If $x^2 - 5\sqrt{2}x - 1 = 0, $ then what will be the value of $x^3 -\frac{1}{x^3}$ ?

$265\sqrt{2}$

We know that,

If x - \(\frac{1}{x}\)  = n

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

If $x^2 - 5\sqrt{2}x - 1 = 0, $

then what will be the value of $x^3 -\frac{1}{x^3}$ = ?

If $x^2 - 5\sqrt{2}x - 1 = 0, $

Divide by x on the both sides of the equation,

x - \(\frac{1}{x}\) = $5\sqrt{2}$

$x^3 -\frac{1}{x^3}$  = ($5\sqrt{2}$)³ + 3 × $5\sqrt{2}$ = $250\sqrt{2}$ + $15\sqrt{2}$ = $265\sqrt{2}$