// Numbas version: finer_feedback_settings
{"name": "Surds: rationalising the denominator conjugate", "extensions": [], "custom_part_types": [], "resources": [["question-resources/sqrt_Irff7Ni.png", "sqrt_Irff7Ni.png"], ["question-resources/fracsqrts.png", "fracsqrts.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"name": "Surds: rationalising the denominator conjugate", "tags": ["irrationals", "surds", "Surds"], "metadata": {"description": "", "licence": "Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International"}, "statement": "", "advice": "", "rulesets": {}, "extensions": [], "builtin_constants": {"e": true, "pi,\u03c0": true, "i": true, "j": false}, "constants": [], "variables": {"tempden": {"name": "tempden", "group": "using", "definition": "surd1-surd2", "description": "", "templateType": "anything", "can_override": false}, "surdlist": {"name": "surdlist", "group": "using", "definition": "sort(shuffle([2,3,5,6,7,10,11,13,14,15,17])[0..2])", "description": "", "templateType": "anything", "can_override": false}, "surd2": {"name": "surd2", "group": "using", "definition": "surdlist[0]", "description": "", "templateType": "anything", "can_override": false}, "surd1": {"name": "surd1", "group": "using", "definition": "surdlist[1]", "description": "", "templateType": "anything", "can_override": false}, "ran": {"name": "ran", "group": "using", "definition": "random(-1,1)", "description": "", "templateType": "anything", "can_override": false}, "densign": {"name": "densign", "group": "using", "definition": "if(ran=1,'$\\\\,+\\\\,$','$\\\\,-\\\\,$')", "description": "", "templateType": "anything", "can_override": false}, "ansden": {"name": "ansden", "group": "using", "definition": "tempden/cancel", "description": "", "templateType": "anything", "can_override": false}, "secondnummult": {"name": "secondnummult", "group": "using", "definition": "if(ran=1,-ansnummult,ansnummult)", "description": "", "templateType": "anything", "can_override": false}, "ansnummult": {"name": "ansnummult", "group": "using", "definition": "questnum/cancel", "description": "", "templateType": "anything", "can_override": false}, "conjsign": {"name": "conjsign", "group": "using", "definition": "if(ran=-1,'$\\\\,+\\\\,$','$\\\\,-\\\\,$')", "description": "", "templateType": "anything", "can_override": false}, "questnum": {"name": "questnum", "group": "using", "definition": "random(2..12)", "description": "", "templateType": "anything", "can_override": false}, "cancel": {"name": "cancel", "group": "using", "definition": "gcd(questnum,tempden)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": [], "variable_groups": [{"name": "using", "variables": ["questnum", "surdlist", "surd1", "surd2", "tempden", "cancel", "ansnummult", "ansden", "ran", "densign", "conjsign", "secondnummult"]}], "functions": {}, "preamble": {"js": "", "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"}, "parts": [{"type": "gapfill", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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&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>", "stepsPenalty": "5", "steps": [{"type": "information", "useCustomName": false, "customName": "", "marks": 0, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "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><span data-jme-visible=\"cancel&gt;1 and ansden&lt;&gt;1\">$\\,$</span></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><span data-jme-visible=\"cancel&gt;1 and ansden=1\">$\\,$</span></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><span data-jme-visible=\"cancel=1 and ansden=1\">$\\,$</span></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>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansnummult", "maxValue": "ansnummult", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "surd1", "maxValue": "surd1", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "secondnummult", "maxValue": "secondnummult", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "surd2", "maxValue": "surd2", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}, {"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "unitTests": [], "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "nextParts": [], "suggestGoingBack": false, "adaptiveMarkingPenalty": 0, "exploreObjective": null, "minValue": "ansden", "maxValue": "ansden", "correctAnswerFraction": false, "allowFractions": false, "mustBeReduced": false, "mustBeReducedPC": 0, "displayAnswer": "", "showFractionHint": true, "notationStyles": ["plain", "en", "si-en"], "correctAnswerStyle": "plain"}], "sortAnswers": false}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always", "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}], "resources": ["question-resources/sqrt_Irff7Ni.png", "question-resources/fracsqrts.png"]}]}], "contributors": [{"name": "Ben Brawn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/605/"}]}