Question

find the values of 'a' and 'b' when 5 + √6 ÷ 5 - √6 = a + b√6

Answer 1

Identities used :

${(a + b)}^{2} = {a}^{2} + {b}^{2} + 2ab \\ \\ (a + b)(a - b) = {a}^{2} - {b}^{2}$

Now,

$\frac{5 + \sqrt{6} }{5 - \sqrt{6} } = a + b \sqrt{6}$

L.H.S,

On rationalizing the denominator we get,

$= \frac{5 + \sqrt{6} }{5 - \sqrt{6} } \times \frac{5 + \sqrt{6} }{5 + \sqrt{6} } \\ \\ = \frac{ {(5)}^{2} + {( \sqrt{6} )}^{2} + 2(5)( \sqrt{6} ) }{ {(5)}^{2} - {( \sqrt{6} )}^{2} } \\ \\ = \frac{25 + 6 + 10 \sqrt{6} }{25 - 6} \\ \\ = \frac{31 + 10 \sqrt{6} }{19}$

On comparing L.H.S and R.H.S we get

$\frac{31 + 10 \sqrt{6} }{19} = a + b \sqrt{6} \\ \\ a = \frac{31}{19} \: : \: b = \frac{10}{19}$