Question

( rationalise the denominator ) Find the values of the rational numbers a and b if a + b = √15

Answer 1

Hope it helps..........
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Answer 2

Heya !!!

Identities used :

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


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

L.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 ☺