Rationalise the denominator and simplify:$ \frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}} $

Given:

\( \frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}} \)

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,

$\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}=\frac{(4 \sqrt{3}+5 \sqrt{2})(\sqrt{48}-\sqrt{18})}{(\sqrt{48}+\sqrt{18})(\sqrt{48}-\sqrt{18})}$

$=\frac{4 \sqrt{144}-4 \sqrt{54}+5 \sqrt{96}-5 \sqrt{36}}{(\sqrt{48})^{2}-(\sqrt{18})^{2}}$

$=\frac{4 \times 12-4 \sqrt{9 \times 6}+5 \sqrt{16 \times 6}-5 \times 6}{48-18}$

$=\frac{48-4 \times 3 \sqrt{6}+5 \times 4 \sqrt{6}-30}{30}$

$=\frac{48-30-12 \sqrt{6}+20 \sqrt{6}}{30}$

$=\frac{18+8 \sqrt{6}}{30}$

$=\frac{9+4 \sqrt{6}}{15}$

Hence, $\frac{4 \sqrt{3}+5 \sqrt{2}}{\sqrt{48}+\sqrt{18}}=\frac{9+4 \sqrt{6}}{15}$.

Tutorialspoint

Updated on: 10-Oct-2022

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