Question

5+2root3/7+root3 =a-root 3b

Answer 1

Heya !!!
$\frac{5 + 2 \sqrt{3} }{7 + \sqrt{3} } = a - b\sqrt{3}$

L.H.S

On rationalizing the denominator we get,

$= \frac{5 + 2 \sqrt{3} }{7 + \sqrt{3} } \times \frac{7 - \sqrt{3} }{7 - \sqrt{3} } \\ \\ = \frac{5(7 - \sqrt{3}) + 2 \sqrt{3}(7 - \sqrt{3} ) }{ {(7)}^{2} - {( \sqrt{3}) }^{2} } \\ \\ = \frac{35 - 5 \sqrt{3} + 14 \sqrt{3} - 6 }{49 - 3} \\ \\ = \frac{29 + 9 \sqrt{3} }{46}$

On comparing we get

a = 29/46

b = 9/46

Hope this helps ☺