\\var{f2}}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "5^f1", "minValue": "5^f1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$49^{\\frac{3}{2}}$
", "allowFractions": false, "variableReplacements": [], "maxValue": "343", "minValue": "343", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "steps": [{"prompt": "Hint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$(1/\\var{f4})^{\\simplify{{f2}/{f1}}}$
", "allowFractions": true, "variableReplacements": [], "maxValue": "5^-{f2}", "minValue": "5^-{f2}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "Make sure to consider the brackets in this question.
\nHint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}, {"stepsPenalty": 0, "prompt": "$(\\var{t1}/\\var{t2})^{\\simplify{{f1}/{f2}}}$
", "allowFractions": true, "variableReplacements": [], "maxValue": "(2/3)^{f1}", "minValue": "(2/3)^{f1}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "steps": [{"prompt": "Make sure to consider the brackets in this question.
\nHint: $x^{y/z}$ can also be considered as $(x^{1/z})^{y}$.
\nAs you know, $x^{1/z} = \\sqrt[z]{x}$ therefore an alternative way of looking at the expression is $(\\sqrt[z]{x})^{y}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "marks": 1, "scripts": {}, "showCorrectAnswer": true, "type": "numberentry", "showPrecisionHint": false}], "statement": "Simplify the following expressions, giving your answer in its simplest form.
\nGive your answer as either a fraction or an integer.
\nTry doing these questions without a calculator if you can to practise your use of powers and roots.
", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "fraction {\n display: inline-block;\n vertical-align: middle;\n}\nfraction > numerator, fraction > denominator {\n float: left;\n width: 100%;\n text-align: center;\n line-height: 2.5em;\n}\nfraction > numerator {\n border-bottom: 1px solid;\n padding-bottom: 5px;\n}\nfraction > denominator {\n padding-top: 5px;\n}\nfraction input {\n line-height: 1em;\n}\n\nfraction .part {\n margin: 0;\n}\n\n.table-responsive, .fractiontable {\n display:inline-block;\n}\n.fractiontable {\n padding: 0; \n border: 0;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n border-bottom: 1px solid black; \n text-align: center;\n}\n\n\n.fractiontable tr {\n height: 3em;\n}\n\n", "js": "document.createElement('fraction');\ndocument.createElement('numerator');\ndocument.createElement('denominator');"}, "variables": {"a": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "a", "description": ""}, "f1": {"definition": "random(2..f2-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f1", "description": ""}, "c": {"definition": "2^b", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(2..4)", "templateType": "anything", "group": "Ungrouped variables", "name": "b", "description": ""}, "f4": {"definition": "5^f1", "templateType": "anything", "group": "Ungrouped variables", "name": "f4", "description": ""}, "d": {"definition": "3^a", "templateType": "anything", "group": "Ungrouped variables", "name": "d", "description": ""}, "g": {"definition": "random(3..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "g", "description": ""}, "f": {"definition": "random(2..g-1)", "templateType": "anything", "group": "Ungrouped variables", "name": "f", "description": ""}, "h": {"definition": "3^g", "templateType": "anything", "group": "Ungrouped variables", "name": "h", "description": ""}, "k": {"definition": "random(2..5 except j)", "templateType": "anything", "group": "Ungrouped variables", "name": "k", "description": ""}, "j": {"definition": "random(2..5)", "templateType": "anything", "group": "Ungrouped variables", "name": "j", "description": ""}, "t2": {"definition": "3^f2", "templateType": "anything", "group": "Ungrouped variables", "name": "t2", "description": ""}, "l": {"definition": "2^j", "templateType": "anything", "group": "Ungrouped variables", "name": "l", "description": ""}, "f2": {"definition": "random(3..6)", "templateType": "anything", "group": "Ungrouped variables", "name": "f2", "description": ""}, "t1": {"definition": "2^f2", "templateType": "anything", "group": "Ungrouped variables", "name": "t1", "description": ""}, "f3": {"definition": "5^f2", "templateType": "anything", "group": "Ungrouped variables", "name": "f3", "description": ""}}, "metadata": {"description": "Simplifying indices.
