Question

Write the reciprocal of $\left(2+\sqrt{3}\right)$.

Answer 1

The reciprocal of $\left(2+\sqrt{3}\right)$

$$\begin{gathered} =\frac{1}{\left(2+\sqrt{3}\right)} \\ =\frac{1}{\left(2+\sqrt{3}\right)}\times \frac{\left(2-\sqrt{3}\right)}{\left(2-\sqrt{3}\right)} \\ =\frac{\left(2-\sqrt{3}\right)}{\left(2+\sqrt{3}\right)\left(2-\sqrt{3}\right)} \\ =\frac{\left(2-\sqrt{3}\right)}{2^2-{\left(\sqrt{3}\right)}^2}\left[\left(a+b\right)\left(a-b\right)=a^2-b^2\right] \end{gathered}$$

$$\begin{gathered} =\frac{\left(2-\sqrt{3}\right)}{4-3} \\ =\frac{\left(2-\sqrt{3}\right)}{1} \\ =\left(2-\sqrt{3}\right) \end{gathered}$$