Question

Rationalize the denominators of $\frac{1}{\sqrt{5}+\sqrt{2}}$

Answer 1

Given expression

$\frac{1}{\sqrt{5}+\sqrt{2}}$

To rationalize it we have to multiply the denominator and numerator by $\frac{\sqrt{5}-\sqrt{2}}{\sqrt{5}-\sqrt{2}}$

We have to use identity here $(a+b)(a-b)=a^2-b^2$

$$\begin{gathered} \Rightarrow \frac{1}{(\sqrt{5}+\sqrt{2})}\times \frac{\sqrt{5}-\sqrt{2}}{(\sqrt{5}-\sqrt{2})}=\frac{\sqrt{5}-\sqrt{2}}{{\left(\sqrt{5}\right)}^2-{\left(\sqrt{2}\right)}^2} \\ =\frac{\sqrt{5}-\sqrt{2}}{5-2}=\frac{\sqrt{5}-\sqrt{2}}{3} \end{gathered}$$

Hence, we get $\frac{\sqrt{5}-\sqrt{2}}{3}$ after rationalization.