// Numbas version: finer_feedback_settings
{"name": "Surds: rationalising the denominator conjugate", "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": "Surds: rationalising the denominator conjugate", "tags": ["irrationals", "surds"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "5", "prompt": "<p>Given the fraction&nbsp;\\[\\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&nbsp;</p>\n<div class=\"fractiontable\">\n<table cellpadding=\"0\" style=\"text-align: right;\" width=\"800\" height=\"120\">\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> If you are marked incorrect, please try swapping the order of terms in the numerator, this question requires the larger surd part on the left and the smaller surd part on the right.&nbsp;</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"}], "statement": "", "variable_groups": [{"variables": ["questnum", "surdlist", "surd1", "surd2", "tempden", "cancel", "ansnummult", "ansden", "ran", "densign", "conjsign", "secondnummult"], "name": "using"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "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": ""}, "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": {"notes": "<p>Not sure how to stop squishing of table when the window is small.</p>\n<p>using background image in table to get a good looking square root symbol</p>\n<p></p>\n<table cellpadding=\"0\" style=\"text-align: right;\">\n<tbody>\n<tr>\n<td>[[0]]</td>\n<td style=\"background-image: url('resources/question-resources/sqrt_Irff7Ni.png'); background-repeat: no-repeat;\">&nbsp; &nbsp;$\\,\\,\\,\\,$[[1]]</td>\n</tr>\n</tbody>\n</table>", "description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "type": "question", "showQuestionGroupNames": false, "question_groups": [{"name": "", "pickingStrategy": "all-ordered", "pickQuestions": 0, "questions": []}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}