Question

Simplify : $\frac{1}{1+\sqrt{2}}$ + $\frac{1}{\sqrt{2}+\sqrt{3}}$ + $\frac{1}{\sqrt{3}+\sqrt{5}}$

Answer 1

$\frac{1}{1+\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}$

$=\frac{1}{1+\sqrt{2}}\times \frac{1-\sqrt{2}}{1-\sqrt{2}}+\frac{1}{\sqrt{2}+\sqrt{3}}\times \frac{\sqrt{2}-\sqrt{3}}{\sqrt{2}-\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{5}}\times \frac{\sqrt{3}-\sqrt{5}}{\sqrt{3}-\sqrt{5}}$

$=\frac{1-\sqrt{2}}{-1}+\frac{\sqrt{2}-\sqrt{3}}{-1}+\frac{\sqrt{3}-\sqrt{5}}{-2}$

$=\frac{\sqrt{2}-1}{1}+\frac{\sqrt{3}-\sqrt{2}}{1}+\frac{\sqrt{5}-\sqrt{3}}{2}$

$=\frac{2\sqrt{2}-2+2\sqrt{3}-2\sqrt{2}+\sqrt{5}-\sqrt{3}}{2}$

$=\frac{\sqrt{5}+\sqrt{3}-2}{2}$