// Numbas version: exam_results_page_options {"name": "Rationalising the denominator - surds", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false}, "question_groups": [{"questions": [{"name": "Rationalising the denominator - surds", "parts": [{"scripts": {}, "showFeedbackIcon": true, "prompt": "

Rationalise the denominator of the following surds to simplify them down to their integer value.

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i)

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$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[0]}{prime_nums[0]}}}}{\\sqrt{\\var{prime_nums[0]}}}=$ [[0]]

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ii)

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$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[1]}{prime_nums[1]}}}}{\\sqrt{\\var{prime_nums[1]}}}=$ [[1]]

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Rationalise the denominator and rewrite as a fraction in its simplest form.

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\n\n\n\n\n\n\n\n\n\n\n
 $\\displaystyle\\frac{\\sqrt{\\var{n_lcm}}}{\\sqrt{\\var{m_lcm}}}=$ $\\surd$[[0]] [[1]]
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Rationalise the denominator of this expression and reduce to lowest terms.

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\$\\frac{\\var{c}}{\\var{d}\\sqrt{\\var{f}}} = \\frac{a\\sqrt{\\var{f}}}{b}\\text{,} \$

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where

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$a =$ [[0]]

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$b =$ [[1]]

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$\\displaystyle\\var{o}\\sqrt{\\var{h}}+\\var{l}$

", "

$\\displaystyle-\\var{l}-\\var{o}\\sqrt{\\var{h}}$

", "

$\\displaystyle\\var{o}\\sqrt{\\var{h}}-\\var{l}$

", "

$\\displaystyle\\var{l}-\\sqrt{\\var{h}}$

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Express the following in the form $a\\sqrt{b}+m$, where $a$, $b$ and $c$ are integers.

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$\\displaystyle\\frac{\\var{g}}{(\\sqrt{\\var{h}}+\\var{j})}$ =

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Rationalise the denominator of this expression and reduce down to its simplest form.

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\$\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}+\\sqrt{\\var{u}}} = \\frac{a-b\\sqrt{\\var{u}}}{c} \\text{,} \$

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where:

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$a =$ [[0]]

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$b =$ [[1]]

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$c =$ [[2]]

$\\displaystyle\\frac{\\sqrt{\\var{p}}+\\var{r_coprime*p_p1}}{\\var{p+1}}$

", "

$\\displaystyle\\frac{\\sqrt{\\var{p}}(\\sqrt{\\var{p}}+\\var{r_coprime*p_p1})}{\\var{p+1}\\sqrt{\\var{p}}}$

", "

$\\displaystyle\\frac{\\var{p}+\\var{qwerty}\\sqrt{\\var{p}}}{\\var{p}\\sqrt{\\var{p}}}$

", "

$\\displaystyle\\frac{\\sqrt{\\var{p}}+\\var{r_coprime*p_p1}}{\\var{p}}$

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Complete this addition by firstly rationalising the denominator of the first fraction. Your final answer should be a fraction with integer denominator.

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\$\\frac{\\var{p}}{\\sqrt{\\var{p}}+\\sqrt{\\var{p^3}}}+\\frac{\\var{r_coprime}}{\\var{s_coprime}} \$

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#### a)

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The rule that should be used is $\\displaystyle\\frac{\\sqrt{a}}{\\sqrt{b}} = \\sqrt{\\frac{a}{b}}$.

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i)

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$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[0]}{prime_nums[0]}}}}{\\sqrt{\\var{prime_nums[0]}}} = \\sqrt{\\frac{{\\simplify{{square_nums[0]}{prime_nums[0]}}}}{{\\var{prime_nums[0]}}}} = \\sqrt{\\var{square_nums[0]}}= \\simplify{{sqrt(square_nums[0])}}$; and

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ii)

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$\\displaystyle\\frac{\\sqrt{\\simplify{{square_nums[1]}{prime_nums[1]}}}}{\\sqrt{\\var{prime_nums[1]}}} = \\sqrt{\\frac{{\\simplify{{square_nums[1]}{prime_nums[1]}}}}{{\\var{prime_nums[1]}}}} = \\sqrt{\\var{square_nums[1]}} = \\simplify{{sqrt(square_nums[1])}}$.

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#### b)

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To rationalise the denominator, you have to use the rule $\\displaystyle\\sqrt{a}\\times\\sqrt{a}=a$.

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You are asked to rationalise the denominator of the expression $\\displaystyle\\frac{\\sqrt{\\var{n_lcm}}}{\\sqrt{\\var{m_lcm}}}$.

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As ${\\sqrt{\\var{m_lcm}}}$ is the denominator, we must multiply the whole fraction by ${\\sqrt{\\var{m_lcm}}}$.

