Question

If $x=\frac{5-\sqrt{11}}{3-2\sqrt{11}},x=a+b\sqrt{11}$, then find the value of $a$ and $b$.

Answer 1

Consider the given surds

$x=\frac{5-\sqrt{11}}{3-2\sqrt{11}}$

$x=\frac{5-\sqrt{11}}{3-2\sqrt{11}}\times \frac{3+2\sqrt{11}}{3+2\sqrt{11}}onrationalize$

$x=\frac{15-3\sqrt{11}+10\sqrt{11}-22}{3^2-4\times 11}$

$x=\frac{-7+7\sqrt{11}}{9-44}$

$x=\frac{-7(1-\sqrt{11})}{-35}$

$x=\frac{1}{5}(1-\sqrt{11})......\left(1\right)$

Now, given that

$x=a+b\sqrt{11}.......\left(2\right)$

Comparing by equation (1) and (2) to, we get,

$a+b\sqrt{11}=\frac{1}{5}(1-\sqrt{11})$

$a+b\sqrt{11}=\frac{1}{5}-\frac{1}{5}\sqrt{11}$

Then,

$a=\frac{1}{5}$ $b=-\frac{1}{5}$

Hence, this is the answer.