Question

Find the value of a and b, when a+b root 15= root 5+root 3upon root 5-root3

Answer 1

Identity used :

${(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

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

R.H.S,

On rationalizing the denominator we get,

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

On comparing L.H.S and R.H.S we get,

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