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If x = 2 + 2^2/3+ 2^1/3, then find the value of x³- 6x² + 6x.

$Given x = 2 + 2^{\frac{1}{3}} + 2^{\frac{2}{3}} \Rightarrow x - 2 = 2^{\frac{1}{3}} + 2^{\frac{2}{3}} = 2^{\frac{1}{3}} (2^{\frac{1}{3}} + 1) Now cubing on both sides we get (x - 2)^{3} = (2^{\frac{1}{3}})^{3} (2^{\frac{1}{3}} + 1)^{3} \Rightarrow x^{3} - 8 + 12 x - 6 x^{2} = 2 (2 + 1 + 3 (2^{\frac{1}{3}} + 2^{\frac{2}{3}})) \Rightarrow x^{3} - 8 + 12 x - 6 x^{2} = 2 (3 + 3 (x - 2)) \Rightarrow x^{3} - 8 + 12 x - 6 x^{2} = 6 + 6 x - 12 \Rightarrow x^{3} - 6 x^{2} + 6 x - 2 = 0 \Rightarrow x^{3} - 6 x^{2} + 6 x = 2$

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