Apply the sum-of-cubes identity with decimals: [(0.87)^3 + (0.13)^3] / [(0.87)^2 − 0.87×0.13 + (0.13)^2] = ?

Introduction / Context:
Recognizing algebraic identities is the fastest route to accurate results. This problem encodes the sum-of-cubes identity in decimal form and asks for a clean simplification without heavy computation.

Given Data / Assumptions:

• a = 0.87, b = 0.13
• Numerator: a^3 + b^3
• Denominator: a^2 − ab + b^2

Concept / Approach:
The identity a^3 + b^3 = (a + b)(a^2 − ab + b^2) applies. Dividing by (a^2 − ab + b^2) cancels that factor, leaving simply a + b. With the given decimals, the answer is exact and elegant: 1.00.

Step-by-Step Solution:
Use a^3 + b^3 = (a + b)(a^2 − ab + b^2).Therefore [(a^3 + b^3)] / [(a^2 − ab + b^2)] = a + b.Compute a + b = 0.87 + 0.13 = 1.00.

Verification / Alternative check:
Directly raising decimals to powers also works but is slower and prone to rounding; the identity is exact and error-proof.

Why Other Options Are Wrong:

• 0.87, 0.13, 0.74: These are component values or differences, not the sum demanded by the identity.
• 0.5: Arbitrary midpoint; unrelated to the identity here.

Common Pitfalls:
Confusing the identity with (a − b)^3 or attempting brute-force decimal exponentiation.

Final Answer:
1