Question

Given: $x=5-4\sqrt{6}$.

If $x-\frac{1}{x}$ is equal to $\frac{360}{71}-\frac{280}{71}\sqrt{m}$, then the value of $m$ is ?

Answer 1

$x=5-4\sqrt{6}$ and $\frac{1}{x}=\frac{1}{5-4\sqrt{6}}$.

We rationalize the denominator,

$\frac{1}{x}=\frac{1}{5-4\sqrt{6}}\times \frac{5+4\sqrt{6}}{5+4\sqrt{6}}$

$=\frac{5+4\sqrt{6}}{25-16\times 6}=-\left(\frac{5+4\sqrt{6}}{71}\right)$

Now, $x-\frac{1}{x}=5-4\sqrt{6}+\left(\frac{5+4\sqrt{6}}{71}\right)$

$=\frac{355+5-(284-4)\sqrt{6}}{71}$

$=\frac{360-280\sqrt{6}}{71}$ .

Therefore, the value of $m=6$.