Simplify the expression (0.73^3 + 0.27^3) divided by (0.73^2 + 0.27^2 - 0.73 × 0.27) and give the exact value.

Introduction / Context:

This algebraic simplification problem uses decimals but is really about standard identities for cubes and squares. It tests whether you can recognize patterns such as a^3 + b^3 and a^2 - ab + b^2, even when the numbers are decimals instead of simple integers.

Given Data / Assumptions:

• The expression is (0.73^3 + 0.27^3) / (0.73^2 + 0.27^2 - 0.73 × 0.27).
• We must simplify it to an exact value.
• a = 0.73 and b = 0.27 are real numbers with a + b = 1.

Concept / Approach:

Recall the identity a^3 + b^3 = (a + b)(a^2 - ab + b^2). Notice that the denominator is a^2 + b^2 - ab, which is the same as a^2 - ab + b^2. Therefore the numerator and denominator share a common factor, leading to a simple result.

Step-by-Step Solution:

Verification / Alternative check:

You can compute the numerator and denominator numerically. 0.73^3 + 0.27^3 is approximately 0.4087, and 0.73^2 + 0.27^2 - 0.73 × 0.27 is also about 0.4087. Their ratio is therefore 1, confirming the algebraic result.

Why Other Options Are Wrong:

The option 0.4087 corresponds to the value of the numerator alone or denominator alone, not the ratio. The options 0.73 and 0.27 come from wrongly taking only a or only b as the final answer. The value 0.5 is a random average that does not follow from the identity.

Common Pitfalls:

Many learners are tempted to do long decimal calculations directly instead of spotting the identity. Others misinterpret the denominator and think it is a^2 + b^2 + ab, which would change the structure completely.

Final Answer:

The simplified value of the expression is 1.