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

Given:

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

To do: 

We have to simplify the given expression.

Solution:

We know that,

Rationalising factor of a fraction with denominator ${\sqrt{a}}$ is ${\sqrt{a}}$.

Rationalising factor of a fraction with denominator ${\sqrt{a}-\sqrt{b}}$ is ${\sqrt{a}+\sqrt{b}}$.

Rationalising factor of a fraction with denominator ${\sqrt{a}+\sqrt{b}}$ is ${\sqrt{a}-\sqrt{b}}$.

Therefore,

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

Therefore,

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

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

$=0$

Hence, $\frac{2}{\sqrt{5}+\sqrt{3}}+\frac{1}{\sqrt{3}+\sqrt{2}}-\frac{3}{\sqrt{5}+\sqrt{2}}=0$.