Simplify the algebraic expression exactly: (69 × 69 × 69 − 65 × 65 × 65) ÷ (69 × 69 + 69 × 65 + 65 × 65) What is the exact numerical value?

Introduction / Context: This question tests recognition of the difference-of-cubes identity. The expression is structured exactly like (a^3 − b^3) divided by (a^2 + ab + b^2), which simplifies neatly to (a − b).

Given Data / Assumptions:

• Numerator: 69^3 − 65^3 • Denominator: 69^2 + 69*65 + 65^2 • Required: exact value (no approximation)

Concept / Approach: Use the identity: a^3 − b^3 = (a − b)(a^2 + ab + b^2) So: (a^3 − b^3) / (a^2 + ab + b^2) = a − b, provided the denominator is not zero (it is not zero for these positive integers).

Step-by-Step Solution: 1) Identify a = 69 and b = 65 2) Recognize numerator as a^3 − b^3 3) Recognize denominator as a^2 + ab + b^2 4) Apply identity: (a^3 − b^3) = (a − b)(a^2 + ab + b^2) 5) Divide by (a^2 + ab + b^2): (a^3 − b^3) / (a^2 + ab + b^2) = a − b 6) Compute a − b: 69 − 65 = 4

Verification / Alternative check: Since 69 and 65 are close, the simplified result being 4 is reasonable. If you expanded the identity, you would see the denominator exactly matches the factor that cancels, leaving only (a − b).

Why Other Options Are Wrong: • 1 or 2: common simplification guesses but do not match a − b. • 0.216 or 0.164: decimals suggest approximation, but this identity gives an exact integer.

Common Pitfalls: • Trying to compute 69^3 and 65^3 directly (unnecessary). • Forgetting the exact form of the difference-of-cubes factorization.

Final Answer: 4