Evaluate the expression (25.7322)^2 − (15.7322)^2 and choose the correct value rounded to two decimal places.

Introduction / Context:
This question uses the difference of squares identity to simplify an expression involving squares of decimals. Instead of squaring both numbers directly and subtracting, we can apply the algebraic identity to save time and reduce arithmetic complexity.

Given Data / Assumptions:

• Expression: (25.7322)^2 − (15.7322)^2.
• We must compute the result and match it with the closest value among the options.
• The decimals are given with four decimal places, and we work with standard arithmetic.

Concept / Approach:
We use the identity a^2 − b^2 = (a − b) * (a + b). This identity holds for all real numbers, including decimals. Here a = 25.7322 and b = 15.7322. The identity reduces the need for two separate squarings and a subtraction, and instead turns the expression into a product of a simple difference and sum of the two numbers.

Step-by-Step Solution:
Step 1: Let a = 25.7322 and b = 15.7322. Step 2: Apply the identity a^2 − b^2 = (a − b) * (a + b). Step 3: Compute a − b: 25.7322 − 15.7322 = 10.0000. Step 4: Compute a + b: 25.7322 + 15.7322 = 41.4644. Step 5: Now compute the product: (a − b) * (a + b) = 10.0000 * 41.4644. Step 6: Multiplying by 10 shifts the decimal point one place, giving 414.644. Step 7: Rounding 414.644 to two decimal places gives 414.64.

Verification / Alternative check:
As a rough check, we can approximate the numbers. 25.7322 is near 25.73 and 15.7322 is near 15.73. Their difference is still about 10 and their sum is about 41.46, giving a product close to 414.6. This matches our more precise calculation. Therefore the option 414.64 is consistent with both exact algebraic reasoning and approximate numerical checks.

Why Other Options Are Wrong:
214.64 is about half the correct value and corresponds to an incorrect difference or sum in the identity.
314.64 is closer but still significantly below the accurate result of about 414.64.
412.64 is close but does not match the exact product of 10 and 41.4644 rounded to two decimal places.

Common Pitfalls:
Without the identity, many learners attempt to compute each square directly, which is lengthy and prone to decimal errors. Others may subtract first and then square, incorrectly computing (a − b)^2 instead of a^2 − b^2. Always read the expression carefully and apply the appropriate identity to simplify the work.

Final Answer:
The value of (25.7322)^2 − (15.7322)^2, rounded to two decimal places, is 414.64.