Apply the cube difference identity to simplify Compute (117^3 − 98^3) / (117^2 + 117×98 + 98^2) by using standard algebraic formulas.

Introduction / Context:This problem is a direct application of the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). The denominator matches the quadratic factor of the identity, allowing an immediate simplification to a − b.

Given Data / Assumptions:

• a = 117, b = 98.
• Expression: (a^3 − b^3) / (a^2 + ab + b^2).

Concept / Approach:From the identity, (a^3 − b^3) / (a^2 + ab + b^2) = (a − b). No further arithmetic is required beyond subtracting b from a.

Step-by-Step Solution:Use identity: a^3 − b^3 = (a − b)(a^2 + ab + b^2).Divide by (a^2 + ab + b^2): result = a − b.Compute a − b = 117 − 98 = 19.

Verification / Alternative check:If expanded numerically, the ratio will indeed equal 19. The algebraic route is more efficient and error-resistant.

Why Other Options Are Wrong:215 and 311 are unrelated large values; 29 and 39 are plausible-looking integers but do not match a − b.

Common Pitfalls:Forgetting the identity and trying to cube large numbers; arithmetic slips during lengthy calculations.

Final Answer:19