Use sum-of-cubes factorization to simplify exactly: [725^3 + 371^3] / [725^2 − 725×371 + 371^2] = ?

Introduction / Context:Recognizing algebraic identities is crucial for simplification. Here the numerator is a sum of cubes, and the denominator is the quadratic factor associated with that identity.

Given Data / Assumptions:

• Numerator: a^3 + b^3 with a = 725 and b = 371.
• Denominator: a^2 − ab + b^2.
• We assume exact integer arithmetic.

Concept / Approach:Use the identity a^3 + b^3 = (a + b)(a^2 − ab + b^2). Dividing by (a^2 − ab + b^2) cancels that factor, leaving just (a + b).

Step-by-Step Solution:

Verification / Alternative check:Multiplying back: (a + b)(a^2 − ab + b^2) reconstructs a^3 + b^3, confirming the correctness.

Why Other Options Are Wrong:9610, 1960, 1016, and 1090 are distractors derived from mis-adding or misordering digits; none equal 725 + 371.

Common Pitfalls:Using the difference-of-cubes formula incorrectly, or trying to expand cubes numerically instead of leveraging the identity.

Final Answer:1096