Question

Rationalise the denominator of each of the following:
$\frac{4}{\left(\sqrt{7}+\sqrt{3}\right)}$

Answer 1

On multiplying the numerator and denominator of the given number by $\sqrt{7}-\sqrt{3}$, we get:
$$\begin{gathered} \frac{4}{\sqrt{7}+\sqrt{3}}=\frac{4}{\sqrt{7}+\sqrt{3}}\times \frac{\sqrt{7}-\sqrt{3}}{\sqrt{7}-\sqrt{3}} \\ =\frac{4\left(\sqrt{7}-\sqrt{3}\right)}{{\left(\sqrt{7}\right)}^2-{\left(\sqrt{3}\right)}^2} \\ =\frac{4\left(\sqrt{7}-\sqrt{3}\right)}{7-3} \\ =\frac{4\left(\sqrt{7}-\sqrt{3}\right)}{4} \\ =\sqrt{7}-\sqrt{3} \end{gathered}$$