Question

Rationalise the denominator and simplify
$\frac{2\sqrt{6}-\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$

Answer 1

$\frac{2\sqrt{6}+\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$

$=\frac{2\sqrt{6}+\sqrt{5}}{3\sqrt{5}-2\sqrt{6}}$$\times$$\frac{3\sqrt{5}+2\sqrt{6}}{3\sqrt{5}+2\sqrt{6}}$ (Rationalisation of denominator)

$=\frac{(2\sqrt{6}+\sqrt{5})(3\sqrt{5}+2\sqrt{6})}{9\times 5-4\times 6}$ $[\because a^2-b^2=(a-b\left)\right(a+b\left)\right]$

$=\frac{(2\sqrt{6}\times 3\sqrt{5}+2\sqrt{6}\times 2\sqrt{6}+\sqrt{5}\times 3\sqrt{5}+\sqrt{5}\times 2\sqrt{6})}{45-24}$

$=\frac{(6\sqrt{30}+24+15+2\sqrt{30})}{21}$

$=\frac{39+8\sqrt{3}}{21}$