Question

√11 -√17/√11+√7 = a-b√√77

Answer 1

Answer:

$\bf a = \frac{9}{2} \: \: and \: \: b = \frac{1}{2}$

Step-by-step explanation:

$\frac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} } = a - b \sqrt{77}$

L.H.S,

Rationalizing the denominator we get:

$= \frac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} + \sqrt{7} } \times \frac{ \sqrt{11} - \sqrt{7} }{ \sqrt{11} - \sqrt{7} } \\ \\ = \frac{ {( \sqrt{11}) }^{2} + {( \sqrt{7} )}^{2} - 2. \sqrt{11} . \sqrt{7} }{ {( \sqrt{11}) }^{2} - {( \sqrt{7} )}^{2} } \\ \\ = \frac{11 + 7 - 2 \sqrt{77} }{11 - 7} \\ \\ = \frac{18 - 2 \sqrt{77} }{4} \\ \\ \bf = \frac{9 - \sqrt{77} }{2}$

Comparing L.H.S and R.H.S we get:

$\frac{9 - \sqrt{77} }{2} = a - b \sqrt{77} \\ \\ \bf a = \frac{9}{2} \: \: and \: \: b = \frac{1}{2}$