Question

Answer the question attached.

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Answer 1

$\frac{1}{ \sqrt{9} + \sqrt{7} } + \frac{ 1 }{ \sqrt{7} + \sqrt{5} } + \frac{1}{ \sqrt{5} + \sqrt{3} } + \frac{1}{ \sqrt{3} + \sqrt{1} }$

By rationalising each term, we get:

$\frac{1}{ \sqrt{9} + \sqrt{7} } \times \frac{ \sqrt{9} - \sqrt{7} }{ \sqrt{9} - \sqrt{7} } = \frac{ \sqrt{9} - \sqrt{7} }{2}$

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

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

$\frac{1}{ \sqrt{3} + \sqrt{1} } \times \frac{ \sqrt{3} - \sqrt{1} }{ \sqrt{3} - \sqrt{1} } = \frac{ \sqrt{3} - \sqrt{1} }{2}$

By adding the terms, we get:

$\frac{ \sqrt{9} - \sqrt{7} }{2} + \frac{ \sqrt{7} - \sqrt{5} }{2} + \frac{ \sqrt{5} - \sqrt{3} }{2} + \frac{ \sqrt{3} - \sqrt{1} }{2}$

$= \frac{ \sqrt{9} - \sqrt{7} + \sqrt{7} - \sqrt{5} + \sqrt{5} - \sqrt{3} + \sqrt{3} - \sqrt{1} }{2}$

$= \frac{ \sqrt{9} - \sqrt{1} }{2} \\ = \frac{3 - 1}{2} \\ = \frac{2}{2} = 1$

Hence, Proved

Answer 2

$hope \: it \: helps \: you$

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