Question

1/root5+root3 + 1/2^(root5-root3)

Answer 1

answer of these question is 1

Answer 2

Step-by-step explanation:

$\bf \underline{Solution-}$

$\textsf{Given that-}$

$\sf{ \frac{1}{ \sqrt{5} + \sqrt{3} } + \frac{1}{2} ( \sqrt{5} - \sqrt{3} ) }$

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

$\sf{ = \frac{1( \sqrt{5} - \sqrt{3} )}{( \sqrt{5} + \sqrt{3} )( \sqrt{5} - \sqrt{3}) } + \frac{1}{2}( \sqrt{5} - \sqrt{3} ) }$

$\sf{ = \frac{ \sqrt{5} - \sqrt{3} }{( \sqrt{5} + \sqrt{3} )( \sqrt{5} - \sqrt{3}) } + \frac{1}{2}( \sqrt{5} - \sqrt{3} ) }$

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

$\sf{ = \frac{ \sqrt{5} - \sqrt{3} }{5- 3 } + \frac{1}{2}( \sqrt{5} - \sqrt{3} ) }$

$\sf{ = \frac{ \sqrt{5} - \sqrt{3} }{2 } + \frac{1}{2}( \sqrt{5} - \sqrt{3} ) }$

$\sf{ = \frac{1}{2} ( \sqrt{5} - \sqrt{3} ) + \frac{1}{2} ( \sqrt{5} - \sqrt{3} )}$

$\sf{ = \bigg( \frac{1}{2} + \frac{1}{2} \bigg) \Big( \sqrt{5} - \sqrt{3} \Big)}$

$\sf{ = 1 \times \Big( \sqrt{5} - \sqrt{3} \Big)}$

$\sf{ = \Big( \sqrt{5} - \sqrt{3} \Big)} \: \: \: \bf{Ans.}$

$\textsf{Hope this helps.}$