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{"name": "Numbers VI: surds (rationalising the denominator - conjugate EXAMPLE)", "extensions": [], "custom_part_types": [], "resources": [["question-resources/sqrt_Irff7Ni.png", "/srv/numbas/media/question-resources/sqrt_Irff7Ni.png"], ["question-resources/fracsqrts.png", "/srv/numbas/media/question-resources/fracsqrts.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": [], "name": "Numbers VI: surds (rationalising the denominator - conjugate EXAMPLE)", "tags": [], "preamble": {"css": ".fractiontable table {\n  width: 40%; \n  padding: 0px; \n  border-width: 0px; \n  layout: fixed;\n}\n\n.fractiontable  .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n  width: 15%; \n  border-bottom: 1px solid black; \n  text-align: center;\n}\n\n.fractiontable .tdeq \n{\n  width: 5%; \n  border-bottom: 0px;\n  font-size: x-large;\n}\n\n\n.fractiontable th {\n    background-color:#aaa;\n}\n/*Fix the height of all cells EXCEPT table-headers to 40px*/\n.fractiontable td {\n    height:40px;\n}\n", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "5", "prompt": "<p>Given the fraction \\[\\dfrac{\\var{questnum}}{\\sqrt{\\var{surd1}}\\var{densign}\\sqrt{\\var{surd2}}}\\] we can rationalise the denominator and rewrite the fraction in the simplified equivalent form.</p>\n<p></p>\n<div class=\"fractiontable\">\n<table cellpadding=\"0\" style=\"text-align: right;\" width=\"345\" height=\"132\">\n<tbody>\n<tr>\n<td class=\"tdnum\" style=\"text-align: center;\">[[0]]</td>\n<td class=\"tdnum\" style=\"text-align: center; background-image: url('resources/question-resources/sqrt_Irff7Ni.png'); background-repeat: no-repeat;\">&nbsp; &nbsp; $\\,\\,\\,\\,$[[1]]</td>\n<td class=\"tdnum\" style=\"text-align: center;\">$+$&nbsp;[[2]]</td>\n<td class=\"tdnum\" style=\"text-align: center; background-image: url('resources/question-resources/sqrt_Irff7Ni.png'); background-repeat: no-repeat;\">&nbsp; &nbsp; $\\,\\,\\,\\,$[[3]]</td>\n</tr>\n<tr>\n<td class=\"tddenom\" colspan=\"4\" style=\"text-align: center;\">[[4]]</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p><strong>Note:</strong>&nbsp;This question requires the larger surd term to be entered&nbsp;on the left and the smaller one on the right e.g. $\\frac{2\\sqrt{7} + 5\\sqrt{3}}{6}$.</p>\n</div>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "ansnummult", "minValue": "ansnummult", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "surd1", "minValue": "surd1", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "secondnummult", "minValue": "secondnummult", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "surd2", "minValue": "surd2", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "ansden", "minValue": "ansden", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": false, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>In the above&nbsp;case, to rationalise the denominator we can multiply the top and bottom of the fraction by the conjugate surd of the denominator. This will rationalise the denominator since $\\left(\\sqrt{a}+\\sqrt{b}\\right)\\times\\left(\\sqrt{a}-\\sqrt{b}\\right)=a-\\sqrt{ab}+\\sqrt{ab}-b=a-b$.&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>$\\dfrac{\\var{questnum}}{\\sqrt{\\var{surd1}}\\var{densign}\\sqrt{\\var{surd2}}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\dfrac{\\var{questnum}}{\\sqrt{\\var{surd1}}\\var{densign}\\sqrt{\\var{surd2}}}\\times\\dfrac{\\sqrt{\\var{surd1}}\\var{conjsign}\\sqrt{\\var{surd2}}}{\\sqrt{\\var{surd1}}\\var{conjsign}\\sqrt{\\var{surd2}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (multiplying top and bottom by the conjugate surd of the