Express each one of the following with rational denominator:$ \frac{1}{\sqrt{6}-\sqrt{5}} $


Given:

\( \frac{1}{\sqrt{6}-\sqrt{5}} \)

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,

$\frac{1}{\sqrt{6}-\sqrt{5}}=\frac{1(\sqrt{6}+\sqrt{5})}{(\sqrt{6}-\sqrt{5})(\sqrt{6}+\sqrt{5})}$

$=\frac{\sqrt{6}+\sqrt{5}}{(\sqrt{6})^{2}-(\sqrt{5})^{2}}$                 [Since $(a+b)(a-b)=a^{2}-b^{2}$]

$=\frac{\sqrt{6}+\sqrt{5}}{6-5}$

$=\frac{\sqrt{6}+\sqrt{5}}{1}$

$=\sqrt{6}+\sqrt{5}$

Hence, $\frac{1}{\sqrt{6}-\sqrt{5}}=\sqrt{6}+\sqrt{5}$.

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Updated on: 10-Oct-2022

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