simplifying, multiplication, fraction of square roots resulting in rational, square root of fraction resulting in rational, rationalising the denominator, simple rationalising the denominator conjugate
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\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
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.
\nA key fact we use here is that $\\sqrt{a\\times b}=\\sqrt{a}\\times\\sqrt{b}$ for non-negative real numbers $a,b$.
\n| $\\sqrt{\\var{easyargument}}$ | \n$=$ | \n$\\sqrt{\\var{easysquareinfo[1]}\\times\\var{easyprime}}$ | \n(you find that $\\var{easysquareinfo[1]}$ is a square factor of $\\var{easyargument}$) | \n
| \n | $=$ | \n$\\sqrt{\\var{easysquareinfo[1]}}\\times\\sqrt{\\var{easyprime}}$ | \n\n |
| \n | $=$ | \n$\\var{easysquareinfo[0]}\\times\\sqrt{\\var{easyprime}}$ | \n\n |
| \n | $=$ | \n$\\var{easysquareinfo[0]}\\sqrt{\\var{easyprime}}$ | \n\n |
Simplify the surd $\\sqrt{\\var{harderargument}}$, entering your numbers in the gaps provided.
\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
This question is similar to the last one but has larger numbers so it is harder to spot the square factors. Because of this, a calculator, factor trees, divisibility tests and/or long division may be useful.
\nThe idea is to find all the factors of $\\var{harderargument}$ which are squares, e.g. 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, ... 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.
\n\n| $\\sqrt{\\var{harderargument}}$ | \n$=$ | \n$\\sqrt{\\var{hardersquareinfo[0][1]}\\times\\var{stillunder}}$ | \n(you might find that $\\var{hardersquareinfo[0][1]}$ is a square factor of $\\var{harderargument}$) | \n
| \n | $=$ | \n$\\sqrt{\\var{hardersquareinfo[0][1]}}\\times\\sqrt{\\var{stillunder}}$ | \n\n |
| \n | $=$ | \n$\\var{hardersquareinfo[0][0]}\\sqrt{\\var{stillunder}}$ | \n\n |
| \n | $=$ | \n$\\var{hardersquareinfo[0][0]}\\times\\sqrt{\\var{hardersquareinfo[1][1]}\\times\\var{leftovers}}$ | \n(you might find that $\\var{hardersquareinfo[1][1]}$ is a square factor of $\\var{stillunder}$) | \n
| \n | $=$ | \n$\\var{hardersquareinfo[0][0]}\\times\\sqrt{\\var{hardersquareinfo[1][1]}}\\times\\sqrt{\\var{leftovers}}$ | \n\n |
| \n | $=$ | \n$\\var{hardersquareinfo[0][0]}\\times\\var{hardersquareinfo[1][0]}\\times\\sqrt{\\var{leftovers}}$ | \n\n |
| \n | $=$ | \n$\\var{hardermult}\\sqrt{\\var{leftovers}}$ | \n(there are no square factors of $\\var{leftovers}$ so we are finished) | \n
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using background image in table to get a good looking square root symbol
\n\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
Simplify the surd product $\\var{square1[0]}\\sqrt{\\var{prime1}}\\times\\var{extra}\\sqrt{\\var{arg2}}$, entering your numbers in the gaps provided.