denominator)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\dfrac{\\var{questnum}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{questnum}\\sqrt{\\var{surd2}}}{\\var{surd1}-\\var{surd2}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp;&nbsp;</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\dfrac{\\var{questnum}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{questnum}\\sqrt{\\var{surd2}}}{\\var{tempden}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp;&nbsp;</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"cancel&gt;1 and ansden&lt;&gt;1\">$=$</span></td>\n<td><span data-jme-visible=\"cancel&gt;1 and ansden&lt;&gt;1\">$\\dfrac{\\var{ansnummult}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{ansnummult}\\sqrt{\\var{surd2}}}{\\var{ansden}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"cancel&gt;1 and ansden&lt;&gt;1\">&nbsp; &nbsp; (cancelling the common factor of $\\var{cancel}$)</span></td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"cancel&gt;1 and ansden=1\">$=$</span></td>\n<td><span data-jme-visible=\"cancel&gt;1 and ansden=1\">$\\var{ansnummult}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{ansnummult}\\sqrt{\\var{surd2}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"cancel&gt;1 and ansden=1\">&nbsp; &nbsp; (cancelling the common factor of $\\var{cancel}$)</span></td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"cancel=1 and ansden=1\">$=$</span></td>\n<td><span data-jme-visible=\"cancel=1 and ansden=1\">$\\var{ansnummult}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{ansnummult}\\sqrt{\\var{surd2}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"cancel=1 and ansden=1\"></span></td>\n</tr>\n</tbody>\n</table>\n<p></p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "extensions": [], "statement": "", "variable_groups": [{"variables": ["questnum", "surdlist", "surd1", "surd2", "tempden", "cancel", "ansnummult", "ansden", "ran", "densign", "conjsign", "secondnummult"], "name": "using"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"tempden": {"definition": "surd1-surd2", "templateType": "anything", "group": "using", "name": "tempden", "description": ""}, "surdlist": {"definition": "sort(shuffle([2,3,5,6,7,10,11,13,14,15,17])[0..2])", "templateType": "anything", "group": "using", "name": "surdlist", "description": ""}, "surd2": {"definition": "surdlist[0]", "templateType": "anything", "group": "using", "name": "surd2", "description": ""}, "surd1": {"definition": "surdlist[1]", "templateType": "anything", "group": "using", "name": "surd1", "description": ""}, "ran": {"definition": "random(-1,1)", "templateType": "anything", "group": "using", "name": "ran", "description": ""}, "densign": {"definition": "if(ran=1,'$\\\\,+\\\\,$','$\\\\,-\\\\,$')", "templateType": "anything", "group": "using", "name": "densign", "description": ""}, "ansden": {"definition": "tempden/cancel", "templateType": "anything", "group": "using", "name": "ansden", "description": ""}, "secondnummult": {"definition": "if(ran=1,-ansnummult,ansnummult)", "templateType": "anything", "group": "using", "name": "secondnummult", "description": ""}, "ansnummult": {"definition": "questnum/cancel", "templateType": "anything", "group": "using", "name": "ansnummult", "description": ""}, "conjsign": {"definition": "if(ran=-1,'$\\\\,+\\\\,$','$\\\\,-\\\\,$')", "templateType": "anything", "group": "using", "name": "conjsign", "description": ""}, "questnum": {"definition": "random(2..12)", "templateType": "anything", "group": "using", "name": "questnum", "description": ""}, "cancel": {"definition": "gcd(questnum,tempden)", "templateType": "anything", "group": "using", "name": "cancel", "description": ""}}, "metadata": {"description": "<p>Single example with detailed solution for rationalising a binomial&nbsp;denominator which contains surds.</p>\n<p><em>Adapted from&nbsp;</em><strong>'Surds: rationalising the denominator conjugate'</strong><em> by&nbsp;</em><strong>Ben Brawn</strong>.</p>", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}]}]}], "contributors": [{"name": "Luke Park", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/826/"}]}