Surds are square roots that cannot be simplified to a whole number. They have a decimal equivalent but their decimal representations are never-ending. Therefore, it is often easier to leave surds as they are in algebraic calculations.
", "functions": {}, "rulesets": {}, "variables": {"sqrta": {"description": "square root of the squared numbers
", "templateType": "anything", "name": "sqrta", "group": "Ungrouped variables", "definition": "sqrt(a)"}, "h": {"description": "", "templateType": "anything", "name": "h", "group": "Ungrouped variables", "definition": "random(1..10 except 4 except 9 except 1)"}, "doptions": {"description": "List of squared numbers from 1 to 144
", "templateType": "list of numbers", "name": "doptions", "group": "Ungrouped variables", "definition": "[ 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 ]"}, "g": {"description": "d times f
", "templateType": "anything", "name": "g", "group": "Ungrouped variables", "definition": "d*f"}, "root": {"description": "", "templateType": "anything", "name": "root", "group": "Ungrouped variables", "definition": "root(cube,3)"}, "k": {"description": "", "templateType": "anything", "name": "k", "group": "Ungrouped variables", "definition": "random(10..15 except h except j)"}, "j": {"description": "", "templateType": "anything", "name": "j", "group": "Ungrouped variables", "definition": "random(1..10 except 4 except 9 except h except 1)"}, "d": {"description": "Random squared number but not the same number as a.
", "templateType": "anything", "name": "d", "group": "Ungrouped variables", "definition": "random(doptions except a)"}, "square": {"description": "", "templateType": "anything", "name": "square", "group": "Ungrouped variables", "definition": "Random(4,9,16,25,36,49,64,81,100)"}, "b": {"description": "Random number between 1 and 12 except 4 and 9.
", "templateType": "anything", "name": "b", "group": "Ungrouped variables", "definition": "random(2..11 except 4 except 8 except 9)"}, "c": {"description": "a times b
", "templateType": "anything", "name": "c", "group": "Ungrouped variables", "definition": "a*b"}, "cube": {"description": "", "templateType": "anything", "name": "cube", "group": "Ungrouped variables", "definition": "Random(8,27,64,125)"}, "aoptions": {"description": "List of random square number between 1 and 36
", "templateType": "list of numbers", "name": "aoptions", "group": "Ungrouped variables", "definition": "[ 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144 ]"}, "sqrtd": {"description": "square root of the selected square number d.
", "templateType": "anything", "name": "sqrtd", "group": "Ungrouped variables", "definition": "sqrt(d)"}, "f": {"description": "Random number between 2 and 12
", "templateType": "anything", "name": "f", "group": "Ungrouped variables", "definition": "random(2..11 except 4 except 8 except 9)"}, "a": {"description": "Random squared number
", "templateType": "anything", "name": "a", "group": "Ungrouped variables", "definition": "random(aoptions)"}}, "variable_groups": [], "ungrouped_variables": ["aoptions", "b", "a", "c", "sqrta", "doptions", "sqrtd", "d", "f", "g", "square", "cube", "h", "j", "k", "root"], "preamble": {"js": "", "css": ""}, "advice": "$\\sqrt{\\var{square}}$ and $\\sqrt[3]{\\var{cube}}$ are not surds, as they can be simplified to whole integers: $\\simplify{{sqrt(square)}}$ and $\\var{root}$ respectively. They are roots, but not surds. All surds are roots but not all roots are surds.
\n$\\sqrt{\\var{h}}$, $\\sqrt{\\var{j}}$ and $\\sqrt{\\var{k}}$ are surds, as 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, $\\sqrt{\\var{h}}$, $\\sqrt{\\var{j}}$ and $\\sqrt{\\var{k}}$ are both roots and surds.
\n\n\nThe rule that should be used is $\\sqrt{a}\\times\\sqrt{b}=\\sqrt{ab}$.
\nWe need to try to find a square number that divides $ab$ and rewrite this as $\\sqrt{b^2}\\times\\sqrt{a}$.
\ni)
\n$\\sqrt{48}$ = $\\sqrt{16}\\times\\sqrt3$
\n$\\sqrt{16}$ simplifies down to $4$ so the final answer is: $4\\sqrt3$.
\nii)
\n$\\sqrt{56}$ = $\\sqrt{4}\\times\\sqrt{14}$
\n$\\sqrt4$ simplifies down to $2$ so the final answer is: $2\\sqrt{14}$.
\niii)
\n$\\sqrt{32}$ = $\\sqrt{16}\\times\\sqrt{2}$
\n$\\sqrt{16}$ simplifies down to $4$ so the final answer is: $4\\sqrt2$.
\niv)
\n$\\sqrt{44}$ = $\\sqrt{4}\\times\\sqrt{11}$
\n$\\sqrt4$ simplifies down to $2$ so the final answer is: $2\\sqrt{11}$.