\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
Given $\\var{square1[0]}\\sqrt{\\var{prime1}}\\times\\var{extra}\\sqrt{\\var{arg2}}$ we could multiply the integer parts together and the surd parts together and then simplify, that is:
\n| $\\var{square1[0]}\\sqrt{\\var{prime1}}\\times\\var{extra}\\sqrt{\\var{arg2}}$ | \n$=$ | \n$\\var{square1[0]}\\times\\var{extra}\\times\\sqrt{\\var{prime1}\\times\\var{arg2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]*extra}\\times\\sqrt{\\var{prime1*arg2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]*extra}\\times\\sqrt{\\var{square2[1]}\\times\\var{prime1*prime2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]*extra}\\times\\sqrt{\\var{square2[1]}}\\times\\sqrt{\\var{prime1*prime2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]*extra}\\times\\var{square2[0]}\\times\\sqrt{\\var{prime1*prime2}}$ | \n
| \n | $=$ | \n$\\var{ansmult}\\sqrt{\\var{ansarg}}$ | \n
Or we could simplify the surds first, then multiply the integer parts and surd parts, and then simplify (if necessary) at the end, that is:
\n| $\\var{square1[0]}\\sqrt{\\var{prime1}}\\times\\var{extra}\\sqrt{\\var{arg2}}$ | \n$=$ | \n$\\var{square1[0]}\\times\\sqrt{\\var{prime1}}\\times\\var{extra}\\times\\sqrt{\\var{square2[1]}\\times\\var{prime2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]}\\times\\sqrt{\\var{prime1}}\\times\\var{extra}\\times\\sqrt{\\var{square2[1]}}\\times\\sqrt{\\var{prime2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]}\\times\\sqrt{\\var{prime1}}\\times\\var{extra}\\times\\var{square2[0]}\\times\\sqrt{\\var{prime2}}$ | \n
| \n | $=$ | \n$\\var{square1[0]}\\times\\var{extra}\\times\\var{square2[0]}\\times\\sqrt{\\var{prime1}\\times\\var{prime2}}$ | \n
| \n | $=$ | \n$\\var{ansmult}\\sqrt{\\var{ansarg}}$ | \n
The second approach is recommended since it keeps the numbers under the square root as small as possible, but it is important to realise both approaches are valid.
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\n\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
The following surd expression \\[\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}\\] simplifies to [[0]].
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\n| $\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$ | \n$=$ | \n$\\displaystyle\\frac{\\sqrt{\\var{square1[1]}\\times\\var{fcf}}}{\\sqrt{\\var{square2[1]}\\times\\var{fcf}}}$ | \n(find square factors) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\var{square1[0]}\\sqrt{\\var{fcf}}}{\\var{square2[0]}\\sqrt{\\var{fcf}}}$ | \n(simplify surds) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\var{square1[0]}}{\\var{square2[0]}}$ | \n(simplify fraction) | \n
| \n | $=$ | \n$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$ | \n\n |
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:
\n| $\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$ | \n$=$ | \n$\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$ | \n(use surd identity) | \n
| \n | $=$ | \n$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}\\times\\var{gcff}}{\\var{den_wo_cf}\\times\\var{gcff}}}$ | \n(find greatest common factor) | \n
| \n | $=$ | \n$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}}{\\var{den_wo_cf}}}$ | \n(cancel common factors) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\sqrt{\\var{num_wo_cf}}}{\\sqrt{\\var{den_wo_cf}}}$ | \n(use surd identity) | \n
| \n | $=$ | \n$\\displaystyle\\sqrt{\\var{num_wo_cf}}$ | \n\n |
| \n | $=$ | \n$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$ | \n\n |
Realising $\\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{\\frac{a}{b}}$ can often be useful.
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"}, "prime1": {"definition": "listprime[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime1", "description": ""}, "square1": {"definition": "listsquare[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "square1", "description": ""}, "square2": {"definition": "listsquare[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "square2", "description": ""}, "prime2": {"definition": "listprime[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime2", "description": ""}, "num_wo_cf": {"definition": "questnum/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "num_wo_cf", "description": ""}, "gcff": {"definition": "gcd(questnum,questden)", "templateType": "anything", "group": "Ungrouped variables", "name": "gcff", "description": ""}, "listsquares": {"definition": "map([n,n^2],n,1..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquares", "description": ""}}, "metadata": {"notes": "using background image in table to get a good looking square root symbol
\n\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
The following surd expression \\[\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}\\] simplifies to [[0]].
