Question

Rationalise the denominator: $\frac{1}{1+\sqrt{2}-\sqrt{3}}$

Answer 1

Given expression$\frac{1}{1+\sqrt{2}-\sqrt{3}}$

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

$=$ $\frac{1-(\sqrt{2}-\sqrt{3})}{1^2-{(\sqrt{2}-\sqrt{3})}^2}$

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

$=$ $\frac{1-(\sqrt{2}-\sqrt{3})}{-4+2\sqrt{6}}$

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

$=$ $\frac{\sqrt{6}-\sqrt{12}+\sqrt{18}+2-2\sqrt{2}+2\sqrt{3}}{2(6-4)}$

$=$ $\frac{\sqrt{6}-2\sqrt{3}+3\sqrt{2}+2-2\sqrt{2}+2\sqrt{3}}{2(6-4)}$

$=$ $\frac{\sqrt{6}+\sqrt{2}+2}{4}$