`(32)^(3) + (79)^(3) - (111)^(3) + 3 xx 32 xx 79 xx 111` is equal to

Options

A. 10000
B. 0
C. 30007
D. 1

Answer

To solve the expression \( (32)^3 + (79)^3 - (111)^3 + 3 \times 32 \times 79 \times 111 \), we can use a special algebraic identity.

### Step-by-Step Solution:

1. **Identify the Identity**: We can use the identity that states if \( A + B + C = 0 \), then:
\[ A^3 + B^3 + C^3 = 3ABC \]
In our case, let:
- \( A = 32 \)
- \( B = 79 \)
- \( C = -111 \)

2. **Check the Sum**: Calculate \( A + B + C \):
\[ 32 + 79 - 111 = 111 - 111 = 0 \]
Since \( A + B + C = 0 \), we can apply the identity.

3. **Apply the Identity**: According to the identity:
\[ A^3 + B^3 + C^3 = 3ABC \]
Therefore, we can rewrite the expression as:
\[ (32)^3 + (79)^3 + (-111)^3 = 3 \times 32 \times 79 \times (-111) \]

4. **Substituting Back**: Now, we substitute back into our expression:
\[ (32)^3 + (79)^3 - (111)^3 + 3 \times 32 \times 79 \times 111 = 3 \times 32 \times 79 \times (-111) + 3 \times 32 \times 79 \times 111 \]

5. **Combine Like Terms**: Notice that:
\[ 3 \times 32 \times 79 \times (-111) + 3 \times 32 \times 79 \times 111 = 3 \times 32 \times 79 \times (-111 + 111) = 3 \times 32 \times 79 \times 0 \]

6. **Final Result**: Since anything multiplied by zero is zero:
\[ 3 \times 32 \times 79 \times 0 = 0 \]

Thus, the value of the expression \( (32)^3 + (79)^3 - (111)^3 + 3 \times 32 \times 79 \times 111 \) is **0**.