Question

root 5 + root 3 / root 5 - root 3

Answer 1

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

On rationalizing the denominator we get,

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

Using the identities :

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

$= \frac{ {( \sqrt{5}) }^{2} + {( \sqrt{3} )}^{2} + 2( \sqrt{5} )( \sqrt{3}) }{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } \\ \\ = \frac{5 + 3 + 2 \sqrt{15} }{5 - 3} \\ \\ = \frac{8 + 2 \sqrt{15} }{2} \\ \\ = 4 + \sqrt{15}$

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

Hope it will help you