Question

4 + 3√5 / 4 - 3√5 = a - b√5​

Answer 1

Answer:

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

Step-by-step explanation:

$\frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } = a - b \sqrt{5}$

L.H.S,

Rationalizing the denominator we get:

$= \frac{4 + 3 \sqrt{5} }{4 - 3 \sqrt{5} } \times \frac{4 + 3 \sqrt{5} }{4 + 3 \sqrt{5} } \\ \\ = \frac{ {(4)}^{2} + {(3 \sqrt{5} ) + 2.4.3 \sqrt{5} } }{ {(4)}^{2} - {(3 \sqrt{5}) }^{2} } \\ \\ = \frac{16 + 45 + 24 \sqrt{5} }{16 - 45 } \\ \\ \bf = \frac{61 + 24 \sqrt{5} }{29}$

Comparing L.H.S and R.H.S we get:

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

Answer 2

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

comparing both sides we get

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