**Given:**
x = 2^(1/3) * 2^(2/3)

**To find:**
The value of (x^3 + 6x^2 + 6x)

**Solution:**
Let's simplify the expression step by step.

1. Simplify x:
x = 2^(1/3) * 2^(2/3)

Using the rule a^(m/n) = (a^m)^(1/n), we can simplify this further:

x = (2^(1 * 2))^(1/3) = 2^(2/3)

2. Cube x:
(x^3) = (2^(2/3))^3

Using the rule (a^m)^n = a^(m * n), we can simplify this further:

(x^3) = 2^((2/3) * 3) = 2^2 = 4

3. Square x:
(x^2) = (2^(2/3))^2

Using the rule (a^m)^n = a^(m * n), we can simplify this further:

(x^2) = 2^((2/3) * 2) = 2^(4/3)

4. Multiply x by 6:
6x = 6 * 2^(2/3)

5. Simplify 6x^2:
6x^2 = 6 * (2^(4/3))^2

Using the rule (a^m)^n = a^(m * n), we can simplify this further:

6x^2 = 6 * 2^((4/3) * 2) = 6 * 2^(8/3)

6. Add all the terms together:
(x^3 + 6x^2 + 6x) = 4 + 6 * 2^(8/3) + 6 * 2^(2/3)

7. Convert the exponents to a common denominator:
(x^3 + 6x^2 + 6x) = 4 + 6 * (2^(8/3) + 2^(2/3))

8. Simplify the expression:
(x^3 + 6x^2 + 6x) = 4 + 6 * (2^(8/3) + 2^(2/3))

9. Since 2^(8/3) and 2^(2/3) are not easily simplified, we can leave the expression as it is.

Therefore, the value of (x^3 + 6x^2 + 6x) is 4 + 6 * (2^(8/3) + 2^(2/3)).

This does not match any of the options provided, so the answer cannot be determined with the given information.

Given that x = 2 + 2^(2/3) + 2^(1/3)

so x - 2 = 2^(2/3) + 2^(1/3)

=(x-2)^3 = [2^(2/3)+2^(1/3)]^3

use (a-b)^3= a^3 - b^3 - 3ab(a-b)

and (a+b)^3= a^3 + b^3 + 3ab(a+b) formulae.

or x^3 - 8 - 6x(x-2) = 2^2 + 2^1 + 3*[2^{(2/3)+(1/3)}[2^(2/3)+2^(1/3)

or x^3 - 8 - 6x^2 + 12x = 4 + 2 + 6(x-2)

or x^3 - 8 -6x^2 + 12x = 6 + 6x - 12

or x^3 - 6x^2 +6x = 2

hope it helps...