Question

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

Answer 1

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

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

$=\frac{2-\sqrt{3}}{4-3}+\frac{\sqrt{5}+\sqrt{3}}{5-3}+\frac{2+\sqrt{5}}{4-5}$ ...{By Rationalizing all parts}

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

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

$=\frac{4-2\sqrt{3}+\sqrt{5}+\sqrt{3}-4-2\sqrt{5}}{2}$

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

Hence, this is the answer.