// Numbas version: exam_results_page_options
{"name": "Indices: multiplying powers (algebraic)", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: multiplying powers (algebraic)", "tags": ["exponent", "exponents", "index", "index laws", "Index Laws", "indices", "multiplying", "nested power", "power", "power of a power", "powers"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Simplify the following without the use of a calculator. Write your answer in index form using ^ to signify powers.</p>", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"nint": {"name": "nint", "group": "Ungrouped variables", "definition": "random(-12..-2)", "description": "", "templateType": "anything", "can_override": false}, "ndec": {"name": "ndec", "group": "Ungrouped variables", "definition": "random(-0.9..-0.1#0.1)", "description": "", "templateType": "anything", "can_override": false}, "prodpow3": {"name": "prodpow3", "group": "Ungrouped variables", "definition": "powers3[0][0]*powers3[1][0]*powers3[2][0]", "description": "", "templateType": "anything", "can_override": false}, "prodpow4": {"name": "prodpow4", "group": "Ungrouped variables", "definition": "nint*ndec*num/den", "description": "", "templateType": "anything", "can_override": false}, "num": {"name": "num", "group": "Ungrouped variables", "definition": "random(-9..-1)", "description": "", "templateType": "anything", "can_override": false}, "prodpow1": {"name": "prodpow1", "group": "Ungrouped variables", "definition": "2*powers1", "description": "", "templateType": "anything", "can_override": false}, "den": {"name": "den", "group": "Ungrouped variables", "definition": "random(10,20,25)", "description": "", "templateType": "anything", "can_override": false}, "prodpow2": {"name": "prodpow2", "group": "Ungrouped variables", "definition": "powers2[0][0]*powers2[1][0]", "description": "", "templateType": "anything", "can_override": false}, "powers1": {"name": "powers1", "group": "Ungrouped variables", "definition": "random(3..6)", "description": "<p>2..6</p>", "templateType": "anything", "can_override": false}, "powers3": {"name": "powers3", "group": "Ungrouped variables", "definition": "shuffle([[2,\"two\"],[3,\"three\"],[4,\"four\"],[5,\"five\"],[6,\"six\"]])[0..3]", "description": "", "templateType": "anything", "can_override": false}, "powers2": {"name": "powers2", "group": "Ungrouped variables", "definition": "shuffle([[3,\"three\"],[4,\"four\"],[5,\"five\"],[6,\"six\"]])[0..2]", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["powers1", "prodpow1", "powers2", "prodpow2", "powers3", "prodpow3", "ndec", "nint", "num", "den", "prodpow4"], "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": "<p>$\\left(d^\\var{powers1}\\right)^2$ =&nbsp;[[0]]</p>", "stepsPenalty": "1", "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": "<p><em>Interpret the powers in expanded form and then write the result in index form.<br/></em></p>\n<hr/>\n<p>Suppose you had to simplify $(5^3)^2$. Well cubed means there are three 5s all being multiplied, so we have</p>\n<p>\\[(5^3)^2=(5\\times 5\\times 5)^2\\]</p>\n<p>and squared means multiplied by itself (in this case the square is <g class=\"gr_ gr_12 gr-alert gr_gramm Punctuation replaceWithoutSep\" id=\"12\" data-gr-id=\"12\">acting on the whole</g> bracketed term) so we get</p>\n<p>\\[(5^3)^2=(5\\times 5\\times 5)^2=(5\\times 5\\times 5)\\times (5\\times 5\\times 5)\\]</p>\n<p>&nbsp;but this is just six 5s all being multiplied together, that is</p>\n<p>\\[(5^3)^2=5^6.\\]</p>\n<hr/>\n<p>In general, we have $\\displaystyle(a^b)^c=a^{bc}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// Parse the student's answer as a syntax tree\n///var studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\n//var rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \n//var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\n//if(!m) {\n  //this.multCredit(0,'Your answer is not in the form $x^y$.');\n//}\n", "order": "after"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "d^{prodpow1}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "maxlength": {"length": "4", "partialCredit": 0, "message": "<p>Your answer is&nbsp;longer than necessary.</p>"}, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "notallowed": {"strings": ["^0"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "valuegenerators": [{"name": "d", "value": ""}]}], "sortAnswers": false}, {"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": "<p>$\\displaystyle(j^\\var{powers2[0][0]})^\\var{powers2[1][0]}$ =&nbsp;[[0]]</p>\n<p></p>\n<p></p>", "stepsPenalty": "1", "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": "<p><em>Interpret the powers in expanded form and then write the result in index form.<br/></em></p>\n<hr/>\n<p>Suppose you had to simplify $(5^3)^2$. Well cubed means there are three 5s all being multiplied, so we have</p>\n<p>\\[(5^3)^2=(5\\times 5\\times 5)^2\\]</p>\n<p>and squared means multiplied by itself (in this case the square is&nbsp;acting on the whole bracketed term) so we get</p>\n<p>\\[(5^3)^2=(5\\times 5\\times 5)^2=(5\\times 5\\times 5)\\times (5\\times 5\\times 5)\\]</p>\n<p>&nbsp;but this is just six 5s all being multiplied together, that is</p>\n<p>\\[(5^3)^2=5^6.