If `x=2 - 2^((1)/(3)) + 2^((2)/(3))` find the value of `x^(3)-6x^(2) + 18x + 18`

Options

A. 22
B. 33
C. 40
D. 45

Answer

To solve the problem, we need to find the value of \( x^3 - 6x^2 + 18x + 18 \) given that \( x = 2 - 2^{1/3} + 2^{2/3} \).

### Step-by-Step Solution:

1. **Substituting the value of \( x \)**:
\[ x = 2 - 2^{1/3} + 2^{2/3} \]

2. **Rearranging the equation**:
\[ x - 2 = -2^{1/3} + 2^{2/3} \]
This simplifies to:
\[ x - 2 = 2^{2/3} - 2^{1/3} \]

3. **Cubing both sides**:
\[ (x - 2)^3 = (2^{2/3} - 2^{1/3})^3 \]

4. **Applying the formula for the cube of a binomial**:
\[ (a - b)^3 = a^3 - b^3 - 3ab(a - b) \]
Here, let \( a = 2^{2/3} \) and \( b = 2^{1/3} \):
\[ (x - 2)^3 = (2^{2/3})^3 - (2^{1/3})^3 - 3(2^{2/3})(2^{1/3})(2^{2/3} - 2^{1/3}) \]

5. **Calculating the cubes**:
\[ (2^{2/3})^3 = 2^{2} = 4, \quad (2^{1/3})^3 = 2^{1} = 2 \]
Thus,
\[ (x - 2)^3 = 4 - 2 - 3(2^{2/3})(2^{1/3})(2^{2/3} - 2^{1/3}) \]
Simplifying gives:
\[ (x - 2)^3 = 2 - 3(2^{2/3})(2^{1/3})(2^{2/3} - 2^{1/3}) \]

6. **Finding the product**:
\[ 2^{2/3} - 2^{1/3} = 2^{1/3}(2^{1/3} - 1) \]
Therefore,
\[ (x - 2)^3 = 2 - 3(2^{2/3})(2^{1/3})(2^{1/3}(2^{1/3} - 1)) \]
Which simplifies to:
\[ (x - 2)^3 = 2 - 3(2^{1})(2^{1/3} - 1) \]

7. **Substituting back to find \( x^3 - 6x^2 + 18x + 18 \)**:
We can express \( x^3 \) in terms of \( x - 2 \):
\[ x^3 = (x - 2)^3 + 6(x - 2) + 8 \]
Thus,
\[ x^3 - 6x^2 + 18x + 18 = (x - 2)^3 + 6(x - 2) + 8 - 6x^2 + 18x + 18 \]

8. **Final simplification**:
After substituting and simplifying, we find that:
\[ x^3 - 6x^2 + 18x + 18 = 0 \]

### Final Answer:
\[ x^3 - 6x^2 + 18x + 18 = 0 \]