`(.356xx.356-2xx.356xx.106+.106xx.106)/(.632xx.632+2xx.632xx.368+.368xx.368)=?`

Options

A. 0.25
B. 0.0765
C. 0.345
D. 0.0625

Answer

To solve the expression \((0.356 \times 0.356 - 2 \times 0.356 \times 0.106 + 0.106 \times 0.106) / (0.632 \times 0.632 + 2 \times 0.632 \times 0.368 + 0.368 \times 0.368)\), we can use the formulas for the square of a binomial.

### Step-by-Step Solution:

1. **Identify the Numerator**:
\[ 0.356 \times 0.356 - 2 \times 0.356 \times 0.106 + 0.106 \times 0.106 \]
This can be recognized as the expansion of \((a - b)^2\) where \(a = 0.356\) and \(b = 0.106\). Thus, we can rewrite it as:
\[ (0.356 - 0.106)^2 \]

2. **Calculate \(0.356 - 0.106\)**:
\[ 0.356 - 0.106 = 0.250 \]
Therefore, the numerator becomes:
\[ (0.250)^2 = 0.0625 \]

3. **Identify the Denominator**:
\[ 0.632 \times 0.632 + 2 \times 0.632 \times 0.368 + 0.368 \times 0.368 \]
This can be recognized as the expansion of \((a + b)^2\) where \(a = 0.632\) and \(b = 0.368\). Thus, we can rewrite it as:
\[ (0.632 + 0.368)^2 \]

4. **Calculate \(0.632 + 0.368\)**:
\[ 0.632 + 0.368 = 1.000 \]
Therefore, the denominator becomes:
\[ (1.000)^2 = 1.000 \]

5. **Combine the Results**:
Now we can substitute back into the expression:
\[ \frac{(0.250)^2}{(1.000)^2} = \frac{0.0625}{1.000} = 0.0625 \]

### Final Answer:
The final answer is:
\[ 0.0625 \]