Evaluate using the cube identity: (0.47^3 − 0.33^3) / (0.47^2 + 0.47×0.33 + 0.33^2). Compute the exact decimal result.

Introduction / Context:This problem is a classic application of polynomial identities with decimals. Recognizing the form (a^3 − b^3) / (a^2 + ab + b^2) immediately simplifies computation and avoids long decimal multiplications.

Given Data / Assumptions:

• a = 0.47, b = 0.33
• Expression: (a^3 − b^3) / (a^2 + ab + b^2)

Concept / Approach:Use the identity a^3 − b^3 = (a − b)(a^2 + ab + b^2). Dividing by (a^2 + ab + b^2) cancels that factor, leaving simply a − b. This identity works for any real numbers a, b, including decimals.

Step-by-Step Solution:Start with (a^3 − b^3) / (a^2 + ab + b^2).Apply the factorization: a^3 − b^3 = (a − b)(a^2 + ab + b^2).Cancel the common factor (a^2 + ab + b^2): result = a − b.Compute a − b = 0.47 − 0.33 = 0.14.

Verification / Alternative check:Approximate multiplication confirms the same outcome but is longer. The identity ensures exactness with far less effort.

Why Other Options Are Wrong:

• 0.8 and 1: These do not match a − b for the given decimals.
• 15.51: Numerical distraction; it is roughly (0.39)^2/0.0099 style conflation, not the correct identity outcome.
• 0.047: Mistakes a − b with a or misreads decimal places.

Common Pitfalls:Multiplying out cubes and squares directly and committing rounding errors; forgetting the exact identity that dramatically reduces work.

Final Answer:0.14