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{"name": "Surds: Simplifying Expressions 2a", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Surds: Simplifying Expressions 2a", "tags": [], "metadata": {"description": "<p>Rewiting expressions involving sums or differences of surds into the form $a \\sqrt{b}$.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "<p>Rewrite the following expression in the form $a\\sqrt{b}$ :</p>\n<p>\\[ \\simplify{sqrt({b1^2*c})+{sgn}*sqrt({b2^2*c})}\\]</p>", "advice": "<p>To write $\\simplify{sqrt({b1^2*c})+{sgn}*sqrt({b2^2*c})}$ in the form $a\\sqrt{b}$ we need to be able to rewrite $\\simplify{sqrt({b1^2*c})}$ and $\\simplify{sqrt({b2^2*c})}$ such that both have the same surd as a common factor.&nbsp;</p>\n<p>We see that</p>\n<p>\\[ \\sqrt{\\var{b1^2*c}} = \\sqrt{\\var{b1^2}\\times\\var{c}}\\,, \\qquad \\sqrt{\\var{b2^2*c}} = \\sqrt{\\var{b2^2}\\times\\var{c}}. \\]</p>\n<p>If we then apply the rule&nbsp;$\\sqrt{ab} = \\sqrt{a} \\sqrt{b}$:</p>\n<p>\\[ \\begin{split} \\sqrt{\\var{b1^2*c}} &amp;\\,= \\sqrt{\\var{b1^2}} \\times \\sqrt{\\var{c}}\\,, \\qquad&nbsp;\\sqrt{\\var{b2^2*c}} &amp;\\,= \\sqrt{\\var{b2^2}} \\times \\sqrt{\\var{c}} \\\\ &amp;\\,= \\var{b1} \\sqrt{\\var{c}} &amp;\\,= \\var{b2} \\sqrt{\\var{c}} \\end{split} \\]</p>\n<p>Therefore,</p>\n<p>\\[ \\begin{split}&nbsp;\\simplify{sqrt({b1^2*c})+{sgn}*sqrt({b2^2*c})} &amp;\\,= \\simplify[!collectNumbers]{{b1}sqrt({c}) + {sgn*b2}sqrt({c})} \\\\ &amp;\\,= \\simplify{{b1+sgn*b2}sqrt({c})}. \\end{split} \\]</p>\n<p></p>\n<p></p>", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true}, "constants": [], "variables": {"b1": {"name": "b1", "group": "Ungrouped variables", "definition": "random(2..8)", "description": "", "templateType": "anything", "can_override": false}, "c": {"name": "c", "group": "Ungrouped variables", "definition": "random([2,3,5,6,7] except [b1,b2])", "description": "", "templateType": "anything", "can_override": false}, "b2": {"name": "b2", "group": "Ungrouped variables", "definition": "random(2..8 except b1)", "description": "", "templateType": "anything", "can_override": false}, "sgn": {"name": "sgn", "group": "Ungrouped variables", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "b1>b2", "maxRuns": 100}, "ungrouped_variables": ["b1", "c", "b2", "sgn"], "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, "prompt": "<p><em>You can type sqrt(x) to get $\\sqrt{x}$.</em></p>", "answer": "({b1+sgn*b2})sqrt({c})", "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": "`+-$n`?*sqrt($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/"}]}