", "licence": "Creative Commons Attribution 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Algebra IV - Properties of Indices (4) - Further", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Sarah Turner", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/881/"}], "functions": {}, "ungrouped_variables": ["a", "a_1", "a_3", "a_2", "b", "c"], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "Recall the laws of indices to help solve the problems:
\n$x^a \\times x^b = x^{a+b}$
$x^a \\div x^b = x^{a-b}$
$x^{-a} = \\frac{1}{x^a}$
$(x^a)^b = x^{ab}$
$(\\frac{x}{y})^a = \\frac{x^a}{y^a}$
$x^0 = 1$
$bx^{-a} = \\frac{b}{x^a}$
$x^\\frac{a}{b} = (\\sqrt[b]{x})^{a}$
$cx^\\frac{-a}{b} = \\frac{c}{(\\sqrt[b]{x})^{a}}$
Which option from the choices below is another way of expressing the following term: $x^{\\frac{1}{2}}$?
", "matrix": [0, 0, "1", 0], "shuffleChoices": true, "variableReplacements": [], "choices": ["$x^{-2}$
", "$\\frac{1}{x^{-2}}$
", "$\\sqrt{x}$
", "$\\frac{2}{x}$
"], "variableReplacementStrategy": "originalfirst", "maxMarks": "1", "distractors": ["", "", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": "1"}, {"displayColumns": 0, "prompt": "Which option from the choices below is another way of expressing the following term: $x^{-1}$?
", "matrix": ["1", 0, 0, 0], "shuffleChoices": true, "variableReplacements": [], "choices": ["$\\frac{1}{x}$
", "$\\frac{1}{\\sqrt{x}}$
", "$\\sqrt{x}$
", "$x^{1/2}$
"], "variableReplacementStrategy": "originalfirst", "maxMarks": "1", "distractors": ["", "", "", ""], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "1_n_2", "displayType": "radiogroup", "minMarks": "1"}, {"maxAnswers": 0, "prompt": "One expression on the top row matches one expression down the left hand side column. Every expression has only one pair.
\nMatch the expressions using a tick in the checkbox to indicate that the two expressions are equivalent to eachother. This will help you understand the application of indices in different contexts.
\nIt may help to zoom in on your computer screen to see the individual expressions.
", "matrix": [["1", 0, 0, 0, 0, 0, 0, 0], [0, "1", 0, 0, 0, 0, 0, 0], [0, 0, "1", 0, 0, 0, 0, 0], [0, 0, 0, "1", 0, 0, 0, 0], [0, 0, 0, 0, "1", 0, 0, 0], [0, 0, 0, 0, 0, "1", 0, 0], [0, 0, 0, 0, 0, 0, "1", 0], [0, 0, 0, 0, 0, 0, 0, "1"]], "shuffleAnswers": true, "minAnswers": 0, "variableReplacements": [], "displayType": "checkbox", "answers": ["$\\sqrt[-{\\var{a}}]{x}$
", "$x^{\\frac{1}{\\var{a_1}}}$
", "${\\var{a_2}}x^{-1}$
", "$\\frac{1}{x^{\\var{a}}}$
", "${\\var{a_1}}^{-1}$
", "$\\frac{1}{x^{-{\\var{a_2}}}}$
", "${\\var{b}}x^{-{\\var{a}}}$
", "$\\var{c}^{-x}$
"], "warningType": "none", "variableReplacementStrategy": "originalfirst", "maxMarks": "8", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "choices": ["$x^{\\frac{-1}{\\var{a}}}$
", "$\\sqrt[\\var{a_1}]{x}$
", "$\\frac{\\var{a_2}}{x}$
", "$x^{-{\\var{a}}}$
", "$\\frac{1}{\\var{a_1}}$
", "$x^{\\var{a_2}}$
", "$\\frac{\\var{b}}{x^{\\var{a}}}$
", "$\\frac{\\var{b}}{{\\var{a}}^x}$
"], "type": "m_n_x", "shuffleChoices": true, "minMarks": "8", "layout": {"expression": "", "type": "all"}}, {"prompt": "$x^0=$
\n", "allowFractions": false, "variableReplacements": [], "maxValue": "1", "minValue": "1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "statement": "The questions below are a series of multiple choice questions to familiarise yourself with the different ways of expressing a certain term in relation to powers and roots.