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\n| $\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$ | \n$=$ | \n$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}\\times\\var{gcff}}{\\var{den_wo_cf}\\times\\var{gcff}}}$ | \n(find greatest common factor) | \n
| \n | $=$ | \n$\\displaystyle\\sqrt{\\frac{\\var{num_wo_cf}}{\\var{den_wo_cf}}}$ | \n(cancel common factors) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\sqrt{\\var{num_wo_cf}}}{\\sqrt{\\var{den_wo_cf}}}$ | \n(use surd identity) | \n
| \n | $=$ | \n$\\displaystyle\\sqrt{\\var{num_wo_cf}}$ | \n(simplify) | \n
| \n | $=$ | \n$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$ | \n(simplify) | \n
Alternatively, we could use \\[\\sqrt{\\frac{a}{b}}=\\frac{\\sqrt{a}}{\\sqrt{b}},\\] straight away, try to simply the surds and then simplify any remaining fraction:
\n| $\\displaystyle\\sqrt{\\frac{\\var{questnum}}{\\var{questden}}}$ | \n$=$ | \n$\\displaystyle\\frac{\\sqrt{\\var{questnum}}}{\\sqrt{\\var{questden}}}$ | \n(use surd identity) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\sqrt{\\var{square1[1]}\\times\\var{fcf}}}{\\sqrt{\\var{square2[1]}\\times\\var{fcf}}}$ | \n(find square factors) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\var{square1[0]}\\sqrt{\\var{fcf}}}{\\var{square2[0]}\\sqrt{\\var{fcf}}}$ | \n(simplify surds) | \n
| \n | $=$ | \n$\\displaystyle\\frac{\\var{square1[0]}}{\\var{square2[0]}}$ | \n(simplify fraction) | \n
| \n | $=$ | \n$\\displaystyle\\simplify{{square1[0]}/{square2[0]}}$ | \n\n |
Realising $\\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}}=\\sqrt{\\frac{a}{b}}$ can often be useful.
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"}, "prime1": {"definition": "listprime[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime1", "description": ""}, "square1": {"definition": "listsquare[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "square1", "description": ""}, "square2": {"definition": "listsquare[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "square2", "description": ""}, "prime2": {"definition": "listprime[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime2", "description": ""}, "num_wo_cf": {"definition": "questnum/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "num_wo_cf", "description": ""}, "gcff": {"definition": "gcd(questnum,questden)", "templateType": "anything", "group": "Ungrouped variables", "name": "gcff", "description": ""}, "listsquares": {"definition": "map([n,n^2],n,1..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquares", "description": ""}}, "metadata": {"notes": "using background image in table to get a good looking square root symbol
\n\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
Given the fraction \\[\\dfrac{\\var{questnum}}{\\var{questdenmult}\\sqrt{\\var{questdensurd}}}\\] we can rationalise the denominator and rewrite the fraction in the simplified equivalent form
\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
| [[2]] | \n
In the above case, to rationalise the denominator we can multiply the top and bottom of the fraction by the surd part. This will rationalise the denominator since $\\sqrt{a}\\times\\sqrt{a}=a$.
\n| $\\dfrac{\\var{questnum}}{\\var{questdenmult}\\sqrt{\\var{questdensurd}}}$ | \n$=$ | \n$\\dfrac{\\var{questnum}}{\\var{questdenmult}\\sqrt{\\var{questdensurd}}}\\times\\dfrac{\\sqrt{\\var{questdensurd}}}{\\sqrt{\\var{questdensurd}}}$ | \n(multiplying top and bottom by the surd part of the denominator) | \n
| \n | $=$ | \n$\\dfrac{\\var{questnum}\\sqrt{\\var{questdensurd}}}{\\var{questdenmult}\\times\\var{questdensurd}}$ | \n\n |
| \n | $=$ | \n$\\dfrac{\\var{questnum}\\sqrt{\\var{questdensurd}}}{\\var{tempden}}$ | \n\n |
| \n | $=$ | \n$\\dfrac{\\var{ansnummult}\\sqrt{\\var{questdensurd}}}{\\var{ansden}}$ | \n(cancelling the common factor of $\\var{cancel}$) | \n
| \n | $=$ | \n$-\\dfrac{\\sqrt{\\var{questdensurd}}}{\\var{ansden}}$ | \n(cancelling the common factor of $\\var{cancel}$) | \n
| \n | $=$ | \n$\\dfrac{\\sqrt{\\var{questdensurd}}}{\\var{ansden}}$ | \n(cancelling the common factor of $\\var{cancel}$) | \n
A nice result of this process is that $\\dfrac{1}{\\sqrt{n}}=\\dfrac{\\sqrt{n}}{n}$ for all $n>0$.
