Find the value of $f(x)$ for $x = 2 + 2^{2/3} + 2^{1/3}$

Question

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!

Answer 1

Letting $t=2^{1/3}\Rightarrow t^3=2$, we have $$(x-2)^3=(t^2+t)^3$$$$\Rightarrow x^3-6x^2+12x-8=t^6+3t^5+3t^4+t^3=4+6t^2+6t+2.$$ Then, we have $$\begin{align}x^3-6x^2+6x&=(6t^2+6t+6)+8-6x\\&=6t^2+6t+14-6(t^2+t+2)\\&=14-12\\&=2.\end{align}$$

Answer 2

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?