If x-1/x=√77, then one of the values of x^3+1/(x^3) is :

If x - \(\frac{1}{x}\)  = n

then 

then, x + \(\frac{1}{x}\)  = \(\sqrt {n^2 + 4}\)

If $x-\frac{1}{x}=\sqrt{77}$

 x + \(\frac{1}{x}\)  = \(\sqrt {77 + 4}\) = 9

If x + \(\frac{1}{x}\)  = n

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

$x^3 +\frac{1}{x^3}$ = 9³ - 3 × 9 = (+-) 702