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

Options

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

Answer

To solve the expression \((4 \frac{11}{15} + \frac{15}{71})^2 - (4 \frac{11}{15} - \frac{15}{71})^2\), we can use the algebraic identity:

\[ a^2 - b^2 = (a + b)(a - b) \]

In our case, let:
- \( a = 4 \frac{11}{15} + \frac{15}{71} \)
- \( b = 4 \frac{11}{15} - \frac{15}{71} \)

### Step 1: Identify \( a + b \) and \( a - b \)

Calculating \( a + b \):

\[ a + b = (4 \frac{11}{15} + \frac{15}{71}) + (4 \frac{11}{15} - \frac{15}{71}) = 2(4 \frac{11}{15}) = 8 \frac{11}{15} \]

Calculating \( a - b \):

\[ a - b = (4 \frac{11}{15} + \frac{15}{71}) - (4 \frac{11}{15} - \frac{15}{71}) = 2(\frac{15}{71}) \]

### Step 2: Substitute into the identity

Now, substituting into the identity:

\[ a^2 - b^2 = (a + b)(a - b) = (8 \frac{11}{15})(2 \frac{15}{71}) \]

### Step 3: Simplify the expression

Calculating \( (8 \frac{11}{15})(2 \frac{15}{71}) \):

First, convert mixed numbers to improper fractions:

\[ 8 \frac{11}{15} = \frac{8 \times 15 + 11}{15} = \frac{120 + 11}{15} = \frac{131}{15} \]
\[ 2 \frac{15}{71} = \frac{2 \times 71 + 15}{71} = \frac{142 + 15}{71} = \frac{157}{71} \]

Now, multiply the two fractions:

\[ \frac{131}{15} \times \frac{157}{71} = \frac{131 \times 157}{15 \times 71} \]

### Step 4: Calculate the product

Calculating the numerator and denominator:

\[ 131 \times 157 = 20547 \]
\[ 15 \times 71 = 1065 \]

Thus, we have:

\[ \frac{20547}{1065} \]

### Step 5: Simplify the fraction

Now, we can simplify \(\frac{20547}{1065}\). Performing the division gives us:

\[ \frac{20547}{1065} = 19.3 \text{ (approximately)} \]

However, we can also notice that we can directly apply the formula \(4ab\) where \(a = \frac{71}{15}\) and \(b = \frac{15}{71}\):

\[ 4ab = 4 \left(\frac{71}{15}\right) \left(\frac{15}{71}\right) = 4 \cdot 1 = 4 \]

### Final Answer

Thus, the final answer is:

\[ \boxed{4} \]