Question

Find the value of $a$ and $b$ if $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=a-b\sqrt{77}$.

Answer 1

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

$=\frac{{(\sqrt{11}-\sqrt{7})}^2}{11-7}=\frac{11+7-2\sqrt{77}}{4}$

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

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