// Numbas version: exam_results_page_options
{"name": "Simon's copy of Surds simplification", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "question_groups": [{"pickingStrategy": "all-ordered", "questions": [{"parts": [{"variableReplacements": [], "warningType": "none", "marks": 0, "scripts": {}, "prompt": "<p>For the following examples, tick the correct box to determine whether or not they are a surd.</p>\n<p></p>", "useCustomName": false, "choices": ["<p>$\\sqrt{\\var{square}}$</p>", "<p>$\\sqrt{\\var{h}}$</p>", "<p>$^3\\sqrt{\\var{cube}}$</p>", "<p>$\\sqrt{\\var{j}}$</p>", "<p>$\\sqrt{\\var{k}}$</p>"], "showCellAnswerState": true, "layout": {"type": "all", "expression": ""}, "maxAnswers": 0, "displayType": "radiogroup", "matrix": [[0, "1"], ["1", 0], [0, "1"], ["1", "0"], ["1", 0]], "shuffleChoices": false, "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "minAnswers": 0, "type": "m_n_x", "minMarks": 0, "maxMarks": 0, "unitTests": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "answers": ["<p>Surd</p>", "<p>Not a surd</p>"], "customName": "", "showCorrectAnswer": true}, {"variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "<p>Match each surd with the equivalent simplification.</p>\n<p>[[0]]</p>", "useCustomName": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "customName": "", "gaps": [{"variableReplacements": [], "warningType": "none", "marks": 0, "scripts": {}, "useCustomName": false, "choices": ["<p>i) $\\sqrt{48}$</p>", "<p>ii) $\\sqrt{32}$</p>", "<p>iii) $\\sqrt{56}$</p>", "<p>iv) $\\sqrt{44}$</p>"], "showCellAnswerState": true, "layout": {"type": "all", "expression": ""}, "maxAnswers": 0, "displayType": "radiogroup", "matrix": [["1", 0, 0, "0"], [0, "0", "0", "1"], ["0", "0", "1", 0], [0, "1", 0, 0]], "shuffleChoices": true, "shuffleAnswers": true, "variableReplacementStrategy": "originalfirst", "minAnswers": 0, "type": "m_n_x", "minMarks": 0, "maxMarks": 0, "unitTests": [], "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "answers": ["<p>$4\\sqrt3$</p>", "<p>$2\\sqrt{11}$</p>", "<p>$2\\sqrt{14}$</p>", "<p>$4\\sqrt{2}$</p>"], "customName": "", "showCorrectAnswer": true}], "showCorrectAnswer": true, "unitTests": []}, {"variableReplacements": [], "type": "gapfill", "variableReplacementStrategy": "originalfirst", "marks": 0, "scripts": {}, "prompt": "<p>Simplify the following surds:</p>\n<p>$\\displaystyle\\sqrt{\\var{c}}$ =&nbsp;[[0]]$\\displaystyle\\sqrt{\\var{b}}$</p>\n<p>$\\displaystyle\\sqrt{\\var{g}}$ = [[1]]$\\displaystyle\\sqrt{\\var{f}}$</p>\n<p></p>", "useCustomName": false, "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "showFeedbackIcon": true, "sortAnswers": false, "customName": "", "gaps": [{"variableReplacements": [], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "marks": 1, "scripts": {}, "showFractionHint": true, "useCustomName": false, "maxValue": "sqrta", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "sqrta", "showFeedbackIcon": true, "correctAnswerFraction": false, "allowFractions": false, "customName": "", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "unitTests": [], "mustBeReduced": false}, {"variableReplacements": [], "type": "numberentry", "variableReplacementStrategy": "originalfirst", "marks": 1, "scripts": {}, "showFractionHint": true, "useCustomName": false, "maxValue": "sqrtd", "customMarkingAlgorithm": "", "extendBaseMarkingAlgorithm": true, "minValue": "sqrtd", "showFeedbackIcon": true, "correctAnswerFraction": false, "allowFractions": false, "customName": "", "notationStyles": ["plain", "en", "si-en"], "mustBeReducedPC": 0, "correctAnswerStyle": "plain", "showCorrectAnswer": true, "unitTests": [], "mustBeReduced": false}], "showCorrectAnswer": true, "unitTests": []}], "ungrouped_variables": ["aoptions", "b", "a", "c", "sqrta", "doptions", "sqrtd", "d", "f", "g", "square", "cube", "h", "j", "k", "root"], "extensions": [], "variable_groups": [], "name": "Simon's copy of Surds simplification", "preamble": {"js": "", "css": ""}, "variablesTest": {"condition": "", "maxRuns": 100}, "tags": [], "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "<p>This question tests the student's understanding of what is and is not a surd, and on their simplification of surds.</p>"}, "statement": "<p>Surds are square roots that cannot be simplified to a whole number. They have a decimal equivalent but their decimal representations&nbsp;are never-ending. Therefore,&nbsp;it is often easier to leave surds as they are in algebraic calculations.</p>", "functions": {}, "rulesets": {}, "advice": "<h4>(a)</h4>\n<p>$\\sqrt{\\var{square}}$ and $\\sqrt[3]{\\var{cube}}$&nbsp;are not surds, as&nbsp;they can be simplified to whole integers: $\\simplify{{sqrt(square)}}$ and $\\var{root}$ respectively.&nbsp;They are roots, but not surds.&nbsp;All surds are roots but not all roots are&nbsp;surds.