Question

$\sqrt{5 \: \: + \sqrt{3 \div \sqrt{5 - \sqrt{3 = a + b \sqrt{15} } } } }$
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Answer 1

Correct Question: If √5 + √3/√5 - √3 = a + b√15, find the value of a and b.

Answer:

$\large{\underline{\boxed{\sf a = 4 \: \: and \: \: b = 1}}}$

Step-by-step explanation:

Given that-

$\implies \sf \dfrac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = a + b \sqrt{15}$

Consider LHS:

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

On comparing the LHS with RHS , we get-

$\implies \sf 4 + \sqrt{15} = a + \sqrt{15} \\ \\ \boxed{\bf \therefore \: a = 4 \: \: and \: \: b = 1}$