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.

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

More Similar Questions

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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