Question

Rationalise the denominator of each of the following:
$\frac{1}{\left(2+\sqrt{3}\right)}$

Answer 1

On multiplying the numerator and denominator of the given number by $2-\sqrt{3}$, we get:
$$\begin{gathered} \frac{1}{2+\sqrt{3}}=\frac{1}{2+\sqrt{3}}\times \frac{2-\sqrt{3}}{2-\sqrt{3}} \\ =\frac{2-\sqrt{3}}{{\left(2\right)}^2-{\left(\sqrt{3}\right)}^2} \\ =\frac{2-\sqrt{3}}{4-3} \\ =\frac{2-\sqrt{3}}{1} \\ =2-\sqrt{3} \end{gathered}$$