Question

If √5+√3÷√5-√3=a+√15b,find a and b

Answer 1

√5+√3/√5-√3=a+√15b

LHS=

√5+√3/√5-√3

=√5+√3/√5-√3 × √5+√3/√5+√3

=(√5+√3)²/(√5)² -(√3)²

(√5)² + 2×√5×√3 + (√3)²/5-3

5+2√15+3/2

5+3+2√15/2

8+2√15/2

ON COMPARING,

a=8/2

= 4

b=2/2

=1

hence,

√5+√3/√5-√3=4+1√15

Answer 2

$\frac{ \sqrt{5} + \sqrt{3} }{ \sqrt{5} - \sqrt{3} } = a + \sqrt{15}b \\ \\ \\ rationalising \: \: LHS \\ \\ \frac{ \sqrt{5} + \sqrt{3}( \sqrt{5} + \sqrt{3} ) }{ {( \sqrt{5} )}^{2} - {( \sqrt{3} )}^{2} } \\ \\ \frac{5 + 3 + 2 \sqrt{15} }{5 - 3} \\ \\ \frac{8 + 2 \sqrt{15} }{2} \\ \\ \frac{2(4 + \sqrt{15} )}{2} \\ \\ 4 + \sqrt{15}$

Hence,

$a = 4 \: \: \: \: \: \: \: \: \& \: \: \: \: b = 1$