Rationalise the Denominator

2011 May June Paper 21 Q1

Without using a calculator, express $\frac{(5+2 \sqrt{3})^{2}}{2+\sqrt{3}}$ in the form $p+q \sqrt{3}$ , where $p$ and $q$ are integers.

\frac{(5 + 2 \sqrt{3})^{2}}{2 + \sqrt{3}}

$\frac{(5+2 \sqrt{3})^{2}}{2+\sqrt{3}}$

= \frac{(25 + 20 \sqrt{3} + 12) (2 - \sqrt{3})}{(2 + \sqrt{3}) (2 - \sqrt{3})}

$=\frac{(25+20 \sqrt{3}+12)(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$

= \frac{(37 + 20 \sqrt{3}) (2 - \sqrt{3})}{4 - 3}

$=\frac{(37+20 \sqrt{3})(2-\sqrt{3})}{4-3}$

= 74 - 37 \sqrt{3} + 40 \sqrt{3} - 60

$=74-37 \sqrt{3}+40 \sqrt{3}-60$

= 14 + 3 \sqrt{3}

$=14+3 \sqrt{3}$

2014 May june Paper 21 Q2

Without using a calculator, express $6(1+\sqrt{3})^{-2}$ in the form $a+b \sqrt{3}$ , where $a$ and $b$ are integers to be found.

$\begin{aligned} & 6(1+\sqrt{3})^{-2} \\=& \frac{6}{(1+\sqrt{3})^{2}} \\=& \frac{6}{1+2 \sqrt{3}+3} \\=& \frac{6}{4+2 \sqrt{3}} \\=& \frac{3}{2+\sqrt{3}} \cdot \frac{2-\sqrt{3}}{2-\sqrt{3}} \\=& \frac{6-3 \sqrt{3}}{1}=6-3 \sqrt{3} \end{aligned}$

2012 Oct Nov Paper 23 Q3

Without using a calculator, simplify $\frac{(3 \sqrt{3}-1)^{2}}{2 \sqrt{3}-3}$ , giving your answer in the form $\frac{a \sqrt{3}+b}{3}$ , where $a$ and $b$ are integers.

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2013 Oct Nov Paper 21 & 22 Q2

Express $\frac{(4 \sqrt{5}-2)^{2}}{\sqrt{5}-1}$ in the form $p \sqrt{5}+q$ , where $p$ and $q$ are integers.

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2014 May June Paper 22 Q1

Without using a calculator, express $\frac{(2+\sqrt{5})^{2}}{\sqrt{5}-1}$ in the form $a+b \sqrt{5}$ , where $a$ and $b$ are constants to be found.

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2016 May June Paper 21 & 23 Q5

Do not use a calculator in this question.
(a) Express $\frac{\sqrt{8}}{\sqrt{7}-\sqrt{5}}$ in the form $\sqrt{a}+\sqrt{b}$ , where $a$ and $b$ are integers.

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2016 Oct Nov Paper 23 Q1

Without using a calculator, show that $\frac{\sqrt{5}+3 \sqrt{3}}{\sqrt{5}+\sqrt{3}}=\sqrt{k}-2$ where $k$ is an integer to be found.

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Rationalise the Denominator

2011 May June Paper 21 Q1

2014 May june Paper 21 Q2

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2012 Oct Nov Paper 23 Q3

2013 Oct Nov Paper 21 & 22 Q2

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2016 May June Paper 21 & 23 Q5

2016 Oct Nov Paper 23 Q1

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