\nRemeber, $x^{1/z}$ is the same as $\\sqrt[z]{x}$, form example, $x^{1/3}$ is the same as $\\sqrt[3]{x}$.
\nRemeber, $x^{-z}$ is the same as $\\frac1{x^z}$, form example, $x^{-4}$ is the same as $\\frac1{x^4}$.
\n", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"a": {"definition": "random(3..6#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a", "description": ""}, "c": {"definition": "{a}*{b}", "templateType": "anything", "group": "Ungrouped variables", "name": "c", "description": ""}, "b": {"definition": "random(7..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "b", "description": ""}, "a_3": {"definition": "random(3..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a_3", "description": ""}, "a_2": {"definition": "random(3..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a_2", "description": ""}, "a_1": {"definition": "random(3..9#1)", "templateType": "randrange", "group": "Ungrouped variables", "name": "a_1", "description": ""}}, "metadata": {"description": "These questions aim to familiarise the student with the different rules of indices through a series of matching statements and true/false questions.
", "licence": "None specified"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}]}, {"name": "Numbers V: standard index form (conversions and operations)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}], "functions": {}, "ungrouped_variables": [], "tags": [], "preamble": {"css": "", "js": ""}, "advice": "If you need more practice, click below on 'Try another question like this one' for a new set of numbers.
", "rulesets": {}, "parts": [{"stepsPenalty": 0, "prompt": "Convert the following from standard index form to its full number form:
\n$\\var{base_a}\\times10^\\var{{tenpow}[0]}$ = [[0]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "If the power of ten is positive, the decimal point 'moves to the right' by the same number of places: $7.5021\\times10^3=7502.1$
\nLikewise, if the power of ten is negative, the decimal point 'moves to the left' by the same number of places: $7.5021\\times10^{-3}=0.0075021$
\n", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{number_a}", "minValue": "{number_a}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "Convert the following number to standard index form:
\n$\\var{number_b}$ = [[0]] $\\times10$[[1]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "The coefficient (number at the front) must always be a number greater than or equal to 1 and strictly less than 10. For instance, $13.4\\times10^{3}$ should be $1.34\\times10^{4}$, just as $0.75\\times10^{6}$ should be $7.5\\times10^{5}$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{base_b}", "minValue": "{base_b}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{{tenpow}[1]}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "Evaluate the following and express the final answer in standard index form (to two sig. fig.s):
\nPlease note: although you may need to write the power of ten as 0 or 1 to get the mark, these instances would normally require the full number form to be written without the base ten multiplication.
\n$(\\var{{bases_cd}[0]}\\times10^{\\var{{tenpow2}[0]}})\\times(\\var{{bases_cd}[1]}\\times10^{\\var{{tenpow2}[1]}})$
\n= [[0]] $\\times10$[[1]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "The two coefficients (the numbers at the front) can be multiplied first, then the powers can be added according to the index rule of same-base multiplication.
\nRemember to adapt the coefficient to make it greater than or equal to 1 and strictly less than 10 if necessary. If you do, don't forget to compensate in the final power of ten.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "sigfig", "precisionMessage": "You have not given your answer to the correct precision.
", "allowFractions": false, "variableReplacements": [], "maxValue": "{finalbase_c}", "strictPrecision": true, "minValue": "{finalbase_c}", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{finalpow_c}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "Evaluate the following and express the final answer in standard index form (to two sig. fig.s):
\nPlease note: although you may need to write the power of ten as 0 or 1 to get the mark, these instances would normally require the full number form to be written without the base ten multiplication.
