`55^(3)+17^(3)-72^(3)+201960` is equal to

Options

A. `-1`
B. `0`
C. `1`
D. `17`

Answer

To solve the expression \( 55^3 + 17^3 - 72^3 + 201960 \), we can apply the identity for the sum of cubes.

### Step-by-step Solution:

1. **Identify the terms**: We have \( A = 55 \), \( B = 17 \), and \( C = -72 \).

2. **Check the condition**: We need to check if \( A + B + C = 0 \):
\[ 55 + 17 - 72 = 0 \]
Since this is true, we can use the identity for the sum of cubes.

3. **Apply the identity**: The identity states that if \( A + B + C = 0 \), then:
\[ A^3 + B^3 + C^3 = 3ABC \]
Therefore,
\[ 55^3 + 17^3 - 72^3 = 3 \times 55 \times 17 \times (-72) \]

4. **Calculate \( 3ABC \)**:
- First, calculate \( 55 \times 17 \):
\[ 55 \times 17 = 935 \]
- Next, calculate \( 935 \times (-72) \):
\[ 935 \times (-72) = -67260 \]
- Now, multiply by 3:
\[ 3 \times (-67260) = -201780 \]

5. **Combine with the constant**: Now we substitute back into the original expression:
\[ -201780 + 201960 \]

6. **Perform the addition**:
\[ -201780 + 201960 = 180 \]

### Final Answer:
Thus, the expression \( 55^3 + 17^3 - 72^3 + 201960 \) is equal to \( 180 \).