Question

Rationalise the denominator and simplify:

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

Answer 1

Given: $\frac{2\sqrt{3}-\sqrt{5}}{2\sqrt{2}+3\sqrt{3}}$

Multiplying both numerator and denominator by $2\sqrt{2}-3\sqrt{3}$ to rationalise the denominator.

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


$=\frac{4\sqrt{6}-18-2\sqrt{10}+3\sqrt{15}}{(2\sqrt{2}{)}^2-(3\sqrt{3}{)}^2}$

$=\frac{4\sqrt{6}-18-2\sqrt{10}+3\sqrt{15}}{8-27}$

$=\frac{4\sqrt{6}-18-2\sqrt{10}+3\sqrt{15}}{-19}$

$=\frac{-(-4\sqrt{6}+18+2\sqrt{10}-3\sqrt{15})}{-19}$

$=\frac{-4\sqrt{6}+18+2\sqrt{10}-3\sqrt{15}}{19}$

$=\frac{18-4\sqrt{6}+2\sqrt{10}-3\sqrt{15}}{19}$