Minimum value of the given function- Find the minimum value of-
$$\frac{(x+\frac{1}{x})^6-(x^6+\frac{1}{x^6})-2}{(x+\frac{1}{x})^3+(x^3+\frac{1}{x^3})}$$
I tried opening the brackets and trying to cancel the terms and using AM-GM.

Hint.
Calling
$$
u = x^3+\frac{1}{x^3}\\
v = \left(x+\frac{1}{x}\right)^3
$$
we have
$$
\frac{v^2-u^2}{v+u}=v-u = 3\left(x+\frac{1}{x}\right)\ge 6
$$
