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{"name": "Indices: Change of Notation 3", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Indices: Change of Notation 3", "tags": ["category: Indices", "leads to: Bc", "skill: Bb"], "metadata": {"description": "<p>Rewriting fractional expressions involving $\\sqrt[n]{x^m}$ using rules to combine and simplify indices.&nbsp;</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression as a single term, in the form $x^n$, where $n$ is a fraction.</p>\n<p>\\[ \\simplify[!collectNumbers]{root(x^{n}x^{m},{p})/(root(x,{q})root(x^{q},{2q}))} \\]</p>", "advice": "<p>To rewrite $\\dfrac{\\sqrt[\\var{p}]{x^\\var{n}x^\\var{m}}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}}$ in the form $x^n$, we need to use the following rules:</p>\n<ol>\n<li>$a^n a^m = a^{n+m}$;</li>\n<li>$\\sqrt[n]{a} = a^{1/n}$;</li>\n<li>$\\left(a^n\\right)^m = a^{n \\times m}$;</li>\n<li>$\\frac{a^n}{a^m} = a^{n-m}$.</li>\n</ol>\n<p>We will simplify the numerator and denominator sepearately to make the steps clearer. Firstly, applying rule 1, then rule 2, and then rule 3 to the numerator:</p>\n<p>\\[ \\begin{split}\\dfrac{\\sqrt[\\var{p}]{x^\\var{n}x^\\var{m}}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}} &amp;\\,=\\dfrac{\\sqrt[\\var{p}]{x^{\\var{n+m}}}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}} \\\\\\\\&amp;\\,=\\dfrac{\\left(x^\\var{n+m}\\right)^\\simplify[fractionNumbers]{{1/p}}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}} \\\\\\\\ &amp;\\,= \\dfrac{x^\\var{(n+m)/p}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}} \\end{split}\\]</p>\n<p></p>\n<p>To simplify the denominator, we want to apply rule 2, then rule 3, and then rule 1:</p>\n<p>\\[ \\begin{split}&nbsp;\\dfrac{x^\\var{(n+m)/p}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}} &amp;\\,= \\dfrac{x^\\var{(n+m)/p}}{x^{1/\\var{q}}\\left(x^\\var{q}\\right)^{1/\\var{2q}}} \\\\\\\\ &amp;\\,=\\dfrac{x^\\var{(n+m)/p}}{x^{1/\\var{q}} x^{1/2}}\\\\\\\\ &amp;\\,=\\dfrac{x^\\var{(n+m)/p}}{x^\\simplify[fractionNumbers]{{1/q+1/2}}}.\\end{split} \\]</p>\n<p>Finally, applying rule 4 and simplifying,&nbsp;</p>\n<p>\\[ \\begin{split} \\dfrac{x^\\var{(n+m)/p}}{x^\\simplify[fractionNumbers]{{1/q+1/2}}} &amp;\\,= x^{\\var{(n+m)/p}-\\simplify[fractionNumbers]{{1/q+1/2}}} \\\\ &amp;\\,=&nbsp;x^{\\simplify[fractionNumbers]{{(n+m)/p-(1/q+1/2)}}}. \\end{split}\\]</p>\n<p>Therefore,</p>\n<p>\\[&nbsp;\\dfrac{\\sqrt[\\var{p}]{x^\\var{n}x^\\var{m}}}{\\sqrt[\\var{q}]{x}\\sqrt[\\var{2q}]{x^\\var{q}}} =&nbsp;x^{\\simplify[fractionNumbers]{{(n+m)/p-(1/q+1/2)}}}. \\]</p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"n": {"name": "n", "group": "Ungrouped variables", "definition": "random(3..9 #2)", "description": "", "templateType": "anything", "can_override": false}, "m": {"name": "m", "group": "Ungrouped variables", "definition": "random(3..9 #2 except n)", "description": "", "templateType": "anything", "can_override": false}, "p": {"name": "p", "group": "Ungrouped variables", "definition": "(n+m)/2", "description": "", "templateType": "anything", "can_override": false}, "q": {"name": "q", "group": "Ungrouped variables", "definition": "random(3..5 except p)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["n", "m", "p", "q"], "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^{(n+m)/p-(1/q+1/2)}", "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": "x^`+-rational:$n", "partialCredit": 0, "message": "", "nameToCompare": ""}, "valuegenerators": [{"name": "x", "value": ""}]}], "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/"}]}