HELM Book 1.2.5 exercises
", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "duration": 0, "percentPass": 0, "showQuestionGroupNames": false, "shuffleQuestionGroups": false, "showstudentname": true, "question_groups": [{"name": "Group", "pickingStrategy": "all-ordered", "pickQuestions": 1, "questionNames": ["", "", ""], "variable_overrides": [[], [], []], "questions": [{"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"}]}], "allowPrinting": true, "navigation": {"allowregen": true, "reverse": true, "browse": true, "allowsteps": true, "showfrontpage": true, "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": "Work through these practice questions. You are encouraged to try each question multiple times as the expressions in each question will change each time you try it.
\n(Note: To enter an answer of $\\dfrac{1}{m^5}$, type 1/m^(5) )