// Numbas version: finer_feedback_settings
{"name": "Surds: fraction of square roots resulting in rational", "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: fraction of square roots resulting in rational", "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": {"questden": {"name": "questden", "group": "Ungrouped variables", "definition": "fcf*square2[1]", "description": "", "templateType": "anything", "can_override": false}, "questnum": {"name": "questnum", "group": "Ungrouped variables", "definition": "fcf*square1[1]", "description": "", "templateType": "anything", "can_override": false}, "extra": {"name": "extra", "group": "Ungrouped variables", "definition": "random(listprimes)", "description": "", "templateType": "anything", "can_override": false}, "ansmult": {"name": "ansmult", "group": "Ungrouped variables", "definition": "extra*square1[0]*square2[0]", "description": "", "templateType": "anything", "can_override": false}, "arg2": {"name": "arg2", "group": "Ungrouped variables", "definition": "square2[1]*prime2", "description": "", "templateType": "anything", "can_override": false}, "listsquare": {"name": "listsquare", "group": "Ungrouped variables", "definition": "shuffle(listsquares)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "listprimes": {"name": "listprimes", "group": "Ungrouped variables", "definition": "[2,3,5,7,11]", "description": "", "templateType": "anything", "can_override": false}, "den_wo_cf": {"name": "den_wo_cf", "group": "Ungrouped variables", "definition": "questden/gcff", "description": "", "templateType": "anything", "can_override": false}, "listprime": {"name": "listprime", "group": "Ungrouped variables", "definition": "shuffle(listprimes)[0..2]", "description": "", "templateType": "anything", "can_override": false}, "ansarg": {"name": "ansarg", "group": "Ungrouped variables", "definition": "prime1*prime2", "description": "", "templateType": "anything", "can_override": false}, "fcf": {"name": "fcf", "group": "Ungrouped variables", "definition": "random(2..12)", "description": "<p>forced common factor</p>", "templateType": "anything", "can_override": false}, "prime1": {"name": "prime1", "group": "Ungrouped variables", "definition": "listprime[0]", "description": "", "templateType": "anything", "can_override": false}, "square1": {"name": "square1", "group": "Ungrouped variables", "definition": "listsquare[0]", "description": "", "templateType": "anything", "can_override": false}, "square2": {"name": "square2", "group": "Ungrouped variables", "definition": "listsquare[1]", "description": "", "templateType": "anything", "can_override": false}, "prime2": {"name": "prime2", "group": "Ungrouped variables", "definition": "listprime[1]", "description": "", "templateType": "anything", "can_override": false}, "num_wo_cf": {"name": "num_wo_cf", "group": "Ungrouped variables", "definition": "questnum/gcff", "description": "", "templateType": "anything", "can_override": false}, "gcff": {"name": "gcff", "group": "Ungrouped variables", "definition": "gcd(questnum,questden)", "description": "", "templateType": "anything", "can_override": false}, "listsquares": {"name": "listsquares", "group": "Ungrouped variables", "definition": "map([n,n^2],n,1..12)", "description": "", "templateType": "anything", "can_override": false}}, "variablesTest": {"condition": "", "maxRuns": 100}, "ungrouped_variables": ["listsquares", "listprimes", "listsquare", "fcf", "square1", "square2", "listprime", "arg2", "prime1", "prime2", "extra", "ansmult", "ansarg", "questnum", "questden", "gcff", "num_wo_cf", "den_wo_cf"], "variable_groups": [], "functions": {}, "preamble": {"js": "", "css": ""}, "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>The surd expression $\\dfrac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$ simplifies to [[0]].</p>", "stepsPenalty": "1", "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>Given $\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$ we could simplify each surd separately, and then simplify the fraction if possible:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$</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>Or, if we can see a common factor we can use \\[\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{\\frac{a}{b}},\\] and then cancel common factors inside the square root, then convert back to a fraction of square roots and try to simply the remaining surds:</p>\n<table>\n<tbody>\n<tr>\n<td>$\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$</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>$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}\\times\\var{gcff}}{\\var{den_wo_cf}\\times\\var{gcff}}}$</td>\n<td style=\"text-align: left;\">&nbsp; &nbsp; (find greatest common factor)</td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\"><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 style=\"text-align: center;\"><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 style=\"text-align: center;\"><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\"></span></td>\n</tr>\n<tr>\n<td></td>\n<td style=\"text-align: center;\">$=$</td>\n<td>$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n</tbody>\n</table>\n<p>Realising&nbsp;$\\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{\\frac{a}{b}}$ can often be useful.</p>"}], "gaps": [{"type": "numberentry", "useCustomName": false, "customName": "", "marks": 1, "scripts": {}, "customMarkingAlgorithm": "", 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