\\]</p>\n<hr/>\n<p>In general, we have $\\displaystyle(a^b)^c=a^{bc}$.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// Parse the student's answer as a syntax tree\n///var studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\n//var rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \n//var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\n//if(!m) {\n  //this.multCredit(0,'Your answer is not in the form $x^y$.');\n//}\n", "order": "after"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "j^{prodpow2}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "maxlength": {"length": "4", "partialCredit": 0, "message": "<p>Your answer is&nbsp;longer than necessary.</p>"}, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "notallowed": {"strings": ["^0"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "valuegenerators": [{"name": "j", "value": ""}]}], "sortAnswers": false}, {"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": "<p>Use the same approach you used in the above questions to simplify the following in index form.</p>\n<p><br/>$\\displaystyle\\left(\\left(m^\\var{powers3[0][0]}\\right)^\\var{powers3[1][0]}\\right)^\\var{powers3[2][0]}$ =&nbsp;[[0]]</p>\n<p></p>", "stepsPenalty": "1", "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": "<p><em>Interpret the powers in expanded form and then write the result in index form.<br/></em></p>\n<hr/>\n<p>Suppose you had to simplify $((5^3)^2)^4$.&nbsp;We could start by applying the square</p>\n<p>&nbsp;\\[((5^3)^2)^4=(5^3\\times 5^3)^4=(5^6)^4\\]&nbsp;</p>\n<p>and then apply the fourth power</p>\n<p>\\[(5^6)^4=5^6\\times 5^6\\times 5^6\\times 5^6 = 5^{24}.\\]</p>\n<hr/>\n<p>Notice the above is the long way of just applying the rule&nbsp; $\\displaystyle(a^b)^c=a^{bc}$ twice:</p>\n<p>\\[((5^3)^2)^4=(5^6)^4=5^{24}.\\]</p>\n<p>In fact, we could just multiply all the powers together in one go:</p>\n<p>\\[((5^3)^2)^4=5^{3\\times 2\\times 4}=5^{24}.\\]</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// Parse the student's answer as a syntax tree\n///var studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\n//var rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \n//var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\n//if(!m) {\n  //this.multCredit(0,'Your answer is not in the form $x^y$.');\n//}\n", "order": "after"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "m^{prodpow3}", "answerSimplification": "basic", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "maxlength": {"length": "6", "partialCredit": 0, "message": "<p>Your answer is&nbsp;longer than necessary.</p>"}, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "notallowed": {"strings": ["^0"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "valuegenerators": [{"name": "m", "value": ""}]}], "sortAnswers": false}, {"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": "<p><br/>$\\displaystyle\\left(\\left(n^\\var{nint}\\right)^\\var{ndec}\\right)^{\\var{num}/\\var{den}}$ =&nbsp;[[0]]</p>\n<p></p>\n<p>Note: If you want to use a fraction as a power you should use brackets to surround your power, for example, type 12^(2/3) for $12^\\frac{2}{3}$.</p>\n<p></p>", "stepsPenalty": "1", "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": "<p><em>Multiply all the powers together as you have done in previous questions.</em></p>\n<hr/>\n<p>Note the answer can be written using a fractional power or a decimal power but a fractional power is more commonly used.</p>"}], "gaps": [{"type": "jme", "useCustomName": false, "customName": "", "marks": 1, "scripts": {"mark": {"script": "// Parse the student's answer as a syntax tree\n///var studentTree = Numbas.jme.compile(this.studentAnswer,Numbas.jme.builtinScope);\n\n// Create the pattern to match against \n// We just want to check that the student has written \"something to the power of something\"\n//var rule = Numbas.jme.compile('?? ^ ??');\n\n// Check the student's answer matches the pattern. \n//var m = Numbas.jme.display.matchTree(rule,studentTree,true);\n\n// If not, take away marks\n//if(!m) {\n  //this.multCredit(0,'Your answer is not in the form $x^y$.');\n//}\n", "order": "after"}}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "answer": "n^{prodpow4}", "answerSimplification": "fractionNumbers", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": "0.0000001", "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "maxlength": {"length": "13", "partialCredit": 0, "message": "<p>Your answer is longer than it needs to be.</p>"}, "musthave": {"strings": ["^"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "notallowed": {"strings": ["^0"], "showStrings": false, "partialCredit": 0, "message": "<p>Use ^ for powers. Input your answer in index form.</p>"}, "valuegenerators": [{"name": "n", "value": ""}]}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}