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Definition of Relations

Simplify:12 +...

Question

Simplify:

$\frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}}$

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Solution

$\frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}}$

Rationalising the denominators,

$= \frac{2 - \sqrt{3}}{(2 + \sqrt{3}) (2 - \sqrt{3}}) + \frac{2 (\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})} + \frac{2 + \sqrt{5}}{(2 - \sqrt{5}) (2 + \sqrt{5})}$

$= \frac{2 - \sqrt{3}}{2^{2} - (\sqrt{3})^{2}} + \frac{2 (\sqrt{5} + \sqrt{3})}{(\sqrt{5})^{2} - (\sqrt{3})^{2}} + \frac{2 + \sqrt{5}}{2^{2} - (\sqrt{5})^{2}}$

[ using identity, $a^{2} - b^{2} = (a - b) (a + b)$ ]

$= \frac{2 - \sqrt{3}}{4 - 3} + \frac{2 (\sqrt{5} + \sqrt{3})}{5 - 3} + \frac{2 + \sqrt{5}}{4 - 5}$

$= \frac{2 - \sqrt{3}}{1} + \frac{2 (\sqrt{5} + \sqrt{3})}{2} + \frac{2 + \sqrt{5}}{- 1}$

$= 2 - \sqrt{3} + \sqrt{5} + \sqrt{3} - (2 + \sqrt{5})$

$= 2 - \sqrt{3} + \sqrt{5} + \sqrt{3} - 2 - \sqrt{5}$

$= 0$

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Definition of Relations

Standard XII Mathematics

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