// Numbas version: exam_results_page_options {"name": "Using Surds, Rationalising the Denominator", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "preventleave": false, "showfrontpage": false}, "question_groups": [{"questions": [{"variablesTest": {"condition": "n^2 > a", "maxRuns": 100}, "parts": [{"variableReplacementStrategy": "originalfirst", "useCustomName": false, "variableReplacements": [], "showCellAnswerState": true, "choices": ["

Can be simplified further

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Cannot be simplified further

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$\\sqrt{\\var{p}}$

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$\\sqrt{\\simplify{{a}*{n}^2}}$

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$\\sqrt{\\var{a}}$

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Which of the following can be simplified further?

", "shuffleAnswers": true, "minMarks": 0, "maxMarks": "3", "displayType": "checkbox", "showCorrectAnswer": true, "showFeedbackIcon": true, "minAnswers": "3", "scripts": {}, "customMarkingAlgorithm": "", "layout": {"type": "all", "expression": ""}, "unitTests": []}, {"customName": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"showPreview": true, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "marks": "2", "failureRate": 1, "type": "jme", "answer": "{n}", "vsetRange": [0, 1], "valuegenerators": [], "customName": "", "vsetRangePoints": 5, "checkVariableNames": false, "answerSimplification": "all", "mustmatchpattern": {"nameToCompare": "", "pattern": "$n", "message": "You haven't fully simplified.", "partialCredit": 0}, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "checkingType": "absdiff", "unitTests": []}], "variableReplacements": [], "stepsPenalty": "1", "showCorrectAnswer": true, "prompt": " Simplify$\\sqrt{\\simplify{{n}^2*{p}}}$. \n$\\sqrt{\\simplify{{n}^2*{p}}} =$[[0]]$\\sqrt{\\var{p}}$. ", "steps": [{"customName": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "variableReplacements": [], "showCorrectAnswer": true, "prompt": " Recall the first rule of surds \n$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}$. \n \n ", "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "information", "customMarkingAlgorithm": "", "unitTests": []}], "useCustomName": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "sortAnswers": false, "customMarkingAlgorithm": "", "unitTests": []}, {"customName": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"showPreview": true, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "musthave": {"message": " You must simplify your answer further. ", "strings": ["sqrt", "(", ")"], "partialCredit": 0, "showStrings": true}, "marks": "2", "failureRate": 1, "type": "jme", "answer": "sqrt({v})", "vsetRange": [0, 1], "valuegenerators": [], "customName": "", "vsetRangePoints": 5, "checkVariableNames": false, "notallowed": {"message": " You must simplify your answer further. ", "strings": ["/"], "partialCredit": 0, "showStrings": false}, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "checkingType": "absdiff", "unitTests": []}], "variableReplacements": [], "stepsPenalty": "1", "showCorrectAnswer": true, "prompt": " Simplify$\\displaystyle\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}}$. \n$\\displaystyle\\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} =$[[0]]. \n ", "steps": [{"customName": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "variableReplacements": [], "showCorrectAnswer": true, "prompt": " You could use either of the following rules: \n$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}$. \n$\\displaystyle\\sqrt{\\frac{a}{b}} = \\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}$. ", "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "information", "customMarkingAlgorithm": "", "unitTests": []}], "useCustomName": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "sortAnswers": false, "customMarkingAlgorithm": "", "unitTests": []}, {"customName": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"showPreview": true, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "marks": "2", "failureRate": 