`(0.39xx0.39+0.17xx0.17)/((0.39+0.17)^(2)+(0.39-0.17)^(2))` equal to :

Options

A. 1
B. 0
C. `1//2`
D. None of these

Answer

To solve the expression \((0.39 \times 0.39 + 0.17 \times 0.17) / ((0.39 + 0.17)^{2} + (0.39 - 0.17)^{2})\), we can simplify it step by step.

### Step-by-Step Solution:

1. **Substitute Variables**:
Let \( A = 0.39 \) and \( B = 0.17 \). The expression now becomes:
\[ \frac{A \times A + B \times B}{(A + B)^{2} + (A - B)^{2}} \]

2. **Rewrite the Numerator**:
The numerator can be simplified as:
\[ A^2 + B^2 \]

3. **Expand the Denominator**:
The denominator consists of two parts:
- \( (A + B)^{2} = A^2 + 2AB + B^2 \)
- \( (A - B)^{2} = A^2 - 2AB + B^2 \)

Adding these two expansions together gives:
\[ (A + B)^{2} + (A - B)^{2} = (A^2 + 2AB + B^2) + (A^2 - 2AB + B^2) = 2A^2 + 2B^2 \]

4. **Combine the Results**:
Now, we can substitute back into the expression:
\[ \frac{A^2 + B^2}{2A^2 + 2B^2} \]

5. **Factor Out Common Terms**:
The denominator can be factored as:
\[ 2(A^2 + B^2) \]
Thus, the expression simplifies to:
\[ \frac{A^2 + B^2}{2(A^2 + B^2)} \]

6. **Cancel Common Terms**:
Since \( A^2 + B^2 \) is common in both the numerator and the denominator (assuming \( A^2 + B^2 \neq 0 \)), we can cancel it out:
\[ \frac{1}{2} \]

### Final Answer:
The value of the expression is:
\[ \frac{1}{2} \]