Question

Rationalize the denominator of $\frac{1}{\left(2+\sqrt{3}\right)}.$

Answer 1

The rationalising factor of $\frac{1}{2+\sqrt{3}}$ is $2-\sqrt{3}$. So
$$\begin{gathered} \frac{1}{2+\sqrt{3}}=\frac{2-\sqrt{3}}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)} \\ =\frac{2-\sqrt{3}}{2^2-{\left(\sqrt{3}\right)}^2}\left[\because \left(a-b\right)\left(a+b\right)=a^2-b^2\right] \\ =\frac{2-\sqrt{3}}{4-3} \\ =2-\sqrt{3} \end{gathered}$$
Hence, $\frac{1}{\left(2+\sqrt{3}\right)}=2-\sqrt{3}$.