If ` x= 3+ 3^(1//3)+ 3^(2//3)` , then the value of the expression ` x^(3)-9x^(2)+8x-12` is equal to _____

Answer

To solve the problem, we need to evaluate the expression \( x^3 - 9x^2 + 8x - 12 \) given that \( x = 3 + 3^{1/3} + 3^{2/3} \).

### Step-by-Step Solution:

1. **Define the variable**:
\[ x = 3 + 3^{1/3} + 3^{2/3} \]

2. **Rearrange the equation**:
We can rearrange the equation to isolate the cube root terms:
\[ x - 3 = 3^{1/3} + 3^{2/3} \]

3. **Cube both sides**:
To eliminate the cube roots, we cube both sides:
\[ (x - 3)^3 = (3^{1/3} + 3^{2/3})^3 \]

4. **Expand the left-hand side**:
Using the binomial expansion:
\[ (x - 3)^3 = x^3 - 9x^2 + 27x - 27 \]

5. **Expand the right-hand side**:
Let \( y = 3^{1/3} \), then \( 3^{2/3} = y^2 \). Thus:
\[ (y + y^2)^3 = y^3 + 3y^2(y^2) + 3y(y^2) + y^3 = 3 + 3 \cdot 3^{1/3} + 3 \cdot 3^{2/3} \]
This simplifies to:
\[ = 3 + 3 + 3^{1/3} + 3^{2/3} = 3 + 9 \]

6. **Set the two expansions equal**:
Equating both sides gives:
\[ x^3 - 9x^2 + 27x - 27 = 12 \]

7. **Rearranging the equation**:
Move all terms to one side:
\[ x^3 - 9x^2 + 27x - 39 = 0 \]

8. **Evaluate the expression**:
We need to evaluate \( x^3 - 9x^2 + 8x - 12 \):
\[ x^3 - 9x^2 + 8x - 12 = (x^3 - 9x^2 + 27x - 39) + (8x - 27) \]
Since \( x^3 - 9x^2 + 27x - 39 = 0 \):
\[ = 0 + (8x - 27) \]

9. **Substituting \( x \)**:
Substitute \( x = 3 + 3^{1/3} + 3^{2/3} \):
\[ 8(3 + 3^{1/3} + 3^{2/3}) - 27 \]
\[ = 24 + 8 \cdot 3^{1/3} + 8 \cdot 3^{2/3} - 27 \]
\[ = -3 + 8 \cdot 3^{1/3} + 8 \cdot 3^{2/3} \]

10. **Final calculation**:
The expression simplifies to:
\[ = 0 \]

Thus, the value of the expression \( x^3 - 9x^2 + 8x - 12 \) is equal to **0**.