Question

find the value of a and b when a+b√15=√5+√3/√5-√3

Answer 1

Heya there !!!
Here is the answer you were looking for:

Identities used :

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


$a + b \sqrt{15} = \frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} }$

In R.H.S

On rationalizing the denominator we get,

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

On comparing we get

$4 + \sqrt{15} = a + b \sqrt{15} \\ \\ a = 4 \: : \: b = 1$

Hope this helps!!!

If you have any doubt regarding to my answer, please ask in the comment section ^_^

@Mahak24

Thanks...
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