If `(x - 5) (y + 6)(z -8)= 1331`, then the minimum value x + y + z is :

Options

A. 40
B. 33
C. 19
D. not unique

Answer

To find the minimum value of \( x + y + z \) given the equation \( (x - 5)(y + 6)(z - 8) = 1331 \), we can use the Arithmetic Mean-Geometric Mean (AM-GM) inequality.

### Step-by-step Solution:

1. **Rewrite the Expression**:
We start with the equation:
\[ (x - 5)(y + 6)(z - 8) = 1331 \]
Let:
\[ a = x - 5, \quad b = y + 6, \quad c = z - 8 \]
Then, we can rewrite the equation as:
\[ abc = 1331 \]

2. **Use the AM-GM Inequality**:
According to the AM-GM inequality:
\[ \frac{a + b + c}{3} \geq \sqrt[3]{abc} \]
Substituting \( abc = 1331 \):
\[ \frac{a + b + c}{3} \geq \sqrt[3]{1331} \]

3. **Calculate the Cube Root**:
We need to find \( \sqrt[3]{1331} \). Since \( 1331 = 11^3 \):
\[ \sqrt[3]{1331} = 11 \]

4. **Substitute Back into the Inequality**:
Now substituting back into the AM-GM inequality:
\[ \frac{a + b + c}{3} \geq 11 \]
Multiplying both sides by 3 gives:
\[ a + b + c \geq 33 \]

5. **Express \( x + y + z \)**:
Recall that:
\[ a = x - 5, \quad b = y + 6, \quad c = z - 8 \]
Therefore:
\[ x + y + z = (a + 5) + (b - 6) + (c + 8) = a + b + c + 7 \]
Substituting the minimum value of \( a + b + c \):
\[ x + y + z \geq 33 + 7 = 40 \]

6. **Conclusion**:
Thus, the minimum value of \( x + y + z \) is:
\[ \boxed{40} \]