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Rationalise the denominator and simplify:$ \frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}} $
Given:
\( \frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}} \)
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,
$\frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}}=\frac{(2 \sqrt{6}-\sqrt{5})(3 \sqrt{5}+2 \sqrt{6})}{(3 \sqrt{5}-2 \sqrt{6})(3 \sqrt{5}+2 \sqrt{6})}$
$=\frac{6 \sqrt{30}+4 \sqrt{6} \times \sqrt{6}-3 \sqrt{5} \times \sqrt{5}-2 \sqrt{30}}{(3 \sqrt{5})^{2}-(2 \sqrt{6})^{2}}$
$=\frac{4 \sqrt{30}+4 \times 6-3 \times 5}{9 \times 5-4 \times 6}$
$=\frac{24-15+4 \sqrt{30}}{45-24}$
$=\frac{9+4 \sqrt{30}}{21}$
Hence, $\frac{2 \sqrt{6}-\sqrt{5}}{3 \sqrt{5}-2 \sqrt{6}}=\frac{9+4 \sqrt{30}}{21}$.
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