Express each one of the following with rational denominator:$ \frac{30}{5 \sqrt{3}-3 \sqrt{5}} $


Given:

\( \frac{30}{5 \sqrt{3}-3 \sqrt{5}} \)

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,

$\frac{30}{5 \sqrt{3}-3 \sqrt{5}}=\frac{30(5 \sqrt{3}+3 \sqrt{5})}{(5 \sqrt{3}-3 \sqrt{5})(5 \sqrt{3}+3 \sqrt{5})}$

$=\frac{30(5 \sqrt{3}+3 \sqrt{5})}{(5 \sqrt{3})^{2}-(3 \sqrt{5})^{2}}$

$=\frac{30(5 \sqrt{3}+3 \sqrt{5})}{25 \times 3-9 \times 5}$

$=\frac{30(5 \sqrt{3}+3 \sqrt{5})}{75-45}$

$=\frac{30(5 \sqrt{3}+3 \sqrt{5})}{30}$

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

Hence, $\frac{30}{5 \sqrt{3}-3 \sqrt{5}}=5 \sqrt{3}+3 \sqrt{5}$.

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Updated on: 10-Oct-2022

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