If $x^4+\frac{1}{x^4}=194$ and $x^3+\frac{1}{x^3}=?$

52

If x⁴ + \(\frac{1}{x^4}\) = a

then x² + \(\frac{1}{x^2}\) = \(\sqrt {a + 2}\) = b

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

$x^4+\frac{1}{x^4}=194$

= x² + \(\frac{1}{x^2}\) = \(\sqrt {196 + 2}\) = 14

=  x + \(\frac{1}{x}\) = \(\sqrt {14 + 2}\) = 4

$x^3+\frac{1}{x^3}$ = 4³ - 3 × 4 = 52