Rationalising a binomial denominator

Text

Rationalise: $\frac{4}{\sqrt{7}-\sqrt{3}}.$

Solution

Multiply by the conjugate $\sqrt7+\sqrt3$: $\frac{4(\sqrt7+\sqrt3)}{(\sqrt7-\sqrt3)(\sqrt7+\sqrt3)}=\frac{4(\sqrt7+\sqrt3)}{7-3}=\frac{4(\sqrt7+\sqrt3)}{4}=\sqrt7+\sqrt3.$ $\boxed{\,\sqrt{7}+\sqrt{3}\,}$