Question

1 upon 3 + root 7 + 1 upon root 7 + root 5 + 1 upon root 5 + root 3 + 1 upon root 3 + 1

Answer 1

Hola User!!

Your answer :-

$= \frac{1}{3 + \sqrt{7} } + \frac{1}{ \sqrt{7} + \sqrt{5} } + \frac{1}{ \sqrt{5} + \sqrt{3} } + 1 \\ \\ = \frac{1}{3 + \sqrt{7} } \times \frac{3 - \sqrt{7} }{3 - \sqrt{7} } + \frac{1}{ \sqrt{7} + \sqrt{5} } \times \frac{ \sqrt{7} - \sqrt{5} }{ \sqrt{7} - \sqrt{5} } + \frac{1}{ \sqrt{5} + \sqrt{3} } \times \frac{ \sqrt{5} - \sqrt{3} }{ \sqrt{5} - \sqrt{3} } + 1 \\ \\ = \frac{3 - \sqrt{7} }{ {(3)}^{2} - {( \sqrt{7}) }^{2} } + \frac{ \sqrt{7} - \sqrt{5} }{ { (\sqrt{7}) }^{2} - {( \sqrt{5}) }^{2} } + \frac{ \sqrt{5} - \sqrt{3} }{ {(\sqrt{5}) }^{2} - {( \sqrt{3} )}^{2} } + 1\\ \\ = \frac{3 - \sqrt{7} }{9 - 7} + \frac{ \sqrt{7} - \sqrt{5} }{7 - 5} + \frac{ \sqrt{5} - \sqrt{3} }{5 - 3} + 1 \\ \\ = \frac{3 - \sqrt{7} + \sqrt{7} - \sqrt{5} + \sqrt{5} - \sqrt{3} }{2} + 1 \\ \\ = \frac{3 - \sqrt{3} }{2} + 1 \\ \\ = \frac{3 - \sqrt{3} + 2}{2} \\ \\ = \frac{5 - \sqrt{3} }{2}$


Hope it helps!!

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