Evaluate : `((0.35)^(3)+(0.41)^(3)-(0.76)^(3))/(9(0.35)(0.41)(0.76))`

Answer

To evaluate the expression \(\frac{(0.35)^3 + (0.41)^3 - (0.76)^3}{9(0.35)(0.41)(0.76)}\), we can use the identity for the sum of cubes. The identity states that if \(A + B + C = 0\), then:

\[ A^3 + B^3 + C^3 = 3ABC \]

### Step-by-Step Solution:

1. **Identify A, B, and C**:
Let:
- \(A = 0.35\)
- \(B = 0.41\)
- \(C = -0.76\)

2. **Check if \(A + B + C = 0\)**:
Calculate:
\[ A + B + C = 0.35 + 0.41 - 0.76 \]
Simplifying this:
\[ = 0.76 - 0.76 = 0 \]
Since \(A + B + C = 0\), we can use the identity.

3. **Apply the identity**:
According to the identity:
\[ A^3 + B^3 + C^3 = 3ABC \]
Therefore:
\[ (0.35)^3 + (0.41)^3 - (0.76)^3 = 3(0.35)(0.41)(-0.76) \]

4. **Substitute into the original expression**:
Now substitute this back into the original expression:
\[ \frac{3(0.35)(0.41)(-0.76)}{9(0.35)(0.41)(0.76)} \]

5. **Simplify the expression**:
The \(0.35\), \(0.41\), and \(0.76\) in the numerator and denominator can be canceled:
\[ = \frac{3(-1)}{9} = \frac{-3}{9} \]

6. **Final simplification**:
Simplifying \(\frac{-3}{9}\):
\[ = -\frac{1}{3} \]

### Final Answer:
\[ \text{The value of the expression is } -\frac{1}{3}. \]