Assume 57 + 59 + 109 = 0, then find the value of `57^(3) + 59^(3) + 109^(3)`.

Options

A. 1099701
B. 1099601
C. 1099801
D. 1098701

Answer

To solve the problem, we need to find the value of \( 57^3 + 59^3 + 109^3 \) given that \( 57 + 59 + 109 = 0 \).

### Step-by-Step Solution:

1. **Understanding the Given Condition**:
We start with the equation:
\[ 57 + 59 + 109 = 0 \]
This can be rearranged to:
\[ 57 + 59 = -109 \]

2. **Using the Identity for Cubes**:
We can use the identity for the sum of cubes:
\[ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - ac - bc) \]
Here, let \( a = 57 \), \( b = 59 \), and \( c = 109 \).

3. **Substituting the Values**:
Since \( a + b + c = 0 \), we can simplify the identity:
\[ a^3 + b^3 + c^3 = 3abc \]

4. **Calculating \( abc \)**:
Now we need to calculate \( abc \):
\[ abc = 57 \times 59 \times 109 \]

5. **Calculating \( 57 \times 59 \)**:
First, calculate \( 57 \times 59 \):
\[ 57 \times 59 = 3363 \]

6. **Calculating \( 3363 \times 109 \)**:
Now, multiply \( 3363 \) by \( 109 \):
\[ 3363 \times 109 = 366507 \]

7. **Final Calculation of \( 57^3 + 59^3 + 109^3 \)**:
Now substitute back into the equation:
\[ 57^3 + 59^3 + 109^3 = 3 \times 366507 = 1099521 \]

### Conclusion:
Thus, the value of \( 57^3 + 59^3 + 109^3 \) is:
\[ \boxed{1099521} \]