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

Given:

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

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,

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

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

$=\frac{6 \sqrt{10}+9 \sqrt{2}+2 \sqrt{5}+3}{20-9}$

$=\frac{6 \sqrt{10}+9 \sqrt{2}+2 \sqrt{5}+3}{11}$

Hence, $\frac{3 \sqrt{2}+1}{2 \sqrt{5}-3}=\frac{6 \sqrt{10}+9 \sqrt{2}+2 \sqrt{5}+3}{11}$.