If $x = 2+2^{\frac{1}{3}} + 2^{\frac{2}{3}}$, then find the value of $x^3 - 6x^2 + 6x$.

2

( a + b ) = a³ + b³ + 3ab( a + b )

x = 2 + 2^2/3 + 2^1/3

= (x – 2) = (2^2/3 + 2^1/3)

Cubing  on both sides

= x³ – 8 – 6x² + 12x = (2^2/3)³ + (2^1/3)³ + 3 × 2^2/3 × 2^1/3(2^2/3 + 2^1/3)

= x³ – 14 – 6x² + 12x = 6 (x – 2)

= x³ – 14 – 6x² + 12x – 6x + 12 = 0

= x³ – 6x² + 6x = 2