Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$ If $x = 2 + 2^{2/3} + 2^{1/3}$, then find the value of $f(x)=x^3 - 6x^2 + 6x$.
I am unable to get to the answer - end up with more than one term. Please help me solve this!

We have $$x-2=2^{\frac23}+2^{\frac13}$$
Cubing we get, $$x^3-3x^2(2)+3x(2^2)+2^3=(2^{\frac23})^3+(2^{\frac13})^3+3\cdot2^{\frac23}\cdot2^{\frac13}(2^{\frac23}+2^{\frac13})$$
$$\iff x^3-6x^2+12x+8=2+2^2+6(x-2)$$
Can you take it home from here?
