If $ x^4 +\frac{1}{x^4}= 47 $, then what is the value of $ x^3 +\frac{1}{x^3}$?

18

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 x⁴ + \(\frac{1}{x^4}\) = 47

then x² + \(\frac{1}{x^2}\) = \(\sqrt {47 + 2}\) = 7

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

$ x^3 +\frac{1}{x^3}$ = (3)³ – 3(3) = 18