Question

Simplify: $\frac{4\sqrt{3}}{2-\sqrt{2}}-\frac{30}{4\sqrt{3}-3\sqrt{2}}-\frac{3\sqrt{2}}{3+2\sqrt{3}}$

Answer 1

Consider the given expression.

$=\frac{4\sqrt{3}}{2-\sqrt{2}}-\frac{30}{4\sqrt{3}-3\sqrt{2}}-\frac{3\sqrt{2}}{3+2\sqrt{2}}$

$=\frac{4\sqrt{3}}{2-\sqrt{2}}\times \frac{2+\sqrt{2}}{2+\sqrt{2}}-\frac{30}{4\sqrt{3}-3\sqrt{2}}\times \frac{4\sqrt{3}+3\sqrt{2}}{4\sqrt{3}+3\sqrt{2}}-\frac{3\sqrt{2}}{3+2\sqrt{2}}\times \frac{3-2\sqrt{2}}{3-2\sqrt{2}}$

$=\frac{8\sqrt{3}+4\sqrt{6}}{4-2}-\frac{30(4\sqrt{3}+3\sqrt{2})}{48-18}-\frac{9\sqrt{2}-12}{9-8}$

$=\frac{2(4\sqrt{3}+2\sqrt{6})}{2}-\frac{30(4\sqrt{3}+3\sqrt{2})}{30}-\frac{9\sqrt{2}-12}{1}$

$=4\sqrt{3}+2\sqrt{6}-4\sqrt{3}-3\sqrt{2}-9\sqrt{2}+12$

$=12-12\sqrt{2}+2\sqrt{6}$

Hence, the value is $(12-12\sqrt{2}+2\sqrt{6})$.