Express each one of the following with rational denominator:$ \frac{1}{2 \sqrt{5}-\sqrt{3}} $

Given:

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

To do: 

We have to express the given fraction with rational denominator.

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,

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

$=\frac{2 \sqrt{5}+\sqrt{3}}{(2 \sqrt{5})^{2}-(\sqrt{3})^{2}}$           [Since $(a+b)(a-b)=a^2-b^2$]

$=\frac{2 \sqrt{5}+\sqrt{3}}{20-3}$

$=\frac{2 \sqrt{5}+\sqrt{3}}{17}$

Hence, $\frac{1}{2 \sqrt{5}-\sqrt{3}}=\frac{2 \sqrt{5}+\sqrt{3}}{17}$.