If $x^2 +\frac{1}{x^2} = 83, x > 0$, then the value of $x^3 - \frac{1}{x^3}$ is :

756

If $x^2 +\frac{1}{x^2} = 83, x > 0$,

then the value of $x^3 - \frac{1}{x^3}$

We know that,

If x² + \(\frac{1}{x^2}\)  = n

then, x - \(\frac{1}{x}\)  = \(\sqrt {n - 2}\)

and we also know that,

If x - \(\frac{1}{x}\)  = n

then, $x^3 -\frac{1}{x^3}$ = n³ + 3 × n

If $x^2 +\frac{1}{x^2} = 83, x > 0$,

then, x - \(\frac{1}{x}\)  = \(\sqrt {83 - 2}\) = 9

Then, $x^3 -\frac{1}{x^3}$ = 9³ + 3 × 3

$x^3 -\frac{1}{x^3}$ = 729 + 27 = 756