1, "type": "jme", "answer": "{b}", "vsetRange": [0, 1], "valuegenerators": [], "customName": "", "vsetRangePoints": 5, "checkVariableNames": false, "answerSimplification": "all", "showCorrectAnswer": true, "checkingAccuracy": 0.001, "variableReplacementStrategy": "originalfirst", "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "unitTests": []}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": " Simplify$\\displaystyle\\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}}$. \n$\\displaystyle\\frac{\\sqrt{\\simplify{({b}*{m})^2*{s}}}}{\\var{m}} =$[[0]]$\\sqrt{\\var{s}}$. \n ", "useCustomName": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "sortAnswers": false, "customMarkingAlgorithm": "", "unitTests": []}, {"customName": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"customName": "", "useCustomName": false, "showPreview": true, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "marks": "2", "failureRate": 1, "type": "jme", "answer": "{(d-b*v+n*b)}sqrt({a})", "vsetRange": [0, 1], "valuegenerators": [], "variableReplacementStrategy": "originalfirst", "vsetRangePoints": 5, "checkVariableNames": false, "mustmatchpattern": {"nameToCompare": "", "pattern": "$n*sqrt($n)", "message": "You haven't fully simplified.", "partialCredit": 0}, "showCorrectAnswer": true, "checkingAccuracy": 0.001, "scripts": {}, "showFeedbackIcon": true, "customMarkingAlgorithm": "", "checkingType": "absdiff", "unitTests": []}], "variableReplacements": [], "showCorrectAnswer": true, "prompt": " Simplify$\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2*{a})+{n}sqrt({b}^2*{a})}$. \n$\\simplify{{d}sqrt({a}) - {b}sqrt({v}^2*{a})+{n}sqrt({b}^2*{a})} =$[[0]]. \n ", "useCustomName": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "sortAnswers": false, "customMarkingAlgorithm": "", "unitTests": []}, {"customName": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"showPreview": true, "useCustomName": false, "variableReplacements": [], "extendBaseMarkingAlgorithm": true, "musthave": {"message": "", "strings": ["sqrt", "(", ")"], "partialCredit": 0, "showStrings": false}, "marks": "1", "failureRate": 1, "type": "jme", "answer": "sqrt({a})", "vsetRange": [0, 1], "valuegenerators": [], "customName": "", "vsetRangePoints": 5, "checkVariableNames": false, "notallowed": {"message": "", "strings": ["/"], "partialCredit": 0, "showStrings": false}, "showCorrectAnswer": true, "showFeedbackIcon": true, "variableReplacementStrategy": "originalfirst", "scripts": {}, "checkingAccuracy": 0.001, "customMarkingAlgorithm": "", "checkingType": "absdiff", "unitTests": []}, {"customName": "", "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "showCorrectAnswer": true, "variableReplacements": [], "showPreview": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 1, "showFeedbackIcon": true, "failureRate": 1, "answer": "{a}", "variableReplacementStrategy": "originalfirst", "type": "jme", "checkingAccuracy": 0.001, "vsetRange": [0, 1], "unitTests": [], "customMarkingAlgorithm": "", "checkingType": "absdiff", "valuegenerators": []}], "variableReplacements": [], "stepsPenalty": "1", "showCorrectAnswer": true, "prompt": " Rationalise the denominator of the fraction$\\displaystyle\\frac{1}{\\sqrt{\\var{a}}}$. \n$\\displaystyle\\frac{1}{\\sqrt{\\var{a}}} =$[[0]] [[1]] . ", "steps": [{"customName": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "variableReplacements": [], "showCorrectAnswer": true, "prompt": " To rationalise the denominator of fractions in the form$\\frac{1}{\\sqrt{a}}$, multiply the top and bottom by$\\sqrt{a}$. ", "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "information", "customMarkingAlgorithm": "", "unitTests": []}], "useCustomName": false, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 0, "showFeedbackIcon": true, "type": "gapfill", "sortAnswers": false, "customMarkingAlgorithm": "", "unitTests": []}, {"customName": "", "variableReplacementStrategy": "originalfirst", "gaps": [{"customName": "", "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "showCorrectAnswer": true, "variableReplacements": [], "showPreview": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": "1", "showFeedbackIcon": true, "failureRate": 