Question

Simplify : $\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}$

Answer 1

$\frac{2+\sqrt{3}}{\sqrt{2}+\sqrt{2+\sqrt{3}}}+\frac{2-\sqrt{3}}{\sqrt{2}-\sqrt{2-\sqrt{3}}}$

$=\frac{2\sqrt{2}+\sqrt{6}}{2+\sqrt{4+2\sqrt{3}}}+\frac{2\sqrt{2}-\sqrt{6}}{2-\sqrt{4-2\sqrt{3}}}$

$=\frac{2\sqrt{2}+\sqrt{6}}{2+1+\sqrt{3}}+\frac{2\sqrt{2}-\sqrt{6}}{2-\sqrt{3}+1}$

$=\frac{2\sqrt{2}+\sqrt{6}}{3+\sqrt{3}}\times \frac{3-\sqrt{3}}{3-\sqrt{3}}+\frac{2\sqrt{2}-\sqrt{6}}{3-\sqrt{3}}\times \frac{3+\sqrt{3}}{3+\sqrt{3}}$

$=\frac{6\sqrt{2}+3\sqrt{6}-2\sqrt{6}-\sqrt{18}+6\sqrt{2}-3\sqrt{6}+2\sqrt{6}-\sqrt{18}}{9-3}$

$=\frac{12\sqrt{2}-2\sqrt{18}}{6}$

$=\frac{6\sqrt{2}-\sqrt{18}}{3}$