Question

Prove that
$\frac{1}{3 + \sqrt{7} } + \frac{1}{ \sqrt{7 + \sqrt{5} } } + \frac{1}{ \sqrt{5 + \sqrt{3} } } + \frac{1}{ \sqrt{3 + 1} } = 1$
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Answer 1

$\huge{GIVEN} =$

$\frac{1}{3 + \sqrt{7} } + \frac{1}{ \sqrt{7 + \sqrt{5} } } + \frac{1}{ \sqrt{5 + \sqrt{3} } } + \frac{1}{ \sqrt{3 + 1} } = 1</p><p>$

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$\frac{3 - \sqrt{7} }{2} + \frac{1}{ \sqrt{7 + \sqrt{5} } } + \frac{ \sqrt{5 - \sqrt{3} } }{2} + \frac{1}{ \sqrt{4} }$

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$\frac{3 - \sqrt{7} }{2} + \frac{1}{ \sqrt{7 + \sqrt{5} } } + \frac{ \sqrt{5 - \sqrt{3} } }{2} + \frac{1}{2}$

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$\frac{3 - \sqrt{7 + \sqrt{5 - \sqrt{3 + 1} } } }{2} + \frac{ \sqrt{7 + \sqrt{5} } }{7 + \sqrt{5} }$

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$\frac{4 - \sqrt{7 + \sqrt{5 - \sqrt{3} } } }{2} + \frac{ \sqrt{7 + \sqrt{5(7 - \sqrt{5)} } } }{44}$

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$\frac{4 - \sqrt{7 + \sqrt{5 - \sqrt{3} } } }{2} + \frac{7 \sqrt{7 + \sqrt{5 - \sqrt{(7 + \sqrt{5) \times 5} } } } }{44}$

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SOLUTION :-

$\frac{4 - \sqrt{7 + \sqrt{5 - \sqrt{3} } } }{2} + \frac{ 7 \sqrt{7 + \sqrt{5 - \sqrt{35 + 5 \sqrt{5} } } } }{44}$

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HENCE YOUR ANSWER IN DECIMAL :-

$\huge1.25818$

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--------SOLVED-------