# Find the value of $x^3-x^{-3}$ given that $x^2+x^{-2} = 83$

If $$x>1$$ and $$x^2+\dfrac {1}{x^2}=83$$, find the value of the expression$$x^3-\dfrac {1}{x^3}$$

a) $$764$$

b) $$750$$

c) $$756$$

d) $$760$$

In this question from given I tried to approximate the value of $$x$$ which should just above to 9 then I tried to calculate the value of cubic expression but all options are close enough to guess. Any idea to solve it?

$$\left(x-\frac{1}{x}\right)^2=x^2-2+\frac{1}{x^2}=83-2=81$$
$$x^3-\frac{1}{x^3}=\left(x-\frac{1}{x}\right)\left(x^2+1+\frac{1}{x^2}\right)=9\cdot(83+1)=756$$
We take the positive root of $81$ because $x>1$.