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{"name": "Surds: Simplifying Expressions 6 - Rationalising the Denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Surds: Simplifying Expressions 6 - Rationalising the Denominator", "tags": ["Category:Surds", "surds", "Surds"], "metadata": {"description": "<p>Rewriting fractions involving surds by rationalising the denominator.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rationalise the denominator in the following expression:</p>\n<p>\\[\\simplify[unitFactor]{{c}/({a}+{sgn}*sqrt({b}))}\\]</p>", "advice": "<p>To rationalise the denominator of a fraction which has the form $\\frac{n}{a\\pm\\sqrt{b}}$, we must multiply the numerator and denominator by $a \\mp\\sqrt{b}$.</p>\n<p>Therefore,&nbsp;</p>\n<p>{advice}</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(1..9)", "description": "", "templateType": "anything", "can_override": false}, "b": {"name": "b", "group": "Ungrouped variables", "definition": "random(2,3,5,6,7)", "description": "", "templateType": "anything", "can_override": false}, "simplify": {"name": "simplify", "group": "Ungrouped variables", "definition": "\"<p>\\\\[\\\\begin{split} \\\\simplify[unitFactor]{{c}/({a}+{sgn}*sqrt({b}))} &amp;\\\\,=\\\\simplify[unitFactor]{{c}({a}-{sgn}*sqrt({b}))/(({a}+{sgn}*sqrt({b}))({a}-{sgn}*sqrt({b})))} \\\\\\\\\\\\\\\\ &amp;\\\\,= \\\\simplify[unitFactor]{({c*a}-{sgn*c}sqrt({b}))/({a^2}-{sgn*a}sqrt({b})+{sgn*a}sqrt({b}) -{b})} \\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[unitFactor]{({c*a}-{sgn*c}sqrt({b}))/({a^2}-{b})}\\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[unitFactor]{({c*a}-{sgn*c}sqrt({b}))/({(a^2-b)})}\\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[unitFactor]{({c*a/simp}-{sgn*c/simp}sqrt({b}))/({(a^2-b)/simp})}\\\\end{split} \\\\]</p>\"", "description": "", "templateType": "long string", "can_override": false}, "simplified": {"name": "simplified", "group": "Ungrouped variables", "definition": "\"<p>\\\\[\\\\begin{split} \\\\simplify[unitFactor]{{c}/({a}+{sgn}*sqrt({b}))} &amp;\\\\,=\\\\simplify[unitFactor]{{c}({a}-{sgn}*sqrt({b}))/(({a}+{sgn}*sqrt({b}))({a}-{sgn}*sqrt({b})))} \\\\\\\\\\\\\\\\ &amp;\\\\,= \\\\simplify[unitFactor]{({c*a}-{sgn*c}sqrt({b}))/({a^2}-{sgn*a}sqrt({b})+{sgn*a}sqrt({b}) -{b})} \\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[unitFactor]{({c*a}-{sgn*c}sqrt({b}))/({a^2}-{b})}\\\\\\\\\\\\\\\\ &amp;\\\\,=\\\\simplify[unitFactor]{({c*a}-{sgn*c}sqrt({b}))/({a^2-b})}\\\\end{split} \\\\]</p>\"", "description": "", 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"customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "prompt": "<p><em>You can type sqrt(x) to get $\\sqrt{x}$.</em></p>", "answer": "({c*a/simp}-{c*sgn/simp}sqrt({b}))/{(a^2-b)/simp}", "answerSimplification": "all", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": []}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben 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