Recognize and apply algebraic identities Evaluate (885^3 + 115^3) / (885^2 + 115^2 − 885*115) exactly.

Introduction / Context:
This item checks recognition of the sum-of-cubes identity and the common factor it shares with a^2 + b^2 − ab. Spotting these identities drastically reduces computation for large numbers like 885 and 115.

Given Data / Assumptions:

• Expression: (a^3 + b^3) / (a^2 + b^2 − ab) with a = 885 and b = 115.
• Use standard algebraic identities; no numeric expansion is required.

Concept / Approach:
Identity: a^3 + b^3 = (a + b)(a^2 − ab + b^2). Observe that the denominator is exactly a^2 + b^2 − ab, which equals a^2 − ab + b^2. Therefore the fraction simplifies to (a + b)(a^2 − ab + b^2)/(a^2 − ab + b^2) = a + b.

Step-by-Step Solution:
Rewrite numerator: a^3 + b^3 = (a + b)(a^2 − ab + b^2).Denominator: a^2 + b^2 − ab = a^2 − ab + b^2.Cancel common factor: result = a + b.Compute a + b = 885 + 115 = 1000.

Verification / Alternative check:
You can verify numerically with smaller placeholders (e.g., a = 8, b = 1) to confirm the identity-based simplification holds. The structure is general, not specific to 885 and 115.

Why Other Options Are Wrong:
115 and 885 are just the individual terms, not the sum.770 is an attractive distractor (e.g., 885 − 115), but the correct operation is addition after cancellation.

Common Pitfalls:
Misremembering a^3 + b^3 as (a + b)^3; attempting long multiplication; or overlooking that a^2 + b^2 − ab equals a^2 − ab + b^2 exactly, enabling cancellation.

Final Answer:
1000