// Numbas version: exam_results_page_options
{"name": "Surds: square root of fraction resulting in rational", "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", "questnum", "questden", "gcff", "num_wo_cf", "den_wo_cf"], "name": "Surds: square root of fraction resulting in rational", "tags": ["irrationals", "surds"], "preamble": {"css": "", "js": ""}, "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "1", "prompt": "<p>The following surd expression \\[\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}\\] simplifies to&nbsp;[[0]].</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": true, "variableReplacements": [], "maxValue": "square1[0]/square2[0]", "minValue": "square1[0]/square2[0]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": true, "showCorrectAnswer": false, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>Given $\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$ we can cancel any&nbsp;common factors inside the square root, then use \\[\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}}\\] and simplify any&nbsp;remaining surds:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$</td>\n<td>$=$</td>\n<td>$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}\\times\\var{gcff}}{\\var{den_wo_cf}\\times\\var{gcff}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (find greatest common factor)</td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"den_wo_cf&gt;1\">$=$</span></td>\n<td><span data-jme-visible=\"den_wo_cf&gt;1\">$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}}{\\var{den_wo_cf}}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"den_wo_cf&gt;1\">&nbsp; &nbsp; (cancel common factors)</span></td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"den_wo_cf&gt;1\">$=$</span></td>\n<td><span data-jme-visible=\"den_wo_cf&gt;1\">$\\displaystyle\\frac{\\sqrt{\\var{num_wo_cf}}}{\\sqrt{\\var{den_wo_cf}}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"den_wo_cf&gt;1\">&nbsp; &nbsp; (use surd identity)</span></td>\n</tr>\n<tr>\n<td></td>\n<td><span data-jme-visible=\"den_wo_cf=1\">$=$</span></td>\n<td><span data-jme-visible=\"den_wo_cf=1\">$\\displaystyle\\sqrt{\\var{num_wo_cf}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"den_wo_cf=1\">&nbsp; &nbsp; (simplify)</span></td>\n</tr>\n<tr>\n<td></td>\n<td>$=$</td>\n<td>$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (simplify)</td>\n</tr>\n</tbody>\n</table>\n<p></p>\n<p>Alternatively, we could use \\[\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\] straight away,&nbsp;try to simply the surds and then simplify any remaining&nbsp;fraction:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (use surd identity)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\displaystyle\\frac{\\sqrt{\\var{square1[1]}\\times\\var{fcf}}}{\\sqrt{\\var{square2[1]}\\times\\var{fcf}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (find square factors)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\displaystyle\\frac{\\var{square1[0]}\\sqrt{\\var{fcf}}}{\\var{square2[0]}\\sqrt{\\var{fcf}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (simplify surds)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\displaystyle\\frac{\\var{square1[0]}}{\\var{square2[0]}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (simplify fraction)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\"><span data-jme-visible=\"gcd(square1[0],square2[0])&gt;1\">$=$</span></td>\n<td><span data-jme-visible=\"gcd(square1[0],square2[0])&gt;1\">$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$</span></td>\n<td style=\"text-align: left;\"><span data-jme-visible=\"gcd(square1[0],square2[0])&gt;1\"></span></td>\n</tr>\n</tbody>\n</table>\n<p>Realising $\\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{\\frac{a}{b}}$ can often be useful.</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [], "variablesTest": {"maxRuns": 100, "condition": ""}, "variables": {"questden": {"definition": "fcf*square2[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "questden", "description": ""}, "questnum": {"definition": "fcf*square1[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "questnum", "description": ""}, "extra": {"definition": "random(listprimes)", "templateType": "anything", "group": "Ungrouped variables", "name": "extra", "description": ""}, "ansmult": {"definition": "extra*square1[0]*square2[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "ansmult", "description": ""}, "arg2": {"definition": "square2[1]*prime2", "templateType": "anything", "group": "Ungrouped variables", "name": "arg2", "description": ""}, "listsquare": {"definition": "shuffle(listsquares)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquare", "description": ""}, "listprimes": {"definition": "[2,3,5,7,11]", "templateType": "anything", "group": "Ungrouped variables", "name": "listprimes", "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": ""}, "ansarg": {"definition": "prime1*prime2", "templateType": "anything", "group": "Ungrouped variables", "name": "ansarg", "description": ""}, "fcf": {"definition": "random(2..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "fcf", "description": "<p>forced common factor</p>"}, "prime1": {"definition": "listprime[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime1", "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": ""}, "prime2": {"definition": "listprime[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime2", "description": ""}, "num_wo_cf": {"definition": "questnum/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "num_wo_cf", "description": ""}, "gcff": {"definition": "gcd(questnum,questden)", "templateType": "anything", "group": "Ungrouped variables", "name": "gcff", "description": ""}, "listsquares": {"definition": "map([n,n^2],n,1..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquares", "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/"}]}