Question

if 4+√5÷4-√5= a +b√5, find a and b

Answer 1

Hey friend,
Here si the answer you were looking for:
$\frac{ 4+ \sqrt{5} }{4 - \sqrt{5} } = a + b \sqrt{5} \\ \\ on \: rationalizing \: the \: denominator \: we \: get \\ \\ = \frac{ 4+ \sqrt{5} }{4 - \sqrt{5} } \times \frac{4 + \sqrt{5} }{4 + \sqrt{5} } \\ \\ using \: the \: identity \\ {(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ (a + b)(a - b) = {a}^{2} - {b}^{2} \\ \\ = \frac{ {(4)}^{2} + {( \sqrt{5} )}^{2} + 2 \times 4 \times \sqrt{5} }{ {(4)}^{2} - {( \sqrt{5}) }^{2} } \\ \\ = \frac{16 + 5 + 8 \sqrt{5} }{16 - 5} \\ \\ \frac{21 + 8 \sqrt{5} }{11} = a + b \sqrt{5} \\ \\ a = \frac{21}{11} \\ \\ b = \frac{8}{11}$

Hope this helps!!!!

@Mahak24

Thanks....
☺☺