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{"name": "Indices: Simplification 3b", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: Simplification 3b", "tags": [], "metadata": {"description": "<p>Simplifying an expression of the form $\\frac{a^4b^3}{\\sqrt{a^4b^2}}$ to $a^2b^2$, for integers $a$ and $b$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Calculate the following expression without a calculator:</p>\n<p>\\[ \\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\sqrt{\\var{a^4} \\times \\var{b^2}}} \\]</p>", "advice": "<p>To simplify $\\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\sqrt{\\var{a^4} \\times \\var{b^2}}}$, we want to make use of the following 2 rules:</p>\n<ol>\n<li>$\\sqrt{a \\times b} = \\sqrt{a} \\times \\sqrt{b}$;</li>\n<li>$\\frac{a^n}{a^m} = a^{n-m}$.</li>\n</ol>\n<p>Applying rule 1 to the denominator:</p>\n<p>\\[ \\begin{split}\\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\sqrt{\\var{a^4} \\times \\var{b^2}}} &amp;\\,=&nbsp;\\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\sqrt{\\var{a^4}} \\times \\sqrt{\\var{b^2}}} \\\\\\\\ &amp;\\,=&nbsp;\\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\var{a^2} \\times \\var{b}}. \\end{split}\\]</p>\n<p>Since $\\var{a^2} = \\var{a}^2$, we can rewrite the denominator so that the numerator and denominator are both in terms of $\\var{a}$ and $\\var{b}$, allowing us to apply rule 2:</p>\n<p>\\[ \\begin{split} \\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\var{a^2} \\times \\var{b}} &amp;\\,=&nbsp;\\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\var{a}^2 \\times \\var{b}} \\\\\\\\ &amp;\\,= \\var{a}^{\\var{n}-2} \\times \\var{b}^{\\var{m}-1} \\\\ &amp;\\,= \\var{a}^\\var{n-2} \\times \\var{b}^\\var{m-1} . \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split}&nbsp;\\frac{\\var{a}^\\var{n} \\times \\var{b}^\\var{m}}{\\sqrt{\\var{a^4} \\times \\var{b^2}}} &amp;\\,= \\var{a}^\\var{n-2} \\times \\var{b}^\\var{m-1} \\\\ &amp;\\,= \\var{a^(n-2)*b^(m-1)}. \\end{split} \\]</p>\n<p></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}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2..5 except [a,a^2])", "description": "", "templateType": "anything", "can_override": false}, "n": {"name": "n", "group": "Ungrouped variables", "definition": "4", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "3", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["a", "b", "n", "m"], "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^(n-2)*b^(m-1)}", "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/"}]}