Question

simplify =1/√5+√3 + 1/2(√5-√3)

Answer 1

Hey friend,
Here is the answer you were looking for:
$\frac{1}{ \sqrt{5} + \sqrt{3} } + \frac{1}{2( \sqrt{5} - \sqrt{3} ) } \\ \\ = \frac{1}{ \sqrt{5} + \sqrt{3} } + \frac{1}{2 \sqrt{5} - 2 \sqrt{3} }$

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 :

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

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

Hope this helps!!

If you have any doubt regarding to my answer, feel free to ask in the comment section or inbox me if needed.

@Mahak24

Thanks...
☺☺

Answer 2

Linked in image

It's really a short method

HOPE it helps