Recovering the intended identity: (a^3 + b^3)/(a^2 − ab + b^2) Assuming the denominator is 343^2 − 343×257 + 257^2, evaluate (343^3 + 257^3)/(343^2 − 343×257 + 257^2).

Introduction / Context:
The printed denominator appears malformed in the raw text. By the Recovery-First Policy, we minimally repair it to the standard identity form a^2 − ab + b^2 so that the expression aligns with (a^3 + b^3)/(a^2 − ab + b^2). This is a classic result that simplifies to a + b.

Given Data / Assumptions:

• Assume denominator is 343^2 − 343×257 + 257^2 (standard form).
• Numerator: 343^3 + 257^3.
• a = 343, b = 257.

Concept / Approach:
Use the identity a^3 + b^3 = (a + b)(a^2 − ab + b^2). Therefore, (a^3 + b^3)/(a^2 − ab + b^2) = a + b. This bypasses any need to compute large powers explicitly.

Step-by-Step Solution:
Recognize identity match: numerator a^3 + b^3 and denominator a^2 − ab + b^2.Apply formula: (a^3 + b^3)/(a^2 − ab + b^2) = a + b.Compute a + b = 343 + 257 = 600.

Verification / Alternative check:
Expanding a^3 + b^3 and dividing by the quadratic factor yields the linear factor (a + b). The identity guarantees exactness for all real a and b.

Why Other Options Are Wrong:
8600, 2600, and 800 are unrelated sums/products; only 600 equals a + b. The distractors are plausible numerically but inconsistent with the identity.

Common Pitfalls:
Not repairing the misprinted denominator; attempting massive exponent calculations instead of using the identity.

Final Answer:
600