</p>\n<p>$\\sqrt{\\var{h}}$, $\\sqrt{\\var{j}}$&nbsp;and $\\sqrt{\\var{k}}$&nbsp;are surds, as&nbsp;they cannot be simplified to a whole integer. There is no number, $b$, such that $b^2=\\var{h}, \\var{j}$ or $\\var{k}$. Therefore, $\\sqrt{\\var{h}}$, $\\sqrt{\\var{j}}$&nbsp;and $\\sqrt{\\var{k}}$ are&nbsp;both roots and surds.</p>\n<p></p>\n<p></p>\n<p></p>\n<h4>(b)</h4>\n<p>Suppose we want to simplify $\\sqrt x$. If we can find&nbsp;a square number $a^2$ which divides into $x$, we can write $x = a^2b$ where $b$ is an integer.</p>\n<p>Then we can&nbsp;simplify $\\sqrt x=\\sqrt {a^2b}=\\sqrt {a^2} \\times \\sqrt{b}=a&nbsp;\\sqrt {b}$</p>\n<p></p>\n<p><em>i)</em></p>\n<p>$\\sqrt{48}=\\sqrt{16 \\times 3}$&nbsp;= $\\sqrt{16}\\times\\sqrt3$</p>\n<p>$\\sqrt{16}$ simplifies down to $4$ so the final answer is: $4\\sqrt3$.</p>\n<p><em>ii)</em></p>\n<p>$\\sqrt{56}=\\sqrt{4 \\times 14}$ = $\\sqrt{4}\\times\\sqrt{14}$</p>\n<p>$\\sqrt4$ simplifies down to $2$ so the final answer is: $2\\sqrt{14}$.</p>\n<p><em>iii)</em></p>\n<p>$\\sqrt{32}=\\sqrt{16 \\times 2}$&nbsp;=&nbsp;$\\sqrt{16}\\times\\sqrt{2}$</p>\n<p>$\\sqrt{16}$ simplifies down to $4$ so the final answer is: $4\\sqrt2$.</p>\n<p><em>iv)</em></p>\n<p>$\\sqrt{44}=\\sqrt{4 \\times 11}$ = $\\sqrt{4}\\times\\sqrt{11}$</p>\n<p>$\\sqrt4$ simplifies down to $2$ so the final answer is: $2\\sqrt{11}$.</p>\n<p></p>\n<p></p>\n<p></p>\n<h4>(c)</h4>\n<p>This question requires you to notice that $\\var{a}$ and $\\var{d}$&nbsp;are square numbers which are factors of $\\var{c}$ and $\\var{g}$ respectively. We can thus split each square root into a multiplication of two roots, and simplify as in part (b).</p>\n<p></p>\n<p><em>i)</em> $\\sqrt{\\var{c}}=\\sqrt{\\var{a} \\times \\var{b}}$ = $\\sqrt{\\var{a}} \\times \\sqrt{\\var{b}}$ = $\\var{sqrta}\\sqrt{\\var{b}}$</p>\n<p>and</p>\n<p><em>ii)</em> $\\sqrt{\\var{g}}=\\sqrt{\\var{d} \\times \\var{f}}$ = $\\sqrt{\\var{d}}&nbsp;\\times \\sqrt{\\var{f}}$ = $\\var{sqrtd}\\sqrt{\\var{f}}$.</p>\n<h4></h4>\n<h4></h4>", "variables": {"aoptions": {"templateType": "list of numbers", "name": "aoptions", "definition": "[ 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 ]", "group": "Ungrouped variables", "description": "<p>List of random square number between 1 and 36</p>"}, "sqrtd": {"templateType": "anything", "name": "sqrtd", "definition": "sqrt(d)", "group": "Ungrouped variables", "description": "<p>square root of the selected square number d.</p>"}, "h": {"templateType": "anything", "name": "h", "definition": "random(1..10 except 4 except 9 except 1)", "group": "Ungrouped variables", "description": ""}, "c": {"templateType": "anything", "name": "c", "definition": "a*b", "group": "Ungrouped variables", "description": "<p>a times b</p>"}, "square": {"templateType": "anything", "name": "square", "definition": "Random(4,9,16,25,36,49,64,81,100)", "group": "Ungrouped variables", "description": ""}, "cube": {"templateType": "anything", "name": "cube", "definition": "Random(8,27,64,125)", "group": "Ungrouped variables", "description": ""}, "a": {"templateType": "anything", "name": "a", "definition": "random(aoptions)", "group": "Ungrouped variables", "description": "<p>Random squared number</p>"}, "j": {"templateType": "anything", "name": "j", "definition": "random(1..10 except 4 except 9 except h except 1)", "group": "Ungrouped variables", "description": ""}, "b": {"templateType": "anything", "name": "b", "definition": "random(2..12 except 4 except 9)", "group": "Ungrouped variables", "description": "<p>Random number between 1 and 12 except 4 and 9.</p>"}, "root": {"templateType": "anything", "name": "root", "definition": "root(cube,3)", "group": "Ungrouped variables", "description": ""}, "f": {"templateType": "anything", "name": "f", "definition": "random(2..12 except 4 except 9)", "group": "Ungrouped variables", "description": "<p>Random number between 2 and 12</p>"}, "d": {"templateType": "anything", "name": "d", "definition": "random(doptions except a)", "group": "Ungrouped variables", "description": "<p>Random squared number but not the same number as a.</p>"}, "g": {"templateType": "anything", "name": "g", "definition": "d*f", "group": "Ungrouped variables", "description": "<p>d times f</p>"}, "doptions": {"templateType": "list of numbers", "name": "doptions", "definition": "[ 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 ]", "group": "Ungrouped variables", "description": "<p>List of squared numbers from 1 to 144</p>"}, "sqrta": {"templateType": "anything", "name": "sqrta", "definition": "sqrt(a)", "group": "Ungrouped variables", "description": "<p>square root of the squared numbers</p>"}, "k": {"templateType": "anything", "name": "k", "definition": "random(10..15 except h except j)", "group": "Ungrouped variables", "description": ""}}, "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}, {"name": "Simon Thomas", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/3148/"}]}