# 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.

$$\frac{(5+2 \sqrt{3})^{2}}{2+\sqrt{3}}$$
$$=\frac{(25+20 \sqrt{3}+12)(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}$$
$$=\frac{(37+20 \sqrt{3})(2-\sqrt{3})}{4-3}$$
$$=74-37 \sqrt{3}+40 \sqrt{3}-60$$
$$=14+3 \sqrt{3}$$

## 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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