Without actually calculating the cubes, evaluate the expression `(30)^(3)+(-18)^(3)+(-12)^(3)`.

Answer

To evaluate the expression \(30^3 + (-18)^3 + (-12)^3\) without actually calculating the cubes, we can use the identity for the sum of cubes.

### Step-by-Step Solution:

1. **Identify the values of a, b, and c**:
Let \(a = 30\), \(b = -18\), and \(c = -12\).

2. **Check if \(a + b + c = 0\)**:
\[ a + b + c = 30 + (-18) + (-12) = 30 - 18 - 12 = 0 \]
Since \(a + b + c = 0\), we can use the special case of the identity.

3. **Use the identity**:
The identity states that if \(a + b + c = 0\), then:
\[ a^3 + b^3 + c^3 = 3abc \]

4. **Calculate \(abc\)**:
\[ abc = 30 \times (-18) \times (-12) \]
First, calculate \(30 \times (-18)\):
\[ 30 \times (-18) = -540 \]
Now, multiply by \(-12\):
\[ -540 \times (-12) = 6480 \]

5. **Substitute back into the identity**:
Now, substituting \(abc\) back into the identity:
\[ a^3 + b^3 + c^3 = 3 \times 6480 = 19440 \]

### Final Result:
Thus, the value of \(30^3 + (-18)^3 + (-12)^3\) is \(19440\).

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