If $x^2 +\frac{1}{x^2}= 7 $, then what is the value of $ x^3 +\frac{1}{x^3}$ ?

18

x² + \(\frac{1}{x^2}\)  = b

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

If $x^2 +\frac{1}{x^2}= 7 $

Then, 

x + \(\frac{1}{x}\) = \(\sqrt {7 + 2}\) = 3

If x + \(\frac{1}{x}\)  = n

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

$x^3 +\frac{1}{x^3}$ = 3³ - 3 × 3 = 18