Question

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Answer 1

Question:-

Simplify the following:-

• $\sf{\dfrac{4\sqrt{3}}{2 - \sqrt{2}} - \dfrac{30}{4\sqrt{3} - \sqrt{18}} - \dfrac{\sqrt{18}}{3 - \sqrt{12}}}$

Solution:-

Let's solve the each terms separately.

For 1st term,

$\sf{\dfrac{4\sqrt{3}}{2 - \sqrt{2}}}$

By Rationalizing the denominator:-

$= \sf{\dfrac{4\sqrt{3}}{2 - \sqrt{2}} \times \dfrac{2 + \sqrt{2}}{2 + \sqrt{2}}}$

$= \sf{\dfrac{4\sqrt{3}(2 + \sqrt{2})}{(2)^2 - (\sqrt{2})^2}}$

$= \sf{\dfrac{4\sqrt{3}(2 + \sqrt{2})}{4 - 2}}$

$= \sf{\dfrac{4\sqrt{3}(2 + \sqrt{2})}{2}}$

$= \sf{\dfrac{\not{4}\sqrt{3}(2 + \sqrt{2})}{\not{2}}}$

$= \sf{2\sqrt{3}(2 + \sqrt{2})}$

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

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

For 2nd term,

$\sf{\dfrac{30}{4\sqrt{3} - \sqrt{18}}}$

By rationalizing the denominator:-

$= \sf{\dfrac{30}{4\sqrt{3} - \sqrt{18}} \times \dfrac{4\sqrt{3} + \sqrt{18}}{4\sqrt{3} + \sqrt{18}}}$

$= \sf{\dfrac{30(4\sqrt{3} + \sqrt{18})}{(4\sqrt{3})^2 - (\sqrt{18})^2}}$

$= \sf{\dfrac{30(4\sqrt{3} + \sqrt{18})}{48 - 18}}$

$= \sf{\dfrac{30(4\sqrt{3} + \sqrt{18})}{30}}$

$= \sf{\dfrac{\not{30}(4\sqrt{3} + \sqrt{18})}{\not{30}}}$

$= \sf{4\sqrt{3} + 3\sqrt{2}}$

$\therefore \sf{\dfrac{30}{4\sqrt{3} - \sqrt{18}} = 4\sqrt{3} + 3\sqrt{2}}$

For 3rd term,

$\sf{\dfrac{\sqrt{18}}{3 - \sqrt{12}}}$

By Rationalizing the denominator:-

$= \sf{\dfrac{\sqrt{18}}{3 - \sqrt{12}} \times \dfrac{3 + \sqrt{12}}{3 + \sqrt{12}}}$

$= \sf{\dfrac{\sqrt{18}(3 + \sqrt{12})}{(3)^2 - (\sqrt{12})^2}}$

$= \sf{\dfrac{\sqrt{18}(3 + \sqrt{12})}{9 - 12}}$

$= \sf{\dfrac{\sqrt{18}(3 + \sqrt{12})}{-3}}$

$= \sf{\dfrac{3\sqrt{2}(3 + \sqrt{12})}{-3}}$

$= \sf{\dfrac{\not{3\sqrt{2}}(3 + \sqrt{12})}{\not{-3}}}$

$= \sf{-\sqrt{2}(3 + 2\sqrt{3})}$

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

$\therefore \sf{\dfrac{\sqrt{18}}{3 - \sqrt{12}} = -3\sqrt{2} - 2\sqrt{6}}$

Finally:-

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

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

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

$\boxed{\underline{\sf{4\sqrt{6}\:\:\:[Answer]}}}$

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