If $$x^4 - 83 x^2 + 1 = 0,$$ then a value of $$x^3 - x^{-3}$$ can be:
$$x^4 - 83 x^2 + 1 = 0$$
$$x^2 - 83+ \frac{1}{x^2} = 0$$
$$x^2 +Â \frac{1}{x^2} -2 = 81$$
$$(x + \frac{1}{x})^2 = 9^2$$
$$x + \frac{1}{x} = 9$$
$$x^3 - \frac{1}{x^{3}} = (x + \frac{1}{x})^3 + 3(x + \frac{1}{x})Â $$
=Â $$(9)^3 + 3(9)$$ = 729 +Â 27 = 756
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