Question

Simplify the following:
$\frac{\sqrt{6}}{\sqrt{5}}+\frac{2}{\sqrt{5}+\sqrt{2}}$.

Answer 1

We rationalise the denominators of $(\frac{\sqrt{6}}{\sqrt{5}}+\frac{2}{\sqrt{5}+\sqrt{2}})$ and then simplify as follows:

$\left(\frac{\sqrt{6}}{\sqrt{5}}\right)+\left(\frac{2}{\sqrt{5}+\sqrt{2}}\right)=(\frac{\sqrt{6}}{\sqrt{5}}\times \frac{\sqrt{5}}{\sqrt{5}})+(\frac{2}{\sqrt{5}+\sqrt{2}}\times \frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}})$

$=\left(\frac{\sqrt{30}}{\sqrt{25}}\right)+\left(\frac{2(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}\right)=\left(\frac{\sqrt{30}}{5}\right)+\left(\frac{2\sqrt{5}-2\sqrt{2}}{(\sqrt{5}{)}^2-(\sqrt{2}{)}^2}\right)$

$=\frac{\sqrt{30}}{5}+\frac{2\sqrt{5}-2\sqrt{2}}{5-2}=\frac{\sqrt{30}}{5}+\frac{2\sqrt{5}-2\sqrt{2}}{3}=\frac{3\sqrt{30}+\left[\right(5\times 2\sqrt{5})-(5\times 2\sqrt{2}\left)\right]}{5\times 3}$

$=\frac{3\sqrt{30}+10\sqrt{5}-10\sqrt{2}}{15}$ .

Hence, $\left(\frac{\sqrt{6}}{\sqrt{5}}\right)+\left(\frac{2}{\sqrt{5}+\sqrt{2}}\right)=\frac{3\sqrt{30}+10\sqrt{5}-10\sqrt{2}}{15}$ .