Evaluate the rational expression for given values: If a = 16 and b = 5, compute the exact value of (a^2 + b^2 + a*b) / (a^3 − b^3). Show the simplified fraction.

Introduction / Context:
This problem checks fluency with powers, factorization, and reducing rational expressions. By plugging in a = 16 and b = 5 carefully and then simplifying the fraction, we avoid large, unnecessary computations and reach the exact simplified answer.

Given Data / Assumptions:

• a = 16, b = 5.
• Expression: (a^2 + b^2 + a*b) / (a^3 − b^3).
• We must produce the simplest exact fraction (no decimals).

Concept / Approach:
First evaluate the numerator and denominator separately. After substituting, look for common factors. If both numerator and denominator share a nontrivial factor, reduce to lowest terms. Basic identities like a^3 − b^3 = (a − b)(a^2 + a*b + b^2) can also guide simplification logic even if we compute directly from numbers.

Step-by-Step Solution:
Compute numerator: a^2 + b^2 + a*b = 16^2 + 5^2 + 16*5 = 256 + 25 + 80 = 361.Compute denominator: a^3 − b^3 = 16^3 − 5^3 = 4096 − 125 = 3971.Form the fraction: 361 / 3971.Note 361 = 19^2 and 3971 = 19 * 209 (since 19*200 = 3800 and 19*9 = 171; sum = 3971).Reduce: (19^2)/(19*209) = 19/209. Since 209 = 11*19, we get 19/209 = 1/11.

Verification / Alternative check:
Using the identity a^3 − b^3 = (a − b)(a^2 + a*b + b^2) shows immediately that (a^2 + a*b + b^2) cancels, giving 1/(a − b) = 1/(16 − 5) = 1/11. This matches our arithmetic reduction.

Why Other Options Are Wrong:
1/19 and 121/3971 are unreduced or incorrect simplifications; 19/209 is an intermediate unreduced form; “None of these” is invalid because 1/11 is attainable.

Common Pitfalls:
Forgetting the cube-difference identity; stopping at 361/3971 and not reducing; arithmetic slips when computing powers (16^3 or 5^3).

Final Answer:
1/11