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Rationalise the denominator of each of the following:

(i) \(\frac{3}{\sqrt{5}}\)

(ii) \(\frac{3}{2\sqrt{5}}\)

(iii) \(\frac{1}{\sqrt{12}}\)

(iv) \(\frac{\sqrt{2}}{\sqrt{5}}\)

(v) \(\frac{\sqrt{3}+1}{\sqrt{2}}\)

(vi) \(\frac{\sqrt2+\sqrt5}{\sqrt{3}}\)

(vii) \(\frac{3\sqrt2}{\sqrt{5}}\)

1 Answer

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(i) As there is √5 in the denominator and we know that √5 x √5 = 5

So, multiply numerator and denominator by √5,

\(\frac{3}{\sqrt{5}}\) = \(\frac{3\times \sqrt{5}}{\sqrt{5}\times \sqrt{5}}\) = \(\frac{3\sqrt{5}}{5}\) = \(\frac{3}{5}{\sqrt{5}}\)

(ii) \(\frac{3}{2\sqrt{5}}\) x \(\frac{2\sqrt5}{2\sqrt{5}}\) = \(\frac{3\times 2\sqrt5}{(2\sqrt{5})^2}\) = \(\frac{6\sqrt{5}}{20}\) = \(\frac{3}{10}{\sqrt{5}}\)

(iii) \(\frac{1}{\sqrt{12}}\) = \(\frac{1}{\sqrt{12}} \times \frac{\sqrt{12}}{\sqrt{12}}\)

\(\frac{1}{\sqrt{12}} \times \frac{\sqrt{12}}{\sqrt{12}}\)

\(\frac{\sqrt3\sqrt{4}}{12}\)

\(\frac{2\sqrt{3}}{12}\)

\(\frac{\sqrt{3}}{6}\)

(iv) \(\frac{\sqrt{2}}{\sqrt{5}}\) x \(\frac{\sqrt{5}}{\sqrt{5}}\) = \(\frac{\sqrt2\times \sqrt5}{(\sqrt{5})^2}\) = \(\frac{1}{5}{\sqrt{10}}\)

(v) \(\frac{\sqrt{3}+1}{\sqrt{2}}\) x \(\frac{\sqrt{2}}{\sqrt{2}}\) = \(\frac{\sqrt6+\sqrt{2}}{2}\)

(vi) \(\frac{\sqrt2+\sqrt5}{\sqrt{3}}\) x \(\frac{\sqrt{3}}{\sqrt{3}}\)\(\frac{\sqrt6+\sqrt{5}}{3}\)

(vii) \(\frac{3\sqrt2}{\sqrt{5}}\) x \(\frac{\sqrt{5}}{\sqrt{5}}\) = \(\frac{3\sqrt{10}}{5}\)

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