// Numbas version: finer_feedback_settings
{"name": "Surds: simplifying", "extensions": [], "custom_part_types": [], "resources": [["question-resources/sqrt_Irff7Ni.png", "/srv/numbas/media/question-resources/sqrt_Irff7Ni.png"]], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"functions": {}, "ungrouped_variables": ["listsquares", "listprimes", "easysquareinfo", "easyprime", "easyargument", "listleftovers", "leftovers", "harderargument", "hardermult", "stillunder"], "name": "Surds: simplifying", "tags": ["irrationals", "surds"], "advice": "", "rulesets": {}, "parts": [{"stepsPenalty": "2", "prompt": "<p>Simplify the surd $\\sqrt{\\var{easyargument}}$, entering your numbers in the gaps provided.</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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "easysquareinfo[0]", "minValue": "easysquareinfo[0]", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "easyprime", "minValue": "easyprime", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>The idea is to find a factor which is a square number, e.g. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ... We can then take the square root of that factor out the front of the square root, we can do this until the number left under the square root has no square factor.</p>\n<p>A key fact&nbsp;we use here&nbsp;is that $\\sqrt{a\\times b}=\\sqrt{a}\\times\\sqrt{b}$ &nbsp;for non-negative real numbers $a,b$.</p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\sqrt{\\var{easyargument}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\sqrt{\\var{easysquareinfo[1]}\\times\\var{easyprime}}$</td>\n<td style=\"text-align: left;\">(you find that $\\var{easysquareinfo[1]}$ is a square factor of $\\var{easyargument}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\sqrt{\\var{easysquareinfo[1]}}\\times\\sqrt{\\var{easyprime}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{easysquareinfo[0]}\\times\\sqrt{\\var{easyprime}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{easysquareinfo[0]}\\sqrt{\\var{easyprime}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n</tbody>\n</table>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}, {"stepsPenalty": "2", "prompt": "<p>Simplify the surd $\\sqrt{\\var{harderargument}}$, entering your numbers in the gaps provided.</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>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "gaps": [{"allowFractions": false, "variableReplacements": [], "maxValue": "hardermult", "minValue": "hardermult", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}, {"integerPartialCredit": 0, "integerAnswer": true, "allowFractions": false, "variableReplacements": [], "maxValue": "leftovers", "minValue": "leftovers", "variableReplacementStrategy": "originalfirst", "correctAnswerFraction": false, "showCorrectAnswer": true, "scripts": {}, "marks": 1, "type": "numberentry", "showPrecisionHint": false}], "steps": [{"prompt": "<p>This question is similar to the last one but has larger numbers so it is harder to spot the square factors.&nbsp;Because of this, a calculator, factor trees, divisibility tests and/or long division may be useful.</p>\n<p>The idea is to find all the factors of $\\var{harderargument}$ which are&nbsp;squares, e.g. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, ...&nbsp;We can then take the square root of those factors out the front of the square root, we can do this until the number left under the square root has no square factors left.</p>\n<p></p>\n<table>\n<tbody>\n<tr>\n<td style=\"text-align: right;\">$\\sqrt{\\var{harderargument}}$</td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\sqrt{\\var{hardersquareinfo[0][1]}\\times\\var{stillunder}}$</td>\n<td style=\"text-align: left;\">(you might find that $\\var{hardersquareinfo[0][1]}$ is a square factor of $\\var{harderargument}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\sqrt{\\var{hardersquareinfo[0][1]}}\\times\\sqrt{\\var{stillunder}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{hardersquareinfo[0][0]}\\sqrt{\\var{stillunder}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{hardersquareinfo[0][0]}\\times\\sqrt{\\var{hardersquareinfo[1][1]}\\times\\var{leftovers}}$</td>\n<td style=\"text-align: left;\">(you might find that $\\var{hardersquareinfo[1][1]}$ is a square factor of $\\var{stillunder}$)</td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{hardersquareinfo[0][0]}\\times\\sqrt{\\var{hardersquareinfo[1][1]}}\\times\\sqrt{\\var{leftovers}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{hardersquareinfo[0][0]}\\times\\var{hardersquareinfo[1][0]}\\times\\sqrt{\\var{leftovers}}$</td>\n<td style=\"text-align: left;\"></td>\n</tr>\n<tr>\n<td style=\"text-align: right;\"></td>\n<td style=\"text-align: center;\">$=$</td>\n<td style=\"text-align: left;\">$\\var{hardermult}\\sqrt{\\var{leftovers}}$</td>\n<td style=\"text-align: left;\">(there are no square factors of $\\var{leftovers}$ so we are finished)</td>\n</tr>\n</tbody>\n</table>\n<p>&nbsp;</p>", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [{"variables": ["listprimesquares", "k", "hardersquareinfo"], "name": "Unnamed group"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": "", "js": ""}, "variables": {"hardermult": {"definition": "hardersquareinfo[0][0]*hardersquareinfo[1][0]", "templateType": "anything", "group": "Ungrouped variables", "name": "hardermult", "description": ""}, "leftovers": {"definition": "listleftovers[0]*listleftovers[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "leftovers", "description": ""}, "k": {"definition": "random(1..6)", "templateType": "anything", "group": "Unnamed group", "name": "k", "description": ""}, "easyprime": {"definition": "random(listprimes[0..5])", "templateType": "anything", "group": "Ungrouped variables", "name": "easyprime", "description": ""}, "hardersquareinfo": {"definition": "[random(listprimesquares[0..k]),random(listprimesquares[k..7])]", "templateType": "anything", "group": "Unnamed group", "name": "hardersquareinfo", "description": ""}, "listprimesquares": {"definition": "map([n,n^2],n,[2,3,5,7,11,13,17])", "templateType": "anything", "group": "Unnamed group", "name": "listprimesquares", "description": ""}, "listleftovers": {"definition": "shuffle(listprimes)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "listleftovers", "description": ""}, "easyargument": {"definition": "easysquareinfo[1]*easyprime", "templateType": "anything", "group": "Ungrouped variables", "name": "easyargument", "description": ""}, "easysquareinfo": {"definition": "random(listsquares[0..11])", "templateType": "anything", "group": "Ungrouped variables", "name": "easysquareinfo", "description": ""}, "listprimes": {"definition": "[2,3,5,7,11,13,17]", "templateType": "anything", "group": "Ungrouped variables", "name": "listprimes", "description": ""}, "harderargument": {"definition": "leftovers*hardersquareinfo[0][1]*hardersquareinfo[1][1]", "templateType": "anything", "group": "Ungrouped variables", "name": "harderargument", "description": ""}, "stillunder": {"definition": "leftovers*hardersquareinfo[1][1]", "templateType": "anything", "group": "Ungrouped variables", "name": "stillunder", "description": ""}, "listsquares": {"definition": "map([n,n^2],n,2..15)", "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/"}]}