\n$(\\var{{bases_cd}[2]}\\times10^{\\var{{tenpow2}[2]}})\\div(\\var{{bases_cd}[3]}\\times10^{\\var{{tenpow2}[3]}})$
\n= [[0]] $\\times10$[[1]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "As with part c), you can divide the coefficients and then subtract the first power of ten from the other, adjusting the final answer as appropriate.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "sigfig", "precisionMessage": "You have not given your answer to the correct precision.
", "allowFractions": false, "variableReplacements": [], "maxValue": "{finalbase_d}", "strictPrecision": true, "minValue": "{finalbase_d}", "variableReplacementStrategy": "originalfirst", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{finalpow_d}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "Evaluate the following and express the final answer in standard index form (to two sig. fig.s):
\n$(\\var{base_el}\\times10^{\\var{{tenpow3}[0]}})+(\\var{base_er} \\times10^{\\var{{tenpow3}[0]}})$
\n= [[0]] $\\times10$[[1]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Adjust the final answer if necessary. Think of the numbers in their full number form if it helps. Since both of the numbers have the same power, we can simply add the coefficients up to form the coefficient of the answer, and the power will be the same. Note that, unlike with multiplication and division, the powers must be the same if we want to add or subtract two numbers in standard form quickly. If not, we need to do some converting beforehand.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "{finalbase_e}", "minValue": "{finalbase_e}", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "answer": "{finalpow_e}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}, {"stepsPenalty": 0, "prompt": "Evaluate the following and express the final answer in standard index form (to two sig. fig.s):
\n$(\\var{base_fl}\\times10^{\\var{{tenpow3}[1]}})+(\\var{base_fr} \\times10^\\var{scndpow_f})$
\n\n= [[0]] $\\times10$[[1]]
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "steps": [{"prompt": "Beware of the different powers of ten! Try: converting the numbers first to the full number forms, carrying out the operation and then converting the answer back into standard index form. After doing this several times, you may start to see quicker ways of reaching the answer. If the coefficient is negative, remember to ensure that its positive value is greater than or equal to 1 and strictly less than 10.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "gaps": [{"precisionType": "sigfig", "precisionMessage": "You have not given your answer to the correct precision.
", "allowFractions": false, "variableReplacements": [], "maxValue": "{absfinalbase_f}", "strictPrecision": true, "minValue": "{absfinalbase_f}", "variableReplacementStrategy": "alwaysreplace", "precisionPartialCredit": 0, "correctAnswerFraction": false, "showCorrectAnswer": true, "precision": "2", "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"vsetrangepoints": 5, "expectedvariablenames": [], "checkingaccuracy": 0.001, "vsetrange": [0, 1], "showpreview": true, "variableReplacements": [], "variableReplacementStrategy": "alwaysreplace", "showCorrectAnswer": true, "scripts": {}, "answer": "{absfinalpow_f}", "marks": 1, "checkvariablenames": false, "checkingtype": "absdiff", "type": "jme"}], "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "gapfill"}], "statement": "A number in standard index form is of the form: $a \\times 10^n$, where $1 \\leqslant |a| < 10$ and $n$ is an integer.
\n($a$ is called the coefficient and $n$ is called the exponent or power. $|a|$ means the positive value of $a$. For example, $|-3| = 3$, $|5| = 5$. It is called the modulus or absolute value function.)
\nComplete the following questions, clicking on 'Show steps' for extra guidance where necessary:
\nGive all answers correct to two significant figures.
