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{"name": "Indices: Simplification 1b", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: Simplification 1b", "tags": [], "metadata": {"description": "<p>Simplifying expressions from $\\left(\\frac{x^m}{x^n}\\right)^p$ to $x^{(m-n)p}$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Simplify the following&nbsp;expression:</p>\n<p>\\[\\left(\\frac{x^\\var{m}}{x^\\var{n}}\\right)^\\var{p}\\]</p>", "advice": "<p>To simplify $\\left(\\frac{x^\\var{m}}{x^\\var{n}}\\right)^\\var{p}$, we want to make use of the following 2 rules:</p>\n<ol>\n<li>$\\frac{a^m}{a^n}=a^{m-n}$;</li>\n<li>$(a^n)^m=a^{n\\times m}$.</li>\n</ol>\n<p>Applying rule 1 to the terms inside the bracket:</p>\n<p>\\[\\begin{split} \\frac{x^\\var{m}}{x^\\var{n}}&amp;\\,=x^{\\var{m}-\\var{n}}\\\\ &amp;\\,=x^\\var{m-n}. \\end{split}\\]</p>\n<p>Then applying rule 2:</p>\n<p>\\[ \\begin{split}\\left(\\frac{x^\\var{m}}{x^\\var{n}}\\right)^\\var{p} &amp;\\,= \\left(x^\\var{m-n}\\right)^\\var{p} \\\\ &amp;\\,=x^{\\var{m-n} \\times \\var{p}}\\\\ &amp;\\,=x^\\var{m*p-n*p}. \\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..4)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "random(2..8 except m)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "random(2..6)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "m", "n", "p"], "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": "x^{m*p-n*p}", "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": "x^`+-$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "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/"}]}