", "variableReplacements": [], "variableReplacementStrategy": "originalfirst", "showCorrectAnswer": true, "scripts": {}, "marks": 0, "type": "information"}], "scripts": {}, "marks": 0, "showCorrectAnswer": true, "type": "gapfill"}], "statement": "", "variable_groups": [{"variables": ["questnum", "questdenmult", "questdensurd", "tempden", "cancel", "ansnummult", "ansden"], "name": "using"}], "variablesTest": {"maxRuns": 100, "condition": ""}, "preamble": {"css": ".fractiontable table {\n width: 20%; \n padding: 0px; \n border-width: 0px; \n layout: fixed;\n}\n\n.fractiontable .tddenom \n{\n text-align: center;\n}\n\n.fractiontable .tdnum \n{\n width: 15%; \n border-bottom: 1px solid black; \n text-align: center;\n}\n\n.fractiontable .tdeq \n{\n width: 5%; \n border-bottom: 0px;\n font-size: x-large;\n}\n\n\n.fractiontable th {\n background-color:#aaa;\n}\n/*Fix the height of all cells EXCEPT table-headers to 40px*/\n.fractiontable td {\n height:40px;\n}\n", "js": ""}, "variables": {"questnum2": {"definition": "fcf*square1[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "questnum2", "description": ""}, "listsquare": {"definition": "shuffle(listsquares)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquare", "description": ""}, "arg2": {"definition": "square2[1]*prime2", "templateType": "anything", "group": "Ungrouped variables", "name": "arg2", "description": ""}, "gcff": {"definition": "gcd(questnum,questden)", "templateType": "anything", "group": "Ungrouped variables", "name": "gcff", "description": ""}, "ansnummult": {"definition": "questnum/cancel", "templateType": "anything", "group": "using", "name": "ansnummult", "description": ""}, "cancel": {"definition": "gcd(questnum,tempden)", "templateType": "anything", "group": "using", "name": "cancel", "description": ""}, "listsquares": {"definition": "map([n,n^2],n,1..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "listsquares", "description": ""}, "questden": {"definition": "fcf*square2[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "questden", "description": ""}, "extra": {"definition": "random(listprimes)", "templateType": "anything", "group": "Ungrouped variables", "name": "extra", "description": ""}, "listprimes": {"definition": "[2,3,5,7,11]", "templateType": "anything", "group": "Ungrouped variables", "name": "listprimes", "description": ""}, "questnum": {"definition": "random(-12..12 except [-1,0,1])", "templateType": "anything", "group": "using", "name": "questnum", "description": ""}, "num_wo_cf": {"definition": "questnum/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "num_wo_cf", "description": ""}, "tempden": {"definition": "questdenmult*questdensurd", "templateType": "anything", "group": "using", "name": "tempden", "description": ""}, "questdenmult": {"definition": "random(2..12 except questnum)", "templateType": "anything", "group": "using", "name": "questdenmult", "description": ""}, "ansmult": {"definition": "extra*square1[0]*square2[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "ansmult", "description": ""}, "ansarg": {"definition": "prime1*prime2", "templateType": "anything", "group": "Ungrouped variables", "name": "ansarg", "description": ""}, "den_wo_cf": {"definition": "questden/gcff", "templateType": "anything", "group": "Ungrouped variables", "name": "den_wo_cf", "description": ""}, "listprime": {"definition": "shuffle(listprimes)[0..2]", "templateType": "anything", "group": "Ungrouped variables", "name": "listprime", "description": ""}, "fcf": {"definition": "random(2..12)", "templateType": "anything", "group": "Ungrouped variables", "name": "fcf", "description": "forced common factor
