Question

$\frac{\sqrt{11}+\sqrt{7}}{\sqrt{11}-\sqrt{7}}=a-\sqrt{77}b$
Then find the value of $a$ & $b$

Answer 1

Rationalize the Left hand side,

$\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}$

$=\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}\times \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}-\sqrt{7}}$

$=\frac{{(\sqrt{11}-\sqrt{7})}^2}{(\sqrt{11}+\sqrt{7})(\sqrt{11}-\sqrt{7})}$

$=\frac{(\sqrt{11}{)}^2+(\sqrt{7}{)}^2-2\left(\sqrt{11}\right)\left(\sqrt{7}\right)}{(\sqrt{11}{)}^2-(\sqrt{7}{)}^2}$

$=\frac{11+7-2\sqrt{77}}{11-7}$

$=\frac{18-2\sqrt{77}}{4}$

$=\frac{18}{4}-\frac{2}{4}\sqrt{77}$

$=\frac{9}{2}-\frac{1}{2}\sqrt{77}$

Now,

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

$\Rightarrow a=\frac{9}{2},b=\frac{1}{2}$

$a+b=\frac{9}{2}+\frac{1}{2}=\frac{10}{2}=5$