1, "answer": "{n}-sqrt({a})", "variableReplacementStrategy": "originalfirst", "type": "jme", "checkingAccuracy": 0.001, "vsetRange": [0, 1], "unitTests": [], "customMarkingAlgorithm": "", "checkingType": "absdiff", "valuegenerators": []}, {"customName": "", "vsetRangePoints": 5, "useCustomName": false, "checkVariableNames": false, "showCorrectAnswer": true, "variableReplacements": [], "showPreview": true, "scripts": {}, "extendBaseMarkingAlgorithm": true, "marks": 1, "showFeedbackIcon": true, "failureRate": 1, "answer": "{n^2-a}", "variableReplacementStrategy": "originalfirst", "type": "jme", "checkingAccuracy": 0.001, "vsetRange": [0, 1], "unitTests": [], "customMarkingAlgorithm": "", "checkingType": "absdiff", "valuegenerators": []}], "variableReplacements": [], "stepsPenalty": "1", "showCorrectAnswer": true, "prompt": " Rationalise the denominator of the fraction$\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}}$. \n$\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}} =$[[0]] [[1]] . ", "steps": [{"customName": "", "variableReplacementStrategy": "originalfirst", "useCustomName": false, "variableReplacements": [], "showCorrectAnswer": true, "prompt": " To rationalise the denominator of fractions in the form$\\displaystyle\\frac{1}{a+\\sqrt{b}}$, multiply the top and bottom by$a-\\sqrt{b}$. 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"}, "s": {"group": "Ungrouped variables", "templateType": "anything", "name": "s", "definition": "random(2..7 except 4 except 6)", "description": " Short list of primes for part d. "}, "m": {"group": "Ungrouped variables", "templateType": "anything", "name": "m", "definition": "random(2..5 #1 except n^2)", "description": " parts b and d "}, "n": {"group": "Ungrouped variables", "templateType": "anything", "name": "n", "definition": "random(3..10 #1)", "description": " all numbers from 3-10 for parts a, b, e, g "}, "v": {"group": "Ungrouped variables", "templateType": "anything", "name": "v", "definition": "random(2,3,5)", "description": " Parts c and e "}, "t": {"group": "Ungrouped variables", "templateType": "anything", "name": "t", "definition": "random(2,3)", "description": ""}}, "extensions": [], "variable_groups": [], "statement": " To include a square root sign in your answer use sqrt(). For example, to write$\\sqrt{3}$, type sqrt(3) into the answer box. If you are entering a number multiplied by the square root of some other number, for example$3\\sqrt{5}$, type 3*sqrt(5) into the answer box. ", "ungrouped_variables": ["p", "d", "n", "m", "b", "a", "v", "t", "c", "s"], "tags": ["fractions", "Fractions", "rationalise the denominator", "Surds", "surds", "taxonomy"], "contributors": [{"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/7/", "name": "Christian Lawson-Perfect"}, {"profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/1591/", "name": "Elliott Fletcher"}], "advice": " #### a) \n Surds can be manipulated using the rule \n \$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b}.\$ \n We are asked to state which of$\\sqrt{\\var{p}}$,$\\sqrt{\\simplify{{a}*{n}^2}}$, and$\\sqrt{\\var{a}}$can be simplified further. Commonly, surds can be simplified if the number inside of the square root has a square number as a factor. \n Here,$\\var{p}$is a prime number which means that its only divisors are$\\var{p}$and$1$. \n Therefore,$\\sqrt{\\var{p}}$cannot be simplified any further. \n Similarly,$\\var{a}$is also a prime number, so$\\sqrt{\\var{a}}$also cannot be simplified any further. \n On the other hand,$\\simplify{{a}*{n}^2}$is not a prime number and we can use the previous rule to simplify$\\sqrt{\\simplify{{a}*{n}^2}}as \n \\\begin{align} \\sqrt{\\simplify{{a}*{n}^2}} &= \\sqrt{\\simplify{{n}^2}} \\times \\sqrt{\\var{a}}\\\\ &= \\simplify{{n}*sqrt({a})}. \\end{align} \ \n #### b) \n Using the same rule of manipulation as in part a), we can simplify\\sqrt{\\simplify{{n}^2*{p}}}as \n \\\begin{align} \\sqrt{\\simplify{{n}^2*{p}}} &= \\sqrt{\\simplify{{n}^2}} \\times \\sqrt{\\var{p}}\\\\ &= \\simplify{{n}*sqrt({p})}. \\end{align} \ \n #### c) \n Here, we can use both of the rules for manipulating surds: \n \$\\sqrt{(ab)} = \\sqrt{a} \\times \\sqrt{b} \\text{.