Question

Rationalise the denominator and simplify:
$\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

Answer 1

We need to rationalise $\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

Rationalising factor of denominator is $\sqrt{5}+\sqrt{2}$
$\frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}-\sqrt{2}}\times \frac{\sqrt{5}+\sqrt{2}}{\sqrt{5}+\sqrt{2}}$
$=\frac{{(\sqrt{5}+\sqrt{2})}^2}{{\left(\sqrt{5}\right)}^2-{\left(\sqrt{2}\right)}^2}=\frac{{\left(\sqrt{5}\right)}^2+2\times \sqrt{5}\times \sqrt{2}+{\left(\sqrt{2}\right)}^2}{5-2}$
$=\frac{5+2\sqrt{10}+2}{3}$

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