Question

simplify:- 1/√5+√3 + 1/2 (√5 - √3)​

Answer 1

Answer:

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Step-by-step explanation:

1/√5+√3+√5/2-√3/2

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2+2√15+6-√15/2√5

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8+√15/2√5 is correct answer

Answer 2

$\begin{gathered} \frac{1}{ \sqrt{5} + \sqrt{3} } + \frac{1}{2( \sqrt{5} - \sqrt{3} ) } \\ \\ = \frac{1}{ \sqrt{5} + \sqrt{3} } + \frac{1}{2 \sqrt{5} - 2 \sqrt{3} } \\ \end{gathered}$

On rationalizing the denominator we get,

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

Using the identity :

$(x + y)(x - y) = {x}^{2} - {y}^{2}$

$\begin{gathered} = \frac{ \sqrt{5} - \sqrt{3} }{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } - \frac{2 \sqrt{5} + 2 \sqrt{3} }{ {(2 \sqrt{5} })^{2} - {(2 \sqrt{3} )}^{2} } \\ \\ = \frac{ \sqrt{5} - \sqrt{3} }{5 - 3} - \frac{2 \sqrt{5} + 2 \sqrt{3} }{20 - 12} \\ \\ = \frac{ \sqrt{5} - \sqrt{3} }{2} - \frac{2 \sqrt{5} + 2 \sqrt{3} }{8} \\ \\ = \frac{ \sqrt{5} - \sqrt{3} }{2} - \frac{ \sqrt{5} + \sqrt{3} }{4} \\ \\ = \frac{ \sqrt{5} \times 2 - \sqrt{3} \times 2 - ( \sqrt{5} + \sqrt{3} )}{4} \\ \\ = \frac{2 \sqrt{5} - 2 \sqrt{3} - \sqrt{5} - \sqrt{3} }{4} \\ \\ = \frac{ \sqrt{5} - 3 \sqrt{3} }{4} \end{gathered}$