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Addition and Subtraction of Like and Unlike Surds

Simplify:2√5 ...

Question

Simplify:

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

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Solution

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

Rationalising the denominators,

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

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

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

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

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

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

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

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

$= 0$

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Addition and Subtraction of Like and Unlike Surds

Standard IX Mathematics

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