Use the (a − b)^3 pattern to simplify Evaluate (.58)^3 − (.42)^3 − 3 * (.58) * (.42) * (.58 − .42) exactly.

Introduction / Context:This expression is designed for the binomial cube identity. Recognizing the structure a^3 − b^3 − 3ab(a − b) allows immediate simplification to (a − b)^3, which is simple to compute.

Given Data / Assumptions:

• a = 0.58, b = 0.42.
• Expression: a^3 − b^3 − 3ab(a − b).
• a − b = 0.16.

Concept / Approach:Identity: (a − b)^3 = a^3 − b^3 − 3ab(a − b). Hence the entire expression equals (a − b)^3. This avoids tedious cubic computations.

Step-by-Step Solution:Compute a − b = 0.58 − 0.42 = 0.16.By identity, value = (a − b)^3 = (0.16)^3.0.16^2 = 0.0256; multiply by 0.16: 0.0256 * 0.16 = 0.004096.

Verification / Alternative check:Expanding (0.16)^3 directly or evaluating a^3, b^3, and 3ab(a − b) numerically will agree with 0.004096.

Why Other Options Are Wrong:1.3976 and 1 are far too large; 0.16 is (a − b), not its cube; 0.04096 is off by a factor of 10.

Common Pitfalls:Forgetting the exact identity; cubing only the difference once; decimal place errors when cubing small decimals.

Final Answer:0.004096