Rationalise the denominator and simplify:$ \frac{5+2 \sqrt{3}}{7+4 \sqrt{3}} $

Given:

\( \frac{5+2 \sqrt{3}}{7+4 \sqrt{3}} \)

To do: 

We have to rationalise the denominator and 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,

$\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=\frac{(5+2 \sqrt{3})(7-4 \sqrt{3})}{(7+4 \sqrt{3})(7-4 \sqrt{3})}$

$=\frac{35-20 \sqrt{3}+14 \sqrt{3}-8 \sqrt{3} \times \sqrt{3}}{(7)^{2}-(4 \sqrt{3})^{2}}$

$=\frac{35-6 \sqrt{3}-24}{49-48}$

$=\frac{11-6 \sqrt{3}}{1}$

$=11-6 \sqrt{3}$

Hence, $\frac{5+2 \sqrt{3}}{7+4 \sqrt{3}}=11-6 \sqrt{3}$. 

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Simply Easy Learning

Updated on: 10-Oct-2022

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