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\$\\frac{\\sqrt{\\var{n_lcm}}}{\\sqrt{\\var{m_lcm}}}\\times\\frac{\\sqrt{\\var{m_lcm}}}{\\sqrt{\\var{m_lcm}}} \$

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By using the rule $\\sqrt{a}\\times\\sqrt{b}$ = $\\sqrt{ab}$, we can get the numerator as $\\sqrt{\\var{num}}$.

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The denominator is ${\\sqrt{\\var{m_lcm}}}\\times{\\sqrt{\\var{m_lcm}}} = {\\var{m_lcm}}$.

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So the whole expression is

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\$\\frac{\\sqrt{\\var{num}}}{\\var{m_lcm}} \$

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Finally, we can cancel down by finding the greatest common divisor of the top and bottom to obtain

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\$\\frac{\\sqrt{\\var{a[0]*b[0]}}}{\\var{b[0]}} \$

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#### c)

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The first thing to do in this question is multiply the whole fraction by the surd on the denominator, $\\sqrt{\\var{f}}$. Using the rule $\\displaystyle\\sqrt{a}\\times\\sqrt{a}=a$, the new value of the denominator is $\\var{d}\\times\\var{f}=\\var{df}$.

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$\\displaystyle\\frac{\\var{c}}{\\var{d}\\sqrt{\\var{f}}}\\times\\simplify[all,!simplifyFractions,!sqrtDivision]{sqrt({f})/(sqrt({f}))}={\\frac{\\simplify{{c}*sqrt({f})}}{{(\\var{d}\\times\\var{f})}}}=\\frac{\\simplify{{c}*sqrt({f})}}{\\var{d*f}}$.

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Make sure to simplify your fractions down to their simplest form. The fraction can be cancelled down using the highest common divisor, $\\var{gcd_cdf}$, to give a final answer of

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\$\\frac{\\simplify{{c}*sqrt({f})}}{\\var{d*f}}=\\frac{\\simplify{{c_coprime}*sqrt({f})}}{\\var{df_coprime}}\$

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We are asked to find an answer in the form $\\frac{a \\sqrt{\\var{f}}}{b}$, so $a = \\var{c_coprime}$ and $b = \\var{df_coprime}$.

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#### d)

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When rationalising an expression with a compound denominator, like $\\displaystyle\\frac{\\var{g}}{\\sqrt{\\var{h}}+\\var{j}}$, we must multiply the fraction by the conjugate of the denominator.

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The conjugate of $(\\sqrt{a}+b)$ is $(\\sqrt{a}-b)$, and therefore the conjugate of $(\\sqrt{\\var{h}}+\\var{j})$ is $(\\sqrt{\\var{h}}-\\var{j})$.

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\$\\frac{\\var{g}}{\\sqrt{\\var{h}}+\\var{j}}\\times\\frac{(\\sqrt{\\var{h}}-\\var{j})}{(\\sqrt{\\var{h}}-\\var{j})}=\\frac{\\var{g}(\\sqrt{\\var{h}}-\\var{j})}{(\\sqrt{\\var{h}}+\\var{j})(\\sqrt{\\var{h}}-\\var{j})}\\text{.} \$

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From this point, we need to multiply out the brackets on the numerator and denominator, using the rule $\\displaystyle\\sqrt{a}\\times\\sqrt{a}=a$.

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$\\displaystyle\\frac{\\var{g}(\\sqrt{\\var{h}}-\\var{j})}{(\\sqrt{\\var{h}}+\\var{j})(\\sqrt{\\var{h}}-\\var{j})}==\\frac{\\simplify[all,!collectNumbers,!noLeadingMinus]{({-{g}*{j}})+{g}}\\sqrt{\\var{h}}}{\\var{h}{-\\simplify{{j}*sqrt({h})}}+\\simplify{{j}*sqrt({h})}-{{\\var{j^2}}}}$.

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The two $\\simplify{{j}*sqrt({h})}$ terms will cancel on the denominator, so that you are left with

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\$\\frac{ \\simplify{ {g*-j} + {g}*sqrt({h})} }{\\var{h-j^2}}\\text{,} \$

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which simplifies to

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\$\\var{o}\\sqrt{\\var{h}}-{\\var{l}}\\text{.} \$

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#### e)

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$\\displaystyle\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}+\\sqrt{\\var{u}}}$ =

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To rationalise the denominator of this fraction, multiply the whole fraction by the conjugate of the denominator, multiply out the brackets and collect like terms.

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$\\displaystyle\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}+\\sqrt{\\var{u}}}\\times\\frac{\\var{t}-\\sqrt{\\var{u}}}{\\var{t}-\\sqrt{\\var{u}}}=\\frac{(\\var{t}-\\sqrt{\\var{u}})(\\var{t}-\\sqrt{\\var{u}})}{(\\var{t}+\\sqrt{\\var{u}})(\\var{t}-\\sqrt{\\var{u}})}=\\frac{\\var{t^2}-\\var{2t}\\sqrt{\\var{u}}+\\var{u}}{\\var{t^2}-\\var{u}}$.

