Question

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Answer 1

Step-by-step explanation:

15)

${( {2})^{2} - (\sqrt{5}) {}^{2} } </p><p>\frac{1}{2 + \sqrt{3} } + \frac{1}{ \sqrt{5} - \sqrt{ 3 } } + \frac{1}{2 - \sqrt{5} } = \frac{1}{2 + \sqrt{3} } \times \frac{2 - \sqrt{3} }{2 - \sqrt{3} } + \: \frac{1}{ \sqrt{5} - \sqrt{3} } \times \frac{ \sqrt{5 } + \sqrt{3} }{ \sqrt{5} + \sqrt{3} } + \frac{1}{2 - \sqrt{5} } \times \frac{2 + \sqrt{5} }{2 + \sqrt{5} } \\ = \frac{2 - \sqrt{3} }{(2) {}^{2} - ( { { \sqrt{3} )}^{2} } } + \frac{ \sqrt{5} + \sqrt{3} }{ ({ \sqrt{5} )}^{2} - (\sqrt{3}) {}^{2} } + \frac{2 + \sqrt{5} }{( {2})^{2} - (\sqrt{5}) {}^{2} } \\ = \frac{2 - \sqrt{3} }{4 - 3 } + \frac{ \sqrt{5} + \sqrt{3} }{5 - 3} + \frac{2 + \sqrt{5} }{4 - 5} \\ = \frac{2 - \sqrt{3} }{1} + \frac{ \sqrt{5} + \sqrt{3} }{2} + \frac{2 + \sqrt{5} }{ - 1} \\ = lets \: take \: lhs \\ = \frac{ - 4 + 2 \sqrt{3} - \sqrt{5} - \sqrt{3} + 4 + 2 \sqrt{5} }{ - 2} \\ = \frac{ \sqrt{3} - \sqrt{5} }{ - 2}$