Evaluate using identities and exact arithmetic: Compute (.356 × .365 − 2 × .365 × .106 + .106 × .106) / (.632 × .632 + 2 × .632 × .368 + .368 × .368) without rounding until the end.

Introduction / Context:This fraction features two classic algebraic patterns hidden in decimal arithmetic. The denominator is of the form x^2 + 2xy + y^2 = (x + y)^2, which collapses neatly. The numerator resembles a squared-expression expansion but with mixed products, so computing it exactly is prudent. The exercise builds precision and pattern recognition.

Given Data / Assumptions:

• Numerator: 0.356 × 0.365 − 2 × 0.365 × 0.106 + 0.106 × 0.106.
• Denominator: 0.632 × 0.632 + 2 × 0.632 × 0.368 + 0.368 × 0.368.
• Compute the exact decimal value and then simplify the ratio.

Concept / Approach:First, simplify the denominator using the identity (x + y)^2. Here x = 0.632 and y = 0.368, so x + y = 1, hence denominator = 1^2 = 1. Next, compute the numerator by careful multiplication and combination of terms. No further identity fully collapses the numerator because the first term is a product of two different decimals (0.356 and 0.365).

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 0.0765 and 0.345: Result from mis-multiplying or mis-subtracting terms.
• 0.625: Off by an order of magnitude; likely ignores decimal placement.

Common Pitfalls:

• Treating 0.356 × 0.365 as a square (it is not).
• Rounding mid-steps instead of at the end, which can shift the hundredths place.

Final Answer: