Question

7 plus 3 root 5 by 3 plus root 5 plus 7 minus 3 root 5 by 3 minus root 5 . find the value of a and b

Answer 1

Identity used :

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


$\frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } + \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } = a + b\sqrt{5}$

L.H.S,

On rationalizing the denominator we get,

$= \frac{7 + 3 \sqrt{5} }{3 + \sqrt{5} } \times \frac{3 - \sqrt{5} }{3 - \sqrt{5} } + \frac{7 - 3 \sqrt{5} }{3 - \sqrt{5} } \times \frac{3 + \sqrt{5} }{3 + \sqrt{5} } \\ \\ = \frac{7(3 - \sqrt{5} ) + 3 \sqrt{5} (3 - \sqrt{5} )}{ {(3)}^{2} - {( \sqrt{5}) }^{2} } + \frac{7(3 + \sqrt{5} ) - 3 \sqrt{5} (3 + \sqrt{5} )}{ {(3)}^{2} - {( \sqrt{5}) }^{2} } \\ \\ = \frac{21 - 7 \sqrt{5} + 9 \sqrt{5} - 15}{9 - 5} + \frac{21 + 7 \sqrt{5} - 9 \sqrt{5} - 15}{9 - 5} \\ \\ = \frac{6 + 2 \sqrt{5} }{4} + \frac{ 6 - 2 \sqrt{5} }{4} \\ \\ = \frac{3 + \sqrt{5} }{2} + \frac{3 - \sqrt{5} }{2} \\ \\ = \frac{3 + \sqrt{5} + 3 - \sqrt{5} }{2} \\ \\ = \frac{6}{2} \\ \\ = 3$

On comparing we get,

a = 3

b = 0