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$\\displaystyle\\frac{\\var{t^2}-\\var{2t}\\sqrt{\\var{u}}+\\var{u}}{\\var{t^2}-\\var{u}}=\\frac{\\var{t^2+u}-\\var{2t}{\\sqrt{\\var{u}}}}{\\var{t^2-u}}$.

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Dividing all terms by their highest common divisor, ${\\var{gcd_frac}}$, gives

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$\\displaystyle\\frac{\\var{num_simp_1}-\\simplify{{num_simp_2}*sqrt({u})}}{\\var{denom_simp}}$.

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#### f)

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To add $\\displaystyle\\frac{\\var{p}}{\\sqrt{\\var{p}}+\\sqrt{\\var{p^3}}}+\\frac{\\var{r_coprime}}{\\var{s_coprime}}$ you must put both terms over the same denominator.

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It is good to notice that $\\sqrt{\\var{p^3}}$ can be rewritten as $\\var{p}\\sqrt{\\var{p}}$, meaning we can rewrite the first fraction as

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\$\\frac{\\var{p}}{\\sqrt{\\var{p}}+\\var{p}\\sqrt{\\var{p}}} \\text{.} \$

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This can be simplified further by collecting like terms to obtain

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\$\\frac{\\var{p}}{\\var{p+1}\\sqrt{\\var{p}}} \\text{.} \$

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Therefore, the expression is now $\\displaystyle\\frac{\\var{p}}{{\\var{p+1}}\\sqrt{\\var{p}}}+\\frac{\\var{r_coprime}}{\\var{s_coprime}}$.

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In order to complete the addition, the second fraction also has to have the same denominator as the first fraction. We have to multiply it by $\\simplify{{p_p1}sqrt({p})}$ in order to get common denominators across both fractions.

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$\\displaystyle\\frac{\\var{r_coprime}}{\\var{s_coprime}}\\times\\simplify[all,!simplifyFractions,!sqrtDivision]{{p_p1}*sqrt({p})/({p_p1}*sqrt({p}))}=\\frac{\\simplify{{one}*sqrt({p})}}{\\var{two}\\sqrt{\\var{p}}}$,

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Both parts of the expression now have the same denominator and we can add them.

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\$\\frac{\\var{p}}{\\var{p+1}\\sqrt{\\var{p}}}+\\frac{\\simplify{{one}*sqrt({p})}}{\\var{two}\\sqrt{\\var{p}}}= \\frac{\\var{p}+\\simplify{{one}*sqrt({p})}}{\\var{p+1}\\sqrt{\\var{p}}} \$

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When simplifying, if we note that $\\displaystyle\\sqrt{a}\\times\\sqrt{a}=a$, we can pull $\\sqrt{\\var{p}}$ out of the numerator as a common factor:

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\$\\frac{\\sqrt{\\var{p}}(\\sqrt{\\var{p}}+\\var{r_coprime*p_p1})}{\\var{p+1}\\sqrt{\\var{p}}}\\text{.} \$

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We can then cancel the common $\\sqrt{\\var{p}}$ term on the numerator and denominator to simplify the expression further and get a final answer of

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\$\\frac{\\sqrt{\\var{p}}+\\var{one}}{\\var{p+1}} \\text{.} \$

", "statement": "

Rationalising the denominator means removing any surds on the denominator.

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Try the following questions to practise rationalising with expressions in different forms.

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Random numbers, not square.

", "name": "b", "templateType": "anything", "definition": "repeat(random(3..5 except 4),5)", "group": "Part b"}, "u": {"description": "", "name": "u", "templateType": "anything", "definition": "random(2..10 except 4 except 9) ", "group": "Part e"}, "k": {"description": "", "name": "k", "templateType": "anything", "definition": "random(2..4)", "group": "Part d"}, "p_p1": {"description": "", "name": "p_p1", "templateType": "anything", "definition": "ceil(p/(s_coprime))", "group": "Part f"}, "m": {"description": "

Random number.

", "name": "m", "templateType": "anything", "definition": "random(2,3,4)", "group": "Part b"}, "n": {"description": "

Random number.

", "name": "n", "templateType": "anything", "definition": "random(2,4,5)", "group": "Part b"}, "gcd_num": {"description": "

GCD of terms in numerator.

", "name": "gcd_num", "templateType": "anything", "definition": "gcd(t^2+u, 2*t)", "group": "Part e"}, "num_simp_2": {"description": "

Simplified surd term of numerator.

