Question

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

Answer 1

$\frac{ \sqrt{11} + \sqrt{77} }{ \sqrt{11} - \sqrt{77} } = a - b \sqrt{77}$
L.H.S,

On rationalizing the denominator we get,

$= \frac{ \sqrt{11} + \sqrt{77} }{ \sqrt{11} - \sqrt{77} } \times \frac{ \sqrt{11} + \sqrt{77} }{ \sqrt{11} + \sqrt{77} } \\ \\ = \frac{ {( \sqrt{11} )}^{2} + {( \sqrt{77}) }^{2} + 2( \sqrt{11} )( \sqrt{77} )}{ {( \sqrt{11} )}^{2} - {( \sqrt{77} )}^{2} } \\ \\ = \frac{11 + 77 + 2 \sqrt{ {11}^{2} \times 7 } }{11 - 77} \\ \\ = \frac{88 + 22 \sqrt{7} }{ - 66} \\ \\ = \frac{22(4 + \sqrt{7} )}{ 22( - 3)} \\ \\ = \frac{ - 4 - \sqrt{7} }{3} \\ \\ \bf \: a \: = \: - \frac{4}{3} \: and \: b \: = \: \frac{1}{3}$