Question

Simplify
$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

Answer 1

$\frac{2\sqrt{6}}{\sqrt{2}+\sqrt{3}}+\frac{6\sqrt{2}}{\sqrt{6}+\sqrt{3}}-\frac{8\sqrt{3}}{\sqrt{6}+\sqrt{2}}$

$=\frac{2\sqrt{6}(\sqrt{3}-\sqrt{2})}{(\sqrt{3}+\sqrt{2})(\sqrt{3}-\sqrt{2})}+\frac{6\sqrt{2}(\sqrt{6}-\sqrt{3})}{(\sqrt{6}+\sqrt{3})(\sqrt{6}-\sqrt{3})}-\frac{8\sqrt{3}(\sqrt{6}-\sqrt{2})}{(\sqrt{6}+\sqrt{2})(\sqrt{6}-\sqrt{2})}$

$=\frac{2\sqrt{3\times 2\times 3}-2\sqrt{3\times 2\times 2}}{3-2}+\frac{6\sqrt{2\times 3\times 2}-6\sqrt{2\times 3}}{6-3}-\frac{8(\sqrt{3\times 3\times 2}-\sqrt{3\times 2})}{6-2}$

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

$=0$