Question

determine a and b, if 4+3√5 divided by 4-3√5 = a+b√5​

Answer 1

Answer:

$a = -\frac{61}{29}\\\\b = -\frac{21}{29}$

Step-by-step explanation:

⇒ $\frac{4+3\sqrt{5}}{4-3\sqrt{5}}$

Rationalising the denominator by multiplying by the conjugate of the denominator in both numerator and denominator.

⇒ $\frac{(4+3\sqrt{5})(4+3\sqrt{5})}{(4-3\sqrt{5})(4+3\sqrt{5})}$

Using the formula $a^{2} - b^{2} = (a+b)(a-b)$

⇒ $\frac{(4+3\sqrt{5})^{2} }{16 - 45}$

Using the formula $(a+b)^{2} = a^{2} + b^{2} + 2ab$

⇒ $-\frac{16+45+24\sqrt{5}}{29}$ = $a + b\sqrt{5}$

⇒ $(-\frac{61}{29}) + (-\frac{21}{29})\sqrt{5} = a+b\sqrt{5}$

Comparing LHS and RHS we get,

$a = -\frac{61}{29}\\\\b = -\frac{21}{29}$