// Numbas version: finer_feedback_settings
{"name": "Surds: rationalising the denominator simple", "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": ["listsquares", "listprimes", "listsquare", "fcf", "square1", "square2", "listprime", "arg2", "prime1", "prime2", "extra", "ansmult", "ansarg", "questnum2", "questden", "gcff", "num_wo_cf", "den_wo_cf"], "name": "Surds: rationalising the denominator simple", "tags": ["irrationals", "surds"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "3", "prompt": "<p>Given the fraction&nbsp;\\[\\dfrac{\\var{questnum}}{\\var{questdenmult}\\sqrt{\\var{questdensurd}}}\\] 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;\">\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</tr>\n<tr>\n<td class=\"tddenom\" colspan=\"2\" style=\"text-align: center;\">[[2]]</td>\n</tr>\n</tbody>\n</table>\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": "questdensurd", "minValue": "questdensurd", "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 surd part. This will rationalise the denominator since $\\sqrt{a}\\times\\sqrt{a}=a$.&nbsp;</p>\n<table>\n<tbody>\n<tr>\n<td>$\\dfrac{\\var{questnum}}{\\var{questdenmult}\\sqrt{\\var{questdensurd}}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\dfrac{\\var{questnum}}{\\var{questdenmult}\\sqrt{\\var{questdensurd}}}\\times\\dfrac{\\sqrt{\\var{questdensurd}}}{\\sqrt{\\var{questdensurd}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (multiplying top and bottom by the surd part of the denominator)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\dfrac{\\var{questnum}\\sqrt{\\var{questdensurd}}}{\\var{questdenmult}\\times\\var{questdensurd}}$</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>$\\dfrac{\\var{questnum}\\sqrt{\\var{questdensurd}}}{\\var{tempden}}$</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 abs(ansnummult)&lt;&gt;1\">$=$</span></td>\n<td><span data-jme-visible=\"cancel&gt;1 and abs(ansnummult)&lt;&gt;1\">$\\dfrac{\\var{ansnummult}\\sqrt{\\var{questdensurd}}}{\\var{ansden}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"cancel&gt;1 and abs(ansnummult)&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=\"ansnummult=-1\">$=$</span></td>\n<td><span data-jme-visible=\"ansnummult=-1\">$-\\dfrac{\\sqrt{\\var{questdensurd}}}{\\var{ansden}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"ansnummult=-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=\"ansnummult=1\">$=$</span></td>\n<td><span data-jme-visible=\"ansnummult=1\">$\\dfrac{\\sqrt{\\var{questdensurd}}}{\\var{ansden}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"ansnummult=1\">&nbsp; &nbsp; (cancelling the common factor of $\\var{cancel}$)</span></td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>A nice result of this process is that $\\dfrac{1}{\\sqrt{n}}=\\dfrac{\\sqrt{n}}{n}$ for all $n&gt;0$.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [{"variables": ["questnum", "questdenmult", "questdensurd", "tempden", "cancel", "ansnummult", "ansden"], "name": "using"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": ".fractiontable table {\n  width: 20%; \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": {"questnum2": {"definition": "fcf*square1[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "questnum2", "description": ""}, "listsquare": {"definition": "shuffle(listsquares)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquare", "description": ""}, "arg2": {"definition": "square2[1]*prime2", "templateType": "anything", "group": "Ungrouped variables", "name": "arg2", "description": ""}, "gcff": {"definition": "gcd(questnum,questden)", "templateType": "anything", "group": "Ungrouped variables", "name": "gcff", "description": ""}, "ansnummult": {"definition": "questnum/cancel", "templateType": "anything", "group": "using", "name": "ansnummult", "description": ""}, "cancel": {"definition": "gcd(questnum,tempden)", "templateType": "anything", "group": "using", "name": "cancel", "description": ""}, "listsquares": {"definition": "map([n,n^2],n,1..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquares", "description": ""}, "questden": {"definition": "fcf*square2[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "questden", "description": ""}, "extra": {"definition": "random(listprimes)", "templateType": "anything", "group": "Ungrouped variables", "name": "extra", "description": ""}, "listprimes": {"definition": "[2,3,5,7,11]", "templateType": "anything", "group": "Ungrouped variables", "name": "listprimes", "description": ""}, "questnum": {"definition": "random(-12..12 except [-1,0,1])", "templateType": "anything", "group": "using", "name": "questnum", "description": ""}, "num_wo_cf": {"definition": "questnum/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "num_wo_cf", "description": ""}, "tempden": {"definition": "questdenmult*questdensurd", "templateType": "anything", "group": "using", "name": "tempden", "description": ""}, "questdenmult": {"definition": "random(2..12 except questnum)", "templateType": "anything", "group": "using", "name": "questdenmult", "description": ""}, "ansmult": {"definition": "extra*square1[0]*square2[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "ansmult", "description": ""}, "ansarg": {"definition": "prime1*prime2", "templateType": "anything", "group": "Ungrouped variables", "name": "ansarg", "description": ""}, "den_wo_cf": {"definition": "questden/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "den_wo_cf", "description": ""}, "listprime": {"definition": "shuffle(listprimes)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "listprime", "description": ""}, "fcf": {"definition": "random(2..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "fcf", "description": "<p>forced common factor</p>"}, "questdensurd": {"definition": "random(2,3,5,6,7,10,11,13,14,15,17)", "templateType": "anything", "group": "using", "name": "questdensurd", "description": ""}, "prime1": {"definition": "listprime[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime1", "description": ""}, "prime2": {"definition": "listprime[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime2", "description": ""}, "ansden": {"definition": "tempden/cancel", "templateType": "anything", "group": "using", "name": "ansden", "description": ""}, "square1": {"definition": "listsquare[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "square1", "description": ""}, "square2": {"definition": "listsquare[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "square2", "description": ""}}, "metadata": {"notes": "<p>using background image in table to get a good looking square root symbol</p>\n<p></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/"}]}