\n\nThis question requires you to notice that $\\sqrt{\\var{a}}$ and $\\sqrt{\\var{d}}$ are squared numbers and can be simplified to integers.
\n$\\sqrt{\\var{a}}$ = $\\var{sqrta}$ such that:
\ni) $\\sqrt{\\var{c}}$ = $\\sqrt{\\var{a}}$ x $\\sqrt{\\var{b}}$ = $\\var{sqrta}\\sqrt{\\var{b}}$ and
\nii) $\\sqrt{\\var{g}}$ = $\\sqrt{\\var{d}}$ x $\\sqrt{\\var{f}}$ = $\\var{sqrtd}\\sqrt{\\var{f}}$.
\n\n", "parts": [{"matrix": [[0, "1"], ["1", 0], [0, "1"], ["1", "0"], ["1", 0]], "shuffleAnswers": false, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "choices": ["$\\sqrt{\\var{square}}$
", "$\\sqrt{\\var{h}}$
", "$^3\\sqrt{\\var{cube}}$
", "$\\sqrt{\\var{j}}$
", "$\\sqrt{\\var{k}}$
"], "marks": 0, "type": "m_n_x", "variableReplacements": [], "scripts": {}, "minMarks": 0, "minAnswers": 0, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "layout": {"type": "all", "expression": ""}, "displayType": "radiogroup", "showCorrectAnswer": true, "shuffleChoices": false, "prompt": "For the following examples, tick the correct box to determine whether or not they are a surd.
\n", "maxAnswers": 0, "maxMarks": 0, "warningType": "none", "answers": ["Surd
", "Not a surd
"], "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "prompt": "Match each surd with the equivalent simplification.
\n[[0]]
", "marks": 0, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "gaps": [{"matrix": [["1", 0, 0, "0"], [0, "0", "0", "1"], ["0", "0", "1", 0], [0, "1", 0, 0]], "shuffleAnswers": true, "variableReplacementStrategy": "originalfirst", "customMarkingAlgorithm": "", "choices": ["i) $\\sqrt{48}$
", "ii) $\\sqrt{32}$
", "iii) $\\sqrt{56}$
", "iv) $\\sqrt{44}$
"], "marks": 0, "type": "m_n_x", "variableReplacements": [], "scripts": {}, "minMarks": 0, "minAnswers": 0, "showFeedbackIcon": true, "extendBaseMarkingAlgorithm": true, "layout": {"type": "all", "expression": ""}, "displayType": "radiogroup", "showCorrectAnswer": true, "shuffleChoices": true, "maxAnswers": 0, "maxMarks": 0, "warningType": "none", "answers": ["$4\\sqrt3$
", "$2\\sqrt{11}$
", "$2\\sqrt{14}$
", "$4\\sqrt{2}$
"], "unitTests": []}], "showFeedbackIcon": true, "unitTests": []}, {"extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "sortAnswers": false, "customMarkingAlgorithm": "", "prompt": "Simplify the following surds:
\n$\\displaystyle\\sqrt{\\var{c}}$ = [[0]]$\\displaystyle\\sqrt{\\var{b}}$
\n$\\displaystyle\\sqrt{\\var{g}}$ = [[1]]$\\displaystyle\\sqrt{\\var{f}}$
\n", "marks": 0, "type": "gapfill", "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "gaps": [{"notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "allowFractions": false, "customMarkingAlgorithm": "", "marks": 1, "type": "numberentry", "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "correctAnswerFraction": false, "minValue": "sqrta", "maxValue": "sqrta", "mustBeReducedPC": 0, "showFeedbackIcon": true, "unitTests": []}, {"notationStyles": ["plain", "en", "si-en"], "extendBaseMarkingAlgorithm": true, "variableReplacementStrategy": "originalfirst", "correctAnswerStyle": "plain", "allowFractions": false, "customMarkingAlgorithm": "", "marks": 1, "type": "numberentry", "mustBeReduced": false, "variableReplacements": [], "showCorrectAnswer": true, "scripts": {}, "correctAnswerFraction": false, "minValue": "sqrtd", "maxValue": "sqrtd", "mustBeReducedPC": 0, "showFeedbackIcon": true, "unitTests": []}], "showFeedbackIcon": true, "unitTests": []}], "metadata": {"description": "This question tests the student's understanding of what is and is not a surd, and on their simplification of surds.
", "licence": "Creative Commons Attribution 4.0 International"}, "tags": [], "type": "question", "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}]}]}], "contributors": [{"name": "Christian Lawson-Perfect", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/"}, {"name": "Harry Flynn", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/976/"}, {"name": "Lauren Richards", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1589/"}]}