", "name": "num_simp_2", "templateType": "anything", "definition": "2t/gcd_frac", "group": "Part e"}, "f": {"description": "", "name": "f", "templateType": "anything", "definition": "random(2..10 except 4 except 9)", "group": "Part c"}, "prime_nums": {"description": "

List of prime numbers.

", "name": "prime_nums", "templateType": "anything", "definition": "shuffle([3,5,7,11,13])", "group": "Part a"}, "ghi": {"description": "", "name": "ghi", "templateType": "anything", "definition": "t^2-u", "group": "Part e"}, "gcd_frac": {"description": "

GCD of all terms in fraction.

", "name": "gcd_frac", "templateType": "anything", "definition": "gcd(gcd_num, t^2-u)", "group": "Part e"}, "d": {"description": "", "name": "d", "templateType": "anything", "definition": "random(2..10)", "group": "Part c"}, "num": {"description": "", "name": "num", "templateType": "anything", "definition": "m_lcm*n_lcm", "group": "Part b"}, "a": {"description": "

Random numbers, not square.

", "name": "a", "templateType": "anything", "definition": "repeat(random(7..11 except 8 except 9 except 10),5)", "group": "Part b"}, "gcd_cdf": {"description": "", "name": "gcd_cdf", "templateType": "anything", "definition": "gcd(c,df)", "group": "Part c"}, "r_coprime": {"description": "", "name": "r_coprime", "templateType": "anything", "definition": "r/gcd(r,s)", "group": "Part f"}, "t": {"description": "", "name": "t", "templateType": "anything", "definition": "random(1..6)", "group": "Part e"}, "qwerty": {"description": "", "name": "qwerty", "templateType": "anything", "definition": "(r_coprime*p_p1)-1", "group": "Part f"}, "c_coprime": {"description": "", "name": "c_coprime", "templateType": "anything", "definition": "c/gcd_cdf", "group": "Part c"}, "j": {"description": "", "name": "j", "templateType": "anything", "definition": "random(1..5)", "group": "Part d"}, "square_nums": {"description": "

List of square numbers.

", "name": "square_nums", "templateType": "anything", "definition": "shuffle([4,9,16,25])", "group": "Part a"}, "num_simp_1": {"description": "

Simplified numerical term of numerator.

", "name": "num_simp_1", "templateType": "anything", "definition": "(t^2+u)/gcd_frac", "group": "Part e"}, "denom_simp": {"description": "

Simplified denominator.

", "name": "denom_simp", "templateType": "anything", "definition": "ghi/gcd_frac", "group": "Part e"}, "p": {"description": "", "name": "p", "templateType": "anything", "definition": "random(2..5 except 4)", "group": "Part f"}, "one": {"description": "", "name": "one", "templateType": "anything", "definition": "r_coprime*p_p1", "group": "Part f"}, "df": {"description": "", "name": "df", "templateType": "anything", "definition": "d*f", "group": "Part c"}, "c": {"description": "", "name": "c", "templateType": "anything", "definition": "random(1..10)", "group": "Part c"}, "h": {"description": "", "name": "h", "templateType": "anything", "definition": "j^2+random(-3..3 except 0)", "group": "Part d"}, "l": {"description": "", "name": "l", "templateType": "anything", "definition": "(g*j)/(h-j^2)", "group": "Part d"}, "o": {"description": "", "name": "o", "templateType": "anything", "definition": "g/(h-j^2)", "group": "Part d"}, "r": {"description": "", "name": "r", "templateType": "anything", "definition": "random(1..2)", "group": "Part f"}, "s": {"description": "", "name": "s", "templateType": "anything", "definition": "p+1", "group": "Part f"}, "two": {"description": "", "name": "two", "templateType": "anything", "definition": "s_coprime*p_p1", "group": "Part f"}, "n_lcm": {"description": "

Multiple of LCM of n amd m.

", "name": "n_lcm", "templateType": "anything", "definition": "a[0]*lcm(n,m)", "group": "Part b"}, "m_lcm": {"description": "

Multiple of LCM of n amd m.

", "name": "m_lcm", "templateType": "anything", "definition": "b[0]*lcm(n,m)", "group": "Part b"}, "df_coprime": {"description": "", "name": "df_coprime", "templateType": "anything", "definition": "df/gcd_cdf", "group": "Part c"}, "g": {"description": "", "name": "g", "templateType": "anything", "definition": "k*(h-j^2)", "group": "Part d"}, "s_coprime": {"description": "", "name": "s_coprime", "templateType": "anything", "definition": "s/gcd(r,s)", "group": "Part f"}}, "functions": {}, "metadata": {"licence": "Creative Commons Attribution 4.0 International", "description": "

Rationalise the denominator with increasingly difficult examples involving compound denominators.

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