Question

Rationalize the denominator of $\frac{4}{2+\sqrt{3}+\sqrt{7}}$

Answer 1

$\frac{4}{2+\sqrt{3}+\sqrt{7}}$

$=\frac{4}{(2+\sqrt{3})+\sqrt{7}}\times \frac{(2+\sqrt{3})-\sqrt{7}}{(2+\sqrt{3})-\sqrt{7}}$

$=\frac{4\left[\right(2+\sqrt{3})-\sqrt{7}]}{(2+\sqrt{3}{)}^2-(\sqrt{7}{)}^2}$

$=\frac{4\left[\right(2+\sqrt{3})-\sqrt{7}]}{4+4\sqrt{3}+3-7}$

$=\frac{4\left[\right(2+\sqrt{3})-\sqrt{7}]}{4\sqrt{3}}$

$=\frac{\sqrt{3}\left[\right(2+\sqrt{3})-\sqrt{7}]}{3}$