"}, "questdensurd": {"definition": "random(2,3,5,6,7,10,11,13,14,15,17)", "templateType": "anything", "group": "using", "name": "questdensurd", "description": ""}, "prime1": {"definition": "listprime[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime1", "description": ""}, "prime2": {"definition": "listprime[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "prime2", "description": ""}, "ansden": {"definition": "tempden/cancel", "templateType": "anything", "group": "using", "name": "ansden", "description": ""}, "square1": {"definition": "listsquare[0]", "templateType": "anything", "group": "Ungrouped variables", "name": "square1", "description": ""}, "square2": {"definition": "listsquare[1]", "templateType": "anything", "group": "Ungrouped variables", "name": "square2", "description": ""}}, "metadata": {"notes": "using background image in table to get a good looking square root symbol
\n\n\n| [[0]] | \n$\\,\\,\\,\\,$[[1]] | \n
Given the fraction \\[\\dfrac{\\var{questnum}}{\\sqrt{\\var{surd1}}\\var{densign}\\sqrt{\\var{surd2}}}\\] we can rationalise the denominator and rewrite the fraction in the simplified equivalent form
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| [[4]] | \n
Note: If you are marked incorrect, please try swapping the order of terms in the numerator, this question requires the larger surd part on the left and the smaller surd part on the right.
\nIn the above case, to rationalise the denominator we can multiply the top and bottom of the fraction by the conjugate surd of the denominator. This will rationalise the denominator since $\\left(\\sqrt{a}+\\sqrt{b}\\right)\\times\\left(\\sqrt{a}-\\sqrt{b}\\right)=a-\\sqrt{ab}+\\sqrt{ab}-b=a-b$.
\n| $\\dfrac{\\var{questnum}}{\\sqrt{\\var{surd1}}\\var{densign}\\sqrt{\\var{surd2}}}$ | \n$=$ | \n$\\dfrac{\\var{questnum}}{\\sqrt{\\var{surd1}}\\var{densign}\\sqrt{\\var{surd2}}}\\times\\dfrac{\\sqrt{\\var{surd1}}\\var{conjsign}\\sqrt{\\var{surd2}}}{\\sqrt{\\var{surd1}}\\var{conjsign}\\sqrt{\\var{surd2}}}$ | \n(multiplying top and bottom by the conjugate surd of the denominator) | \n
| \n | $=$ | \n$\\dfrac{\\var{questnum}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{questnum}\\sqrt{\\var{surd2}}}{\\var{surd1}-\\var{surd2}}$ | \n\n |
| \n | $=$ | \n$\\dfrac{\\var{questnum}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{questnum}\\sqrt{\\var{surd2}}}{\\var{tempden}}$ | \n\n |
| \n | $=$ | \n$\\dfrac{\\var{ansnummult}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{ansnummult}\\sqrt{\\var{surd2}}}{\\var{ansden}}$ | \n(cancelling the common factor of $\\var{cancel}$) | \n
| \n | $=$ | \n$\\var{ansnummult}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{ansnummult}\\sqrt{\\var{surd2}}$ | \n(cancelling the common factor of $\\var{cancel}$) | \n
| \n | $=$ | \n$\\var{ansnummult}\\sqrt{\\var{surd1}}\\var{conjsign}\\var{ansnummult}\\sqrt{\\var{surd2}}$ | \n\n |
Not sure how to stop squishing of table when the window is small.
\nusing background image in table to get a good looking square root symbol
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