Question

Rationalise the denominator of $\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$

Answer 1

$\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$ (Rationalise the denominator)

$\Rightarrow \frac{(\sqrt{2}+\sqrt{3})-\left(\sqrt{5}\right)}{[\sqrt{2}+\sqrt{3})+\sqrt{5}\left]\right[\sqrt{2}+\sqrt{3})-\sqrt{5}]}$

$\Rightarrow \frac{(\sqrt{2}+\sqrt{3})-\left(\sqrt{5}\right)}{(\sqrt{2}+\sqrt{3}{)}^2-(\sqrt{5}{)}^2}$

$\Rightarrow \frac{(\sqrt{2}+\sqrt{3})-\left(\sqrt{5}\right)}{(5+2\sqrt{6}-5}\Rightarrow \frac{\left[\right(\sqrt{2}+\sqrt{3})-(\sqrt{5}\left)\right].\sqrt{6}}{\left(2\sqrt{6}\right)\left(\sqrt{6}\right)}$

$\Rightarrow \frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{2\times 6}$

$\Rightarrow \frac{2\sqrt{3}+3\sqrt{2}-\sqrt{30}}{12}$