Question

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

Answer 1

Rationalize the elements individually,

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

Using $\left(a+b\right)\left(a-b\right)=a^2-b^2$

$\frac{6\left(2\sqrt{3}+\sqrt{6}\right)}{{\left(2\sqrt{3}\right)}^2-{\left(\sqrt{6}\right)}^2}+\frac{\sqrt{6}\left(\sqrt{3}-\sqrt{2}\right)}{{\left(\sqrt{3}\right)}^2-{\left(\sqrt{2}\right)}^2}-\frac{4\sqrt{3}\left(\sqrt{6}+\sqrt{2}\right)}{{\left(\sqrt{6}\right)}^2-{\left(\sqrt{2}\right)}^2}$

$=\frac{12\sqrt{3}+6\sqrt{6}}{12-6}+\frac{3\sqrt{2}-2\sqrt{3}}{3-2}-\frac{12\sqrt{2}+4\sqrt{6}}{6-2}$

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

$=0$