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{"name": "Indices: Simplification 3a", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: Simplification 3a", "tags": [], "metadata": {"description": "<p>Simplifying an expression of the form $a^3 \\times (a^4)^{1/2}$ to $a^5$, where $a$ is an integer.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Without a calculator, simplify the&nbsp;following expression into a single integer:</p>\n<p>\\[ \\var{a}^3 \\times \\var{a^4}^{1/2}\\]</p>", "advice": "<p>To simplify $\\var{a}^3 \\times \\var{a^4}^{1/2}$, we want to make use of the following 2 rules:</p>\n<ol>\n<li>$\\left(a^n\\right)^m =a^{n\\times m}$;</li>\n<li>$a^n \\times a^m = a^{n+m}$.</li>\n</ol>\n<p>Notice that we can rewrite this expression so that both numbers are in terms of $\\var{a}$, since $\\var{a^4} = \\var{a}^4$:</p>\n<p>\\[\\var{a}^3 \\times \\var{a^4}^{1/2} =&nbsp;\\var{a}^3 \\times \\left(\\var{a}^4\\right)^{1/2}.\\]</p>\n<p>Applying rule 1 to the second term:</p>\n<p>\\[ \\begin{split} \\var{a}^3 \\times \\left(\\var{a}^4\\right)^{1/2} &amp;\\,=&nbsp;\\var{a}^3 \\times \\var{a}^{4 \\times 1/2} \\\\ &amp;\\,=&nbsp;\\var{a}^3 \\times \\var{a}^2. \\end{split} \\]</p>\n<p>Then applying rule 2:</p>\n<p>\\[\\var{a}^3 \\times \\var{a}^2 = \\var{a}^5.\\]</p>\n<p>Therefore,&nbsp;</p>\n<p>\\[ \\begin{split}&nbsp;\\var{a}^3 \\times \\var{a^4}^{1/2} &amp;\\,=&nbsp;\\var{a}^5 \\\\ &amp;\\,= \\var{a^5}. \\end{split} \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2,3)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a"], "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": "{a^5}", "answerSimplification": "fractionNumbers", "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": "$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}