Question

50 points..

$show \: that \: \frac{4 - \sqrt{5} }{4 + \sqrt{5} } + \frac{2}{5 + \sqrt{3} } + \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{2}{5 - \sqrt{3} } = \frac{52}{11}$
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Answer 1

$</p><p>to \: show \: that \: \frac{4 - \sqrt{5} }{4 + \sqrt{5} } + \frac{2}{5 + \sqrt{3} } + \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{2}{5 - \sqrt{3} } = \frac{52}{11} \\ \\ \frac{4 - \sqrt{5} }{4 + \sqrt{5} } \\ \\ = \frac{(4 - \sqrt{5} )}{(4 + \sqrt{5} )} \frac{(4 - \sqrt{5} )}{(4 - \sqrt{5} )} \\ \\ = \frac{ {(4 - \sqrt{5}) }^{2} }{16 - 5} \\ \\ = \frac{21 - 8 \sqrt{5} }{11} \\ \\ \ \frac{4 + \sqrt{5} }{4 - \sqrt{5} } \\ \\ = \frac{(4 + \sqrt{5} )}{(4 - \sqrt{5} )} \frac{(4 + \sqrt{5} )}{(4 + \sqrt{5} )} \\ \\ = \frac{ {(4 + \sqrt{5} )}^{2} }{16 - 5} \\ \\ = \frac{21 + 8 \sqrt{5} }{11} \\ \\ \frac{2}{5 + \sqrt{3} } \\ \\ = \frac{2}{(5 + \sqrt{3} )} \frac{(5 - \sqrt{3}) }{(5 - \sqrt{3}) } \\ \\ = \frac{10 - 2 \sqrt{3} }{25 - 3} \\ \\ = \frac{10 - 2 \sqrt{3} }{22} \\ \\ = \frac{5 - \sqrt{3} }{11} \\ \\ \frac{2}{5 - \sqrt{3} } \\ \\ = \frac{2}{(5 - \sqrt{3} )} \frac{(5 + \sqrt{3}) }{(5 + \sqrt{3}) } \\ \\ = \frac{10 + 2 \sqrt{3} }{25 - 3} \\ \\ = \frac{10 + 2 \sqrt{3} }{22} \\ \\ = \frac{5 + \sqrt{3} }{11} \\ \\ now \\ \\ \frac{4 - \sqrt{5} }{4 + \sqrt{5} } + \frac{2}{5 + \sqrt{3} } + \frac{4 + \sqrt{5} }{4 - \sqrt{5} } + \frac{2}{5 - \sqrt{3} } \\ \\ = \frac{21 - 8 \sqrt{5} }{11} + \frac{5 - \sqrt{3} }{11} + \frac{21 + 8 \sqrt{5} }{11} + \frac{5 + \sqrt{3} }{11} \\ \\ = \frac{21 - 8 \sqrt{5} + 5 - \sqrt{3} + 21 + 8 \sqrt{5} + 5 + \sqrt{3} }{11} \\ \\ = \frac{52}{11}$

hope it helps

Answer 2

Here is your answer ❤️❤️☺️☺️

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Hope this helps you ❤️❤️☺️☺️