Compute a decimal expression with exponents and a linear denominator: Evaluate (0.5^3 × 0.6^3) / (0.5 × 0.5 − 0.3 + 0.6 × 0.6) accurately, simplifying powers first.

Introduction / Context:This item blends small decimal powers with a simple quadratic-like denominator. Calculating the powers first, then simplifying the denominator, and finally performing the division keeps errors in check. It is a good test of decimal place-value discipline and exponent handling.

Given Data / Assumptions:

• Numerator: 0.5^3 × 0.6^3.
• Denominator: 0.5 × 0.5 − 0.3 + 0.6 × 0.6.
• No rounding is needed until the final step.

Concept / Approach:Compute 0.5^3 and 0.6^3 exactly, multiply for the numerator, then evaluate each term in the denominator and combine. The final division can be expressed as a fraction to keep track of place values, then converted to a decimal with appropriate precision.

Step-by-Step Solution:

Verification / Alternative check:

Why Other Options Are Wrong:

• 0.3 and 0.61: Far too large relative to the small numerator 0.027.
• 0.1: Close but still an overestimate; reflects rounding 0.027/0.27 by mistake.

Common Pitfalls:

• Dropping a zero in the decimal when dividing by 0.31.
• Miscomputing 0.6^3 (some mistakenly take 0.6^2 = 0.36 and stop).

Final Answer: