Question

find the value of a and b when 4+3√5/4-3√5=a+b√5​

Answer 1

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a + b \sqrt{5} \\ \\ \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } \times \frac{4 + 3 \sqrt{5} }{4 + 3 \sqrt{5} } = a + b \sqrt{5} \\ \\ \frac{16 + 45 + 24 \sqrt{5} }{16 - 45} = a + b \sqrt{5} \\ \\ \frac{61 + 24 \sqrt{5} }{ - 29} = a + b \sqrt{5} \\ \\ - \frac{61}{29} - \frac{24}{29} \sqrt{5} = a + \sqrt{5}$

comparing both sides we get

$a = - \frac{61}{29} \: \: \: \: and \: \: \: b = - \frac{24}{29}$