Question

Simplify the following by rationalising the denominator:
$\frac{3\sqrt{5}-\sqrt{7}}{3\sqrt{3}+\sqrt{2}}$

Answer 1

$\frac{3\sqrt{5}-\sqrt{7}}{3\sqrt{3}+\sqrt{2}}$

Multiply by $3\sqrt{3}-\sqrt{2}$ to numerator and denominator both

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

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

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

$=\frac{9\times \left(\sqrt{5\times 3}\right)-3\times \left(\sqrt{7\times 3}\right)-3\left(\sqrt{5\times 2}\right)+\left(\sqrt{7\times 2}\right)}{27-2}$

$=\frac{9\times \sqrt{15}-3\times \sqrt{21}-3\times \sqrt{10}+\sqrt{14}}{25}$

$=\frac{9\sqrt{15}-3\sqrt{21}-3\sqrt{10}+\sqrt{14}}{25}$

Hence, $\frac{3\sqrt{5}-\sqrt{7}}{3\sqrt{3}+\sqrt{2}}=\frac{9\sqrt{15}-3\sqrt{21}-3\sqrt{10}+\sqrt{14}}{25}$