If x = $$2 - {2^{\frac{1}{3}}} + {2^{\frac{2}{3}}},$$   then find the value of x³ - 6x² + 18x.

Solution (By Examveda Team)

$$\eqalign{ & x = 2 - {2^{\frac{1}{3}}} + {2^{\frac{2}{3}}} \cr & x - 2 = {2^{\frac{2}{3}}} - {2^{\frac{1}{3}}} \cr & {\left( {x - 2} \right)^3} = {\left( {{2^{\frac{2}{3}}} - {2^{\frac{1}{3}}}} \right)^3} \cr & {x^3} - 8 - 3 \times 2x\left( {x - 2} \right) = 4 - 2 - 3 \times {2^{\frac{2}{3}}} \times {2^{\frac{1}{3}}}\left( {{2^{\frac{2}{3}}} - {2^{\frac{1}{3}}}} \right) \cr & {x^3} - 8 - 6{x^2} + 12x = 4 - 2 - 6\left( {x - 2} \right) \cr & {x^3} - 8 - 6{x^2} + 12x = 2 - 6x + 12 \cr & {x^3} - 6{x^2} + 12x + 6x = 2 + 12 + 8 \cr & {x^3} - 6{x^2} + 18x = 22 \cr} $$