If $x^4 + (\frac{1}{x^4}) = 322,$ then what is the value of $x^3 - (\frac{1}{x^3})$ ?

76

If x⁴ + \(\frac{1}{x^4}\) = a

then x² + \(\frac{1}{x^2}\) = \(\sqrt {a + 2}\) = b

and x - \(\frac{1}{x}\) = \(\sqrt {b - 2}\)

If $x^4 + (\frac{1}{x^4}) = 322,$

then x² + \(\frac{1}{x^2}\) = \(\sqrt {322 + 2}\) = 18

and x - \(\frac{1}{x}\) = \(\sqrt {18 - 2}\) = 4

If x - \(\frac{1}{x}\)  = n

then, $x^3 - \frac{1}{x^3}$ = 4³ + 3 × 4 = 76