Question

Root 5 + root 3 by 2 root 5 minus 3 root 3 is equal to a minus b root 15

Answer 1

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Answer 2

Given :

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

Let's simplify L.H.S by rationalising it's denominator.

\frac{ \sqrt{5} + \sqrt{3} }{2 \sqrt{5} - 3 \sqrt{3} } \\ \\ \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}

Now,

\frac{ - 19}{7} + \frac{5 }{7} \sqrt{15}

On comparing the result of L.H.S from R.H.S , we get ;

\bold{a = \frac{ - 19}{7} \: \: and \: \: b = \frac{5}{7} }