If $x+\frac{1}{x}=4$, then the value of $x^5+\frac{1}{x^5}$ is:

724

 x⁵ + $\frac{1}{x^5}$ = (x² + $\frac{1}{x^2}$) × (x³ + $\frac{1}{x^3}$) – (x + $\frac{1}{x}$)

If $x+\frac{1}{x}=4$,

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

If $K+\frac{1}{K}=n$

then, $K^2+\frac{1}{K^2}$ = n² – 2

$x^2+\frac{1}{x^2}$ = 4² – 2 = 14

 x⁵ + $\frac{1}{x^5}$ = 14 × 52 – 4 = 724