Question

Rationalise the denominators 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) $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}}$

(vii) $\frac{3\sqrt{2}}{\sqrt{5}}$

Answer 1

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

(ii) $\frac{3}{2\sqrt{5}}=\frac{3\times \sqrt{5}}{2\times \sqrt{5}\times \sqrt{5}}=\frac{3\sqrt{5}}{2\times 5}=\frac{3\sqrt{5}}{10}=\frac{3}{10}\sqrt{5}$

(iii) $\frac{1}{\sqrt{12}}=\frac{1}{\sqrt{4\times 3}}=\frac{1}{2\sqrt{3}}=\frac{1\times \sqrt{3}}{2\sqrt{3}\times \sqrt{3}}=\frac{\sqrt{3}}{2\times 3}=\frac{\sqrt{3}}{6}$

(iv) $\frac{\sqrt{2}}{\sqrt{5}}=\frac{\sqrt{2}\times \sqrt{5}}{\sqrt{5}\times \sqrt{5}}=\frac{\sqrt{10}}{5}=\frac{1}{5}\sqrt{10}$

(v) $\frac{\sqrt{3}+1}{\sqrt{2}}=\frac{(\sqrt{3}+1)\sqrt{2}}{\sqrt{2}\times \sqrt{2}}=\frac{\sqrt{6}+\sqrt{2}}{2}$

(vi) $\frac{\sqrt{2}+\sqrt{5}}{\sqrt{3}}=\frac{(\sqrt{2}+\sqrt{5})\times \sqrt{3}}{\sqrt{3}\times \sqrt{3}}=\frac{\sqrt{6}+\sqrt{15}}{3}$

(vii) $\frac{3\sqrt{2}}{\sqrt{5}}=\frac{3\sqrt{2}\times \sqrt{5}}{\sqrt{5}\times \sqrt{5}}=\frac{3\times \sqrt{10}}{5}=\frac{3}{5}\sqrt{10}$