} \$ \n \$\\sqrt{\\frac{a}{b}} = \\frac{\\sqrt{a}}{\\sqrt{b}} \\text{.} \$ \n We can simplify\\displaystyle\\frac{ \\sqrt{\\simplify{{a}*{v}}} }{ \\sqrt{\\var{a}} }as follows. \n \\\begin{align} \\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\frac{\\sqrt{\\var{a}} \\times \\sqrt{\\var{v}}}{\\sqrt{\\var{a}}} \\\\[0.5em] &= \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\times \\sqrt{\\var{v}} \\\\[0.5em] &= \\simplify{{sqrt(a)/sqrt(a)}} \\times \\sqrt{\\var{v}} \\\\[0.5em] &= \\sqrt{\\var{v}} \\text{.} \\end{align} \ \n Or, \n \\\begin{align} \\frac{\\sqrt{\\simplify{{a}*{v}}}}{\\sqrt{\\var{a}}} &= \\sqrt{\\frac{\\simplify{{a}*{v}}}{\\var{a}}} \\\\[0.5em] &= \\sqrt{\\var{v}} \\text{.} \\end{align} \ \n #### d) \n We can simplify the fraction as \n \\\begin{align} \\frac{\\sqrt{\\simplify{({b}{m})^2*{s}}}}{\\var{m}} &= \\frac{\\sqrt{\\simplify{({b*m})^2}} \\times \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em] &= \\frac{\\simplify{{b*m}} \\times \\sqrt{\\var{s}}}{\\var{m}} \\\\[0.5em] &= \\simplify{{b}*sqrt({s})} \\text{.} \\end{align} \ \n #### e) \n \\\begin{align} \\simplify{{d}sqrt({a}) - {b}sqrt({v}^2{a})+{n}sqrt({b}^2*{a})} &= \\var{d}\\sqrt{\\var{a}} - \\var{b}(\\sqrt{\\simplify{{v}^2}} \\times \\sqrt{\\var{a}})+\\var{n}(\\sqrt{\\simplify{{b}^2}} \\times \\sqrt{\\var{a}}) \\\\ &= \\var{d}\\sqrt{\\var{a}} -\\var{b}(\\simplify{{v}*sqrt({a})})+\\var{n}(\\simplify{{b}*sqrt({a})}) \\\\ &= \\simplify{{d}sqrt({a})}-\\simplify{{b}*{v}sqrt({a})}+\\simplify{{n}*{b}sqrt({a})} \\\\ &= \\simplify{({d}-{b}*{v}+{n}*{b})sqrt({a})} \\text{.} \\end{align} \ \n #### f) \n We rationalise the denominator of fractions of the form\\displaystyle\\frac{1}{\\sqrt{a}}$, by multiplying the top and bottom by$\\sqrt{a}$. \n Therefore, to rationalise the denominator of the fraction$\\displaystyle\\frac{1}{\\sqrt{\\var{a}}}$, we multiply top and bottom by$\\sqrt{\\var{a}}. \n \\\begin{align} \\frac{1}{\\sqrt{\\var{a}}} &= \\frac{1}{\\sqrt{\\var{a}}} \\times \\frac{\\sqrt{\\var{a}}}{\\sqrt{\\var{a}}} \\\\[0.5em] &= \\frac{\\sqrt{\\var{a}}}{\\var{a}} \\text{.} \\end{align} \ \n #### g) \n We rationalise the denominator of fractions of the form\\displaystyle\\frac{1}{a+\\sqrt{b}}$by multiplying the top and bottom by$a-\\sqrt{b}$. \n Therefore, to rationalise the denominator of the fraction$\\displaystyle\\frac{1}{\\var{n}+\\sqrt{\\var{a}}}$, we multiply the top and bottom by$\\var{n} - \\sqrt{\\var{a}}. \n \\\begin{align} \\frac{1}{\\var{n}+\\sqrt{\\var{a}}} &= \\frac{1}{\\var{n}+\\sqrt{\\var{a}}} \\times \\frac{\\var{n}-\\sqrt{\\var{a}}}{\\var{n}-\\sqrt{\\var{a}}} \\\\[0.5em] &=\\frac{\\var{n}-\\sqrt{\\var{a}}}{(\\var{n}+\\sqrt{\\var{a}})(\\var{n}-\\sqrt{\\var{a}})} \\\\[0.5em] &=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2}-\\var{a}} \\\\[0.5em] &=\\frac{\\var{n}-\\sqrt{\\var{a}}}{\\simplify{{n}^2-{a}}} \\text{.} \\end{align} \ \n #### h) \n We rationalise the denominator of fractions of the form\\displaystyle\\frac{1}{a-\\sqrt{b}}$by multiplying the top and bottom by$a+\\sqrt{b}$. \n Therefore, to rationalise the denominator of the fraction$\\displaystyle\\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}}$, we multiply the top and bottom by$\\var{d+p}+\\sqrt{\\var{p}}\$.

\n

\\\begin{align} \\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} &= \\frac{\\var{t}}{\\var{d+p}-\\sqrt{\\var{p}}} \\times \\frac{\\var{d+p}+\\sqrt{\\var{p}}}{\\var{d+p}+\\sqrt{\\var{p}}} \\\\[0.5em] &=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{(\\var{d+p}-\\sqrt{\\var{p}})(\\var{d+p}+\\sqrt{\\var{p}})} \\\\[0.5em] &=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2}-\\var{p}} \\\\[0.5em] &=\\frac{\\var{t}(\\var{d+p}+\\sqrt{\\var{p}})}{\\simplify{{d+p}^2-{p}}} \\\\[0.5em] &=\\simplify{{t}/{(d+p)^2-p}}(\\var{d+p}+\\sqrt{\\var{p}}) \\\\[0.5em] &= \\simplify[all,!noleadingMinus]{({t*(d+p)}+{t}*sqrt({p}))/({(d+p)^2-p})} \\text{.} \\end{align} \

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Manipulate surds and rationalise the denominator of a fraction when it is a surd.

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