Question

Rationalise the denominator of the following:
$\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

Answer 1

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

Dividing numerator and denominator by $\sqrt{5}+\sqrt{2}$

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

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

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

$=\frac{7+\sqrt{10}+\sqrt{10}}{3}$

$=\frac{7+2\sqrt{10}}{3}$