Question

Simplify $\frac{1}{7+4\sqrt{3}}+\frac{1}{2+\sqrt{5}}$

Answer 1

The rationalising factor of $7+4\sqrt{3}$ is $7-4\sqrt{3}$ and the rationalising factor of $2+\sqrt{5}$ is $2-\sqrt{5}$.
$=\frac{1}{7+4\sqrt{3}}+\frac{1}{2+\sqrt{5}}$
$=\frac{1}{7+4\sqrt{3}}\times \frac{7-4\sqrt{3}}{7-4\sqrt{3}}+\frac{1}{2+\sqrt{5}}\times \frac{2-\sqrt{5}}{2-\sqrt{5}}$
$=\frac{7-4\sqrt{3}}{49-48}+\frac{2-\sqrt{5}}{(4-5)}$
$=\frac{7-4\sqrt{3}}{1}+\frac{2-\sqrt{5}}{(-1)}$
$=7-4\sqrt{3}-2+\sqrt{5}=5-4\sqrt{3}+\sqrt{5}$