Question: if x=[2+{2^(1/3)}+{2^(2/3)}] then [(x^3)-{6*(x^2)}+(6*x)]=?
Let x = a + a^2 + a^3, where a = 2^(1/3)
Let X = x^3 - 6x^2 + 6x = x(x^2 - 6x + 6)
Using x = a(1 + a + a^2),
x^2 = a^2(1 + a^2 + a^4 + 2a + 2a^2 + 2a^3)
-6x = -6a(1 + a + a^2)
6 = 6
============ adding the above three lines
x^2 - 6x + 6 = a^2 + a^4 + a^6 + 2a^3 + 2a^4 + 2a^5 - 6a - 6a^2 - 6a^3 + 6
x^2 - 6x + 6 = -5a^2 + 2a + 2^2 + 2.2 + 2.2a + 2.2a^2 - 6a - 6.2 + 6
x^2 - 6x + 6 = -5a^2 + 4a^2 + 2a + 4a - 6a + 4 + 4 - 12 + 6
x^2 - 6x + 6 = -a^2 + 2
X = x(x^2 - 6x + 6) = (a + a^2 + a^3)(2 - a^2) = 2a + 2a^2 + 4 - a^3 - a^4 - 2a^2
X = 2a + 4 - 2 - 2a
X = 4 - 2
X = 2
Answer: X = 2