Question

Rationalize the denominator of $\frac{1}{(\sqrt{5}+\sqrt{2})}$ .

Answer 1

Multiply the numerator and denominator of $\frac{1}{(\sqrt{5}+\sqrt{2})}$ with $(\sqrt{5}-\sqrt{2}).$
We get,
$\frac{1}{\sqrt{5}+\sqrt{2}}=\frac{1(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})}\left[\right(a+b\left)\right(a-b)=(a^2-b^2\left)\right]=\frac{\sqrt{5}-\sqrt{2}}{(\sqrt{5}{)}^2-(\sqrt{2}{)}^2}[(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})={\sqrt{5}}^2-{\sqrt{2}}^2=5-2]=\frac{\sqrt{5}-\sqrt{2}}{5-2}=\frac{\sqrt{5}-\sqrt{2}}{3}$