`(4(11)/(15)+(15)/(71))^(2)-(4(11)/(15)-(15)/(71))^(2)` is equal to

Options

A. 1
B. 2
C. 3
D. 4

Answer

To solve the expression \((\frac{4 \cdot 11}{15} + \frac{15}{71})^2 - (\frac{4 \cdot 11}{15} - \frac{15}{71})^2\), we can use the difference of squares formula, which states that \(A^2 - B^2 = (A + B)(A - B)\).

### Step-by-Step Solution:

1. **Identify A and B**:
Let:
\[ A = \frac{4 \cdot 11}{15} + \frac{15}{71} \]
\[ B = \frac{4 \cdot 11}{15} - \frac{15}{71} \]

2. **Apply the Difference of Squares Formula**:
Using the formula \(A^2 - B^2 = (A + B)(A - B)\), we can rewrite the expression:
\[ (\frac{4 \cdot 11}{15} + \frac{15}{71})^2 - (\frac{4 \cdot 11}{15} - \frac{15}{71})^2 = (A + B)(A - B) \]

3. **Calculate \(A + B\)**:
\[ A + B = \left(\frac{4 \cdot 11}{15} + \frac{15}{71}\right) + \left(\frac{4 \cdot 11}{15} - \frac{15}{71}\right) \]
The \(\frac{15}{71}\) terms cancel out:
\[ A + B = 2 \cdot \frac{4 \cdot 11}{15} = \frac{8 \cdot 11}{15} = \frac{88}{15} \]

4. **Calculate \(A - B\)**:
\[ A - B = \left(\frac{4 \cdot 11}{15} + \frac{15}{71}\right) - \left(\frac{4 \cdot 11}{15} - \frac{15}{71}\right) \]
The \(\frac{4 \cdot 11}{15}\) terms cancel out:
\[ A - B = 2 \cdot \frac{15}{71} = \frac{30}{71} \]

5. **Combine \(A + B\) and \(A - B\)**:
Now substitute back into the difference of squares formula:
\[ (A + B)(A - B) = \left(\frac{88}{15}\right) \left(\frac{30}{71}\right) \]

6. **Multiply the Fractions**:
\[ = \frac{88 \cdot 30}{15 \cdot 71} \]

7. **Simplify the Expression**:
First, simplify \(\frac{88 \cdot 30}{15}\):
\[ = \frac{88 \cdot 2}{1} = 176 \quad (\text{since } 30 \div 15 = 2) \]
Now, the expression becomes:
\[ = \frac{176}{71} \]

### Final Answer:
Thus, the value of the expression is:
\[ \frac{176}{71} \]