Question

Simplify: $1+\frac{4\sqrt{3}}{2-\sqrt{2}}-\frac{30}{4\sqrt{3}-\sqrt{18}}-\frac{\sqrt{18}}{3+2\sqrt{3}}$

• A. $0$
• B. $1$
• C. $\sqrt{2}$
• D. $\sqrt{3}$

Answer 1

The correct option is B $1$

$1+\frac{4\sqrt{3}}{2-\sqrt{2}}-\frac{30}{4\sqrt{3}-\sqrt{18}}-\frac{\sqrt{18}}{3+2\sqrt{3}}$

$=1+\frac{4\sqrt{3}}{2-\sqrt{2}}-\frac{30}{4\sqrt{3}-3\sqrt{2}}-\frac{3\sqrt{2}}{3+2\sqrt{3}}$

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

$=1+\frac{4(2\sqrt{3}+\sqrt{6})}{(2{)}^2-(\sqrt{2}{)}^2}-\frac{30(4\sqrt{3}+3\sqrt{2})}{\left(4\sqrt{3}{)}^2\right(3\sqrt{2}{)}^2}$ $-\frac{3(3\sqrt{2}-2\sqrt{6})}{3^3-(2\sqrt{3}{)}^2}$

$=1+\frac{4(2\sqrt{3}+\sqrt{6})}{4-2}-\frac{30(4\sqrt{3}+3\sqrt{2})}{48-18}$$-\frac{3(3\sqrt{2}-2\sqrt{6})}{9-12}$

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

$=1+2(2\sqrt{3}+\sqrt{6})-(4\sqrt{3}+3\sqrt{2})+(3\sqrt{2}-2\sqrt{6})$

$=1+4\sqrt{3}+2\sqrt{6}-4\sqrt{3}-3\sqrt{2}+3\sqrt{2}-2\sqrt{6}$

$=1$