$x=2+2^{\frac{2}{3}}+2^{\frac{1}{3}}\Rightarrow x-2=2^{\frac{2}{3}}+2^{\frac{1}{3}}$

Cubing both sides i.e. ${\left(x-2\right)}^3={\left(2^{\frac{2}{3}}+2^{\frac{1}{3}}\right)}^3$

$\Rightarrow x^3-6x^2+12x-8=2^2+2^1+3·2^{\frac{2}{3}}·2^{\frac{1}{3}}·\left(2^{\frac{2}{3}}+2^{\frac{1}{3}}\right)$ (Using ${\left(a-b\right)}^3=a^3-3a^{2}b+3a{b}^2-b^3, {\left(a+b\right)}^3=a^3+b^3+3ab\left(a+b\right), {\left(a^m\right)}^n=a^{mn}$)

$\Rightarrow x^3-6x^2+12x-8=6+3·2·\left(x-2\right)$ (Using $a^m·a^n=a^{m+n}$)

$\Rightarrow x^3-6x^2+6x=2$