Question

If $\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}=x-y\sqrt{77,}$ find the values $x$ and$y$ ?

Answer 1

Let $x=a$ and $y=b$

By Rationalization,$\frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}+\sqrt{7}}\ast \frac{\sqrt{11}-\sqrt{7}}{\sqrt{11}-\sqrt{7}}$

On Denominator,By Formula:$(a+b)(a-b)=a^2-b^2$

On Nominator,By Formula:$a^2+b^2+2ab=(a+b{)}^2$

Then,$\frac{(\sqrt{11}-\sqrt{7}{)}^2}{(\sqrt{11}{)}^2-(\sqrt{7}{)}^2}$\\ $\frac{11+7-2\sqrt{77}}{11-7}$

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

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

Then,$a=\frac{18}{4}=\frac{9}{2}$

$b\sqrt{77}=\frac{2\sqrt{77}}{4}=\frac{1}{2}$

$b=\frac{1}{2}$