Simplify a ratio of polynomial-like decimal forms: (1.04^2 + 1.04×0.04 + 0.04^2) / (1.04^3 − 0.04^3). Use sum of squares over difference of cubes.

Introduction / Context:The expression mirrors well-known polynomial identities but with decimal numbers. Leveraging those identities lets you avoid tedious multiplication and get an exact result instantly.

Given Data / Assumptions:

• a = 1.04, b = 0.04
• Numerator: a^2 + ab + b^2
• Denominator: a^3 − b^3

Concept / Approach:Recall that a^3 − b^3 = (a − b)(a^2 + ab + b^2). Because the denominator contains (a^2 + ab + b^2) as a factor, the entire fraction simplifies to 1/(a − b).

Step-by-Step Solution:Start: (a^2 + ab + b^2) / (a^3 − b^3).Factor the denominator: a^3 − b^3 = (a − b)(a^2 + ab + b^2).Cancel common factor (a^2 + ab + b^2).Result = 1 / (a − b) = 1 / (1.04 − 0.04) = 1 / 1.00 = 1.

Verification / Alternative check:Direct numeric evaluation also yields 1 but is unnecessarily long. Identity-based simplification is exact and efficient.

Why Other Options Are Wrong:

• 0.10, 0.1, 0.01: These imply treating (a − b) as 10, 10, or 100, which is incorrect.
• 1.04: Would require the denominator to be smaller than the numerator by factor 1/1.04, not the case here.

Common Pitfalls:Attempting to compute a^3 and b^3 directly with decimals, which invites rounding mistakes; overlooking the factorization.

Final Answer:1