Question

If $x-\frac{1}{x}=3+2\sqrt{2}$, find the value of $x^3-\frac{1}{x^3}$

Answer 1

In the given problem, we have to find the value of

Given

Cubing on both sides of we get

${\left(x-\frac{1}{x}\right)}^3={\left(3+2\sqrt{2}\right)}^3$

We shall use identity

$$\begin{gathered} 27+16\sqrt{2}+18\sqrt{2}\times 3+18\sqrt{2}\times 2\sqrt{2}=x^3-\frac{1}{x^3}-9-6\sqrt{2} \\ 27+16\sqrt{2}+54\sqrt{2}+72=x^3-\frac{1}{x^3}-9-6\sqrt{2} \end{gathered}$$

Hence the value of is .