Math, asked by sri77777, 1 year ago


 \frac{ \sqrt{5} +  \sqrt{3}  }{2 \sqrt{5}  - 3 \sqrt{3}  }  = a - b \sqrt{15}

Answers

Answered by DaIncredible
9
Using the identity

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

 \frac{ \sqrt{5} + \sqrt{3} }{2 \sqrt{5} - 3 \sqrt{3} } = a - b \sqrt{15} \\

L.H.S,

On rationalizing the denominator we get,

 = \frac{ \sqrt{5} + \sqrt{3} }{2 \sqrt{5} - 3 \sqrt{3} } \times \frac{ 2\sqrt{5} + 3 \sqrt{3} }{2 \sqrt{5} + 3 \sqrt{3} } \\ \\ = \frac{ \sqrt{5}(2 \sqrt{5} + 3 \sqrt{3}) + \sqrt{3}(2 \sqrt{5} + 3 \sqrt{3} ) }{ {(2 \sqrt{5} )}^{2} - {(3\sqrt{3}) }^{2} } \\ \\ = \frac{10 + 3 \sqrt{15} + 2 \sqrt{15} + 9 }{20 - 27} \\ \\ = \frac{19 + 5 \sqrt{15} }{ - 7} \\ \\ = \frac{ - 19 - 5 \sqrt{15} }{7}

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

a = - 19/7

b = 5/7
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