", "variable_groups": [{"variables": ["tenpow", "bases_ab", "base_a", "number_a", "base_b", "number_b"], "name": "Parts a) and b)"}, {"variables": ["tenpow2", "bases_cd", "finalbase_c", "finalpow_c", "finalbase_d", "finalpow_d"], "name": "Parts c) and d)"}, {"variables": ["tenpow3", "bases_ef", "base_el", "base_er", "finalbase_e", "finalpow_e", "base_fl", "base_fr", "fstpow_f", "scndpowsel", "scndpow_f", "finalpow_f", "finalbase_f", "absfinalpow_f", "absfinalbase_f"], "name": "Parts e) and f)"}], "variablesTest": {"maxRuns": "100", "condition": ""}, "variables": {"scndpow_f": {"definition": "if(scndpowsel=0,tenpow3[1]+1,if(scndpowsel=1,tenpow3[1]-1))", "templateType": "anything", "group": "Parts e) and f)", "name": "scndpow_f", "description": ""}, "base_fr": {"definition": "bases_ef[3]", "templateType": "anything", "group": "Parts e) and f)", "name": "base_fr", "description": ""}, "bases_cd": {"definition": "shuffle(1.1..9.9#0.1)[0..4]", "templateType": "anything", "group": "Parts c) and d)", "name": "bases_cd", "description": ""}, "base_fl": {"definition": "if(bases_ef[2]=-bases_ef[3],bases_ef[4],bases_ef[2],if(-bases_ef[2]=bases_ef[3],bases_ef[4],bases_ef[2]))", "templateType": "anything", "group": "Parts e) and f)", "name": "base_fl", "description": ""}, "bases_ef": {"definition": "shuffle([-4.9,-4.8,-4.7,-4.6,-4.5,-4.4,-4.3,-4.2,-4.1,-4.0,-3.9,-3.8,-3.7,-3.6,-3.5,-3.4,-3.3,-3.2,-3.1,-3.0,-2.9,-2.8,-2.7,-2.6,-2.5,-2.4,-2.3,-2.2,-2.1,-2.0,-1.9,-1.8,-1.7,-1.6,-1.5,-1.4,-1.3,-1.2,-1.1,-1.0,1.0,1.1,1.2,1.3,1.4,1.5,1.6,1.7,1.8,1.8,2.0,2.1,2.2,2.3,2.4,2.5,2.6,2.7,2.8,2.9,3.0,3.1,3.2,3.3,3.4,3.5,3.6,3.7,3.8,3.9,4.0,4.1,4.2,4.3,4.4,4.5,4.6,4.7,4.8,4.9])[0..6]", "templateType": "anything", "group": "Parts e) and f)", "name": "bases_ef", "description": ""}, "finalpow_c": {"definition": "if((bases_cd[0]*bases_cd[1]>10),(tenpow2[0]+tenpow2[1]+1),if((bases_cd[0]*bases_cd[1]<10),(tenpow2[0]+tenpow2[1]),if((bases_cd[0]*bases_cd[1]=10),(tenpow2[0]+tenpow2[1]))))", "templateType": "anything", "group": "Parts c) and d)", "name": "finalpow_c", "description": ""}, "finalpow_e": {"definition": "if(base_el+base_er<1 and base_el+base_er>-1,tenpow3[0]-1,tenpow3[0])", "templateType": "anything", "group": "Parts e) and f)", "name": "finalpow_e", "description": ""}, "finalpow_d": {"definition": "if((bases_cd[2]/bases_cd[3]<1),(tenpow2[2]-tenpow2[3]-1),if((bases_cd[2]/bases_cd[3]>1),(tenpow2[2]-tenpow2[3]),if((bases_cd[2]/bases_cd[3]=1),(tenpow2[2]-tenpow2[3]))))", "templateType": "anything", "group": "Parts c) and d)", "name": "finalpow_d", "description": ""}, "finalpow_f": {"definition": "if(scndpowsel=0,scndpow_f,if(scndpowsel=1,tenpow3[1]))", "templateType": "anything", "group": "Parts e) and f)", "name": "finalpow_f", "description": ""}, "base_a": {"definition": "bases_ab[0]", "templateType": "anything", "group": "Parts a) and b)", "name": "base_a", "description": ""}, "base_b": {"definition": "bases_ab[1]", "templateType": "anything", "group": "Parts a) and b)", "name": "base_b", "description": ""}, "absfinalpow_f": {"definition": "if(finalbase_f>-1 and finalbase_f<1,finalpow_f+1,finalpow_f)", "templateType": "anything", "group": "Parts