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{"name": "Indices: Simplification 2a", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: Simplification 2a", "tags": ["category: Indices"], "metadata": {"description": "<p>Using indices rules to rewrite an expression from $a^\\frac{m}{n}$ to $\\frac{1}{b}$, for integers $a$, $b$, $m$ and $n$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as a fraction:</p>\n<p>\\[\\var{a^n}^{\\frac{\\var{m}}{\\var{n}}}\\]</p>", "advice": "<p>To find $\\var{a^n}^{\\frac{\\var{m}}{\\var{n}}}$, 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>$a^{-n} = \\frac{1}{a^n}$.</li>\n</ol>\n<p>By rewriting the power $\\frac{\\var{m}}{\\var{n}}$ as a product of $\\var{m} \\times \\frac{1}{\\var{n}}$, we can apply rule 1:</p>\n<p>\\[ \\begin{split}&nbsp;\\var{a^n}^{\\frac{\\var{m}}{\\var{n}}} &amp;\\,=&nbsp;\\var{a^n}^{\\left(\\var{m} \\times \\frac{1}{\\var{n}}\\right)} \\\\ &amp;\\,= \\left(\\var{a^n}^\\frac{1}{\\var{n}}\\right)^\\var{m} \\\\ &amp;\\,= \\var{a}^\\var{m}\\end{split} \\]</p>\n<p>Then applying rule 2:</p>\n<p>\\[ \\begin{split}&nbsp;\\var{a}^\\var{m} &amp;\\,= \\frac{1}{\\var{a}^{\\var{-m}}} \\\\ &amp;\\,= \\frac{1}{\\var{a^(-m)}} \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\var{a^n}^{\\frac{\\var{m}}{\\var{n}}} =\\frac{1}{\\var{a^(-m)}}. \\]</p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2,3)*-1", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..3 except -m)", "description": "", "templateType": "anything", "can_override": false}, "a": {"name": "a", "group": "Ungrouped variables", "definition": "random(2,3,4)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["m", "n", "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": "1/{a^-m}", "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": "1/`+-$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}]}], "contributors": [{"name": "Ruth Hand", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3228/"}, {"name": "Ben McGovern", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/4872/"}]}