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Simplify: $ \frac{4 \sqrt{3}}{2-\sqrt{2}}-\frac{30}{4 \sqrt{3}-\sqrt{18}}-\frac{\sqrt{18}}{3-\sqrt{12}} $
Given:
\( \frac{4 \sqrt{3}}{2-\sqrt{2}}-\frac{30}{4 \sqrt{3}-\sqrt{18}}-\frac{\sqrt{18}}{3-\sqrt{12}} \)
To do:
We have to simplify the given expression.
Solution:
$\frac{4 \sqrt{3}}{(2-\sqrt{2})}-\frac{30}{(4 \sqrt{3}-3 \sqrt{2})}-\frac{3 \sqrt{2}}{(3+2 \sqrt{3})}$
Rationalizing the denominators of each term, we get,
$\frac{4 \sqrt{3}}{(2-\sqrt{2})}-\frac{30}{(4 \sqrt{3}-3 \sqrt{2})}-\frac{3 \sqrt{2}}{(3+2 \sqrt{3})}=\frac{4 \sqrt{3}(2+\sqrt{2})}{(2-\sqrt{2})(2+\sqrt{2})}-\frac{30(4 \sqrt{3}+3 \sqrt{2})}{(4 \sqrt{3}-3 \sqrt{2})(4 \sqrt{3}+3 \sqrt{2})}-\frac{3 \sqrt{2}(3-2 \sqrt{3})}{(3+2 \sqrt{3})(3-2 \sqrt{3})}$
$=\frac{8 \sqrt{3}+4 \sqrt{6}}{4-2}-\frac{120 \sqrt{3}+90 \sqrt{2}}{48-18}-\frac{9 \sqrt{2}-6 \sqrt{6}}{9-12}$
$=\frac{8 \sqrt{3}+4 \sqrt{6}}{2}-\frac{120 \sqrt{3}+90 \sqrt{2}}{30}-\frac{9 \sqrt{2}-6 \sqrt{6}}{-3}$
$=\frac{8 \sqrt{3}+4 \sqrt{6}}{2}-\frac{120 \sqrt{3}+90 \sqrt{2}}{30}-\frac{9 \sqrt{2}-6 \sqrt{6}}{3}$
$=4 \sqrt{3}+2 \sqrt{6}-4 \sqrt{3}-3 \sqrt{2}+3 \sqrt{2}-2 \sqrt{6}$
$=0$ .
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