e) and f)", "name": "absfinalpow_f", "description": ""}, "base_er": {"definition": "bases_ef[1]", "templateType": "anything", "group": "Parts e) and f)", "name": "base_er", "description": ""}, "tenpow2": {"definition": "shuffle(-8..8 except[0,1,2])[0..4]", "templateType": "anything", "group": "Parts c) and d)", "name": "tenpow2", "description": ""}, "tenpow3": {"definition": "shuffle(-8..8 except[-1,0,1,2])[0..4]", "templateType": "anything", "group": "Parts e) and f)", "name": "tenpow3", "description": ""}, "finalbase_e": {"definition": "if(base_el+base_er<1 and base_el+base_er>-1, (base_el+base_er)*10,if(base_el+base_er>=1 or base_el+base_er<=-1, (base_el+base_er)))", "templateType": "anything", "group": "Parts e) and f)", "name": "finalbase_e", "description": ""}, "finalbase_d": {"definition": "if((bases_cd[2]/bases_cd[3]<1),((bases_cd[2]/bases_cd[3])*10),if((bases_cd[2]/bases_cd[3]>1),((bases_cd[2]/bases_cd[3]),if((bases_cd[2]/bases_cd[3]=1),((bases_cd[2]/bases_cd[3]))))))", "templateType": "anything", "group": "Parts c) and d)", "name": "finalbase_d", "description": ""}, "finalbase_f": {"definition": "if(scndpowsel=0,(base_fl/10)+base_fr,if(scndpowsel=1,base_fl+(base_fr/10)))", "templateType": "anything", "group": "Parts e) and f)", "name": "finalbase_f", "description": "if(base_fl+base_fr<1 and base_fl+base_fr>-1, (base_fl+base_fr)*10,if(base_fl+base_fr>=1 or base_fl+base_fr<=-1, (base_fl+base_fr)))
"}, "finalbase_c": {"definition": "if((bases_cd[0]*bases_cd[1]>10),((bases_cd[0]*bases_cd[1])/10),if((bases_cd[0]*bases_cd[1]<10),(bases_cd[0]*bases_cd[1]),if((bases_cd[0]*bases_cd[1]=10),(bases_cd[0]*bases_cd[1]))))", "templateType": "anything", "group": "Parts c) and d)", "name": "finalbase_c", "description": ""}, "tenpow": {"definition": "shuffle(-6..6 except[0,1,2])[0..2]", "templateType": "anything", "group": "Parts a) and b)", "name": "tenpow", "description": ""}, "absfinalbase_f": {"definition": "if(finalbase_f>-1 and finalbase_f<1,(finalbase_f*10),finalbase_f)", "templateType": "anything", "group": "Parts e) and f)", "name": "absfinalbase_f", "description": ""}, "number_b": {"definition": "base_b*10^tenpow[1]", "templateType": "anything", "group": "Parts a) and b)", "name": "number_b", "description": ""}, "number_a": {"definition": "base_a*10^tenpow[0]", "templateType": "anything", "group": "Parts a) and b)", "name": "number_a", "description": ""}, "scndpowsel": {"definition": "random(0..1#1)", "templateType": "randrange", "group": "Parts e) and f)", "name": "scndpowsel", "description": ""}, "fstpow_f": {"definition": "tenpow3[1]", "templateType": "anything", "group": "Parts e) and f)", "name": "fstpow_f", "description": ""}, "bases_ab": {"definition": "shuffle(1.1..9.9#0.1)[0..2]", "templateType": "anything", "group": "Parts a) and b)", "name": "bases_ab", "description": ""}, "base_el": {"definition": "if(bases_ef[0]=-bases_ef[1],bases_ef[5],bases_ef[0],if(-bases_ef[0]=bases_ef[1],bases_ef[6],bases_ef[0]))", "templateType": "anything", "group": "Parts e) and f)", "name": "base_el", "description": ""}}, "metadata": {"description": "Practice with standard index form:
\n