Evaluate the identity-based expression exactly: [0.5*0.5*0.5 + 0.2*0.2*0.2 + 0.3*0.3*0.3 − 3*0.5*0.3*0.2] / [0.5*0.5 + 0.2*0.2 + 0.3*0.3 − 0.5*0.2 − 0.2*0.3 − 0.5*0.3].

Introduction / Context:This expression is a classic application of the identity (a^3 + b^3 + c^3 − 3abc) = (a + b + c)(a^2 + b^2 + c^2 − ab − bc − ca). Recognizing this structure turns a messy-looking fraction into a simple sum a + b + c.

Given Data / Assumptions:

• a = 0.5, b = 0.2, c = 0.3.
• Numerator matches a^3 + b^3 + c^3 − 3abc.
• Denominator matches a^2 + b^2 + c^2 − ab − bc − ca.

Concept / Approach:Use the identity directly. Provided a + b + c ≠ 0, the ratio simplifies exactly to a + b + c. Compute that sum with the given decimal values.

Step-by-Step Solution:

Verification / Alternative check:Plugging numbers explicitly gives the same result; however, the identity provides a faster and exact path with less arithmetic risk.

Why Other Options Are Wrong:0.6, 0.4, 0.03, and 0.5 are arbitrary decimals not equal to a + b + c for the given values.

Common Pitfalls:Failing to spot the identity leads to tedious computations and rounding errors. Another frequent error is misplacing minus signs in the denominator terms −ab − bc − ca.

Final Answer:1