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Rationalisation

Simplify: (i)...

Question

Simplify:

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

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

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

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Solution

(i) $\frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} + \frac{\sqrt{5} - \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{(\sqrt{5} + \sqrt{3})^{2} + (\sqrt{5} - \sqrt{3})^{2}}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})} = \frac{2 [(\sqrt{5})^{2} + (\sqrt{3})^{2}]}{(\sqrt{5})^{2} - (\sqrt{3})^{2}} {∵ (a + b)^{2} + (a - b)^{2} = 2 (a^{2} + b^{2})} = \frac{2 (5 + 3)}{5 - 3} = \frac{2 \times 8}{2} = 8$

(ii) $\frac{1}{2 + \sqrt{3}} = \frac{1 (2 - \sqrt{3})}{(2 + \sqrt{3}) (2 - \sqrt{3})} = \frac{1 (2 - \sqrt{3})}{(2)^{2} - (\sqrt{3})^{2}} = \frac{2 - \sqrt{3}}{4 - 3} = \frac{2 - \sqrt{3}}{1} = 2 - \sqrt{3} \frac{2}{\sqrt{5} - \sqrt{3}} = \frac{2 (\sqrt{5} + \sqrt{3})}{(\sqrt{5} - \sqrt{3}) (\sqrt{5} + \sqrt{3})} = \frac{2 (\sqrt{5} + \sqrt{3})}{(\sqrt{5})^{2} - (\sqrt{3})^{2}} = \frac{2 (\sqrt{5} + \sqrt{3})}{5 - 3} = \frac{2 (\sqrt{5} + \sqrt{3})}{2} = \sqrt{5} + \sqrt{3} \frac{1}{2 - \sqrt{5}} = \frac{1 (2 + \sqrt{5})}{(2 - \sqrt{5}) (2 + \sqrt{5})} = \frac{2 + \sqrt{5}}{(2)^{2} - (\sqrt{5})^{2}} = \frac{2 + \sqrt{5}}{4 - 5} = \frac{2 + \sqrt{5}}{- 1} = (- 2 - \sqrt{5}) N o w, \frac{1}{2 + \sqrt{3}} + \frac{2}{\sqrt{5} - \sqrt{3}} + \frac{1}{2 - \sqrt{5}} = (2 - \sqrt{3}) + (\sqrt{5} + \sqrt{3}) + (- 2 - \sqrt{5}) = 2 - \sqrt{3} + \sqrt{5} + \sqrt{3} - 2 - \sqrt{5} = 0$

(iii) $\frac{2}{\sqrt{5} + \sqrt{3}} = \frac{2 (\sqrt{5} - \sqrt{3})}{(\sqrt{5} + \sqrt{3}) (\sqrt{5} - \sqrt{3})} = \frac{2 (\sqrt{5} - \sqrt{3})}{(\sqrt{5})^{2} - (\sqrt{3})^{2}} = \frac{2 (\sqrt{5} - \sqrt{3})}{5 - 3} = \frac{2 (\sqrt{5} - \sqrt{3})}{2} = \sqrt{5} - \sqrt{3} \frac{1}{\sqrt{3} + \sqrt{2}} = \frac{1 \times (\sqrt{3} - \sqrt{2})}{(\sqrt{3} + \sqrt{2}) (\sqrt{3} - \sqrt{2})} = \frac{\sqrt{3} - \sqrt{2}}{(\sqrt{3})^{2} - (\sqrt{2})^{2}} = \frac{\sqrt{3} - \sqrt{2}}{3 - 2} = \frac{\sqrt{3} - \sqrt{2}}{1} = \sqrt{3} - \sqrt{2} \frac{3}{\sqrt{5} + \sqrt{2}} = \frac{3 (\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2}) (\sqrt{5} - \sqrt{2})} = \frac{3 (\sqrt{5} - \sqrt{2})}{(\sqrt{5})^{2} - (\sqrt{2})^{2}} = \frac{3 (\sqrt{5} - \sqrt{2})}{5 - 2} = \frac{3 (\sqrt{5} - \sqrt{2})}{3} = \sqrt{5} - \sqrt{2} N o w, \frac{2}{\sqrt{2} + \sqrt{3}} + \frac{1}{\sqrt{3} + \sqrt{2}} - \frac{3}{\sqrt{5} + \sqrt{2}} = \sqrt{5} - \sqrt{3} + \sqrt{3} - \sqrt{2} - (\sqrt{5} - \sqrt{2}) = \sqrt{5} - \sqrt{3} + \sqrt{3} - \sqrt{2} - \sqrt{5} + \sqrt{2} = 0$

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Q. Simplify:

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\frac{5 + \sqrt{3}}{5 - \sqrt{3}} + \frac{5 - \sqrt{3}}{5 + \sqrt{3}}

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