Question

if a and b are rational numbers and √11-√7/√11+√7=a-b√77,find the value of a and b

Answer 1

$\frac{ \sqrt{11} + \sqrt{7} }{ \sqrt{11} - \sqrt{7} } = a - b \sqrt{77} \\ \frac{ {( \sqrt{11} + \sqrt{7} ) }^{2} }{ { (\sqrt{11} ) }^{2} - {( \sqrt{7} )}^{2} } = a - b \sqrt{77} \\ \frac{11 + 7 + 2 \sqrt{77} }{11 - 7} = a - b \sqrt{77} \\ \frac{18 + 2 \sqrt{77} }{4} = a - b \sqrt{77} \\ \frac{9 + \sqrt{77} }{2} = a - b \sqrt{77} \\ \frac{9}{2} + \frac{ \sqrt{77} }{2} = a - b \sqrt{77} \\ \\ compare \: both \: sides \\ \\ a = \frac{9}{2} \\ b = \frac{-1}{2}$

Answer 2

√11-√7÷ √11+√7 =

√11-√7÷√11+√7× √11-√7÷√11-√7

=( √11-√7 )°2 ÷√11°2 -√7°2

=√11°2 +√7°2 -2×√11×√7÷ 11-7

= 11+7-2√77÷ 4 = 18 -2√77÷4

= 9-√77÷ 2 = a-b√77

= a= 9/2 And b = 1/2