Question

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

Answer 1

We have,

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

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

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

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

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

$=\sqrt{5}-1$

Hence, the value of expression is $\sqrt{5}-1$.