Evaluate the expression (0.96)^3 − (0.1)^3 divided by [ (0.96)^2 + (0.96*0.1) + (0.1)^2 ]. Use the a^3 − b^3 factorization to simplify.

Introduction / Context:This expression is designed for the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). When the denominator matches a^2 + ab + b^2, the overall fraction collapses to a − b. Recognizing such structures converts a potentially tedious decimal computation into a one-step result.

Given Data / Assumptions:

• a = 0.96, b = 0.10.
• Expression: [a^3 − b^3] / [a^2 + ab + b^2].
• Note that ab = 0.96 * 0.10 = 0.096 (as written).

Concept / Approach:Apply the cube difference factorization. Since the denominator equals a^2 + ab + b^2 exactly, the expression simplifies to a − b immediately. Then compute 0.96 − 0.10 to reach the final answer without rounding issues.

Step-by-Step Solution:Use identity: a^3 − b^3 = (a − b)(a^2 + ab + b^2).Given denominator = a^2 + ab + b^2, the ratio = a − b.Compute a − b = 0.96 − 0.10 = 0.86.

Verification / Alternative check:Approximate directly: (0.884736 − 0.001) / (0.9216 + 0.096 + 0.01) ≈ 0.883736 / 1.0276 ≈ 0.86 (rounded), confirming the identity-based result.

Why Other Options Are Wrong:

• 1.06 / 0.95 / 0.97 / 0.80: These values arise from arithmetic slips or from misreading the denominator as a^2 + a + b^2 or using 0.96 instead of 0.096 for ab.

Common Pitfalls:Misreading 0.096 as 0.96; attempting to compute the cubes fully and rounding at intermediate steps; forgetting the identity and overcomplicating the work.

Final Answer:0.86