If $a^4 +\frac{1}{a^4}$ = 194, then what is the value of $a^3 +\frac{1}{a^3}$ ?

52

If x⁴ + \(\frac{1}{x^4}\) = a

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

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

If $a^4 +\frac{1}{a^4}$ = 194

then x² + \(\frac{1}{x^2}\) = \(\sqrt {194 + 2}\) = 14

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

If x + \(\frac{1}{x}\)  = n

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

$x^3 +\frac{1}{x^3}$ = 4³ - 3 × 4  = 52