For a real number $x$ with $x+1/x=4$, the value of $x^5+1/x^5$ is:

Build up in steps: $x^2+1/x^2=16-2=14$ and $x^3+1/x^3=64-3(4)=52$. Then $(x^2+1/x^2)(x^3+1/x^3)=x^5+1/x^5+x+1/x$, so $x^5+1/x^5=14\times 52-4=728-4=724$. (728 forgets to subtract the leftover $x+1/x$; 776 comes from $4\times 194$ without subtracting $x^3+1/x^3$; 1024 is simply $4^5$.) With $x=2$ + √3 ≈ 3.7321, $x^5$ ≈ 723.999 and $1/x^5$ ≈ 0.001.