Solve $\frac{(0.73)^3 + (0.27)^3}{(0.73)^2 + (0.27)^2 - 0.73 \times 0.27} = x$

### Concept & Formula This expression matches the structure of a standard cubic algebraic identity: $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ Rearranging this identity to match our rational expression format yields: $$\frac{a^3 + b^3}{a^2 - ab + b^2} = a + b$$ ### Step-by-Step Solution * **Step 1:** Assign variable mappings for the decimal figures: Let $a = 0.73$ and $b = 0.27$. * **Step 2:** Substitute these variables into the given expression structural template: $$\frac{a^3 + b^3}{a^2 + b^2 - ab}$$ * **Step 3:** Expand the cubic terms in the numerator using the identity: $$\frac{(a + b)(a^2 - ab + b^2)}{(a^2 - ab + b^2)}$$ * **Step 4:** Cancel the identical algebraic expressions from both the numerator and the denominator: $$a + b$$ * **Step 5:** Compute the final numeric sum: $$0.73 + 0.27 = 1.00 = 1$$ ### Exam Strategy & Shortcut **Algebraic Form Matching:** Identify the template $\frac{a^3 + b^3}{a^2 - ab + b^2}$ immediately upon looking at the structure. Recognize that this template always simplifies directly down to $(a + b)$. The calculation becomes a simple addition exercise: $0.73 + 0.27 = 1$. ### Common Pitfall * **Long Multiplication of Decimals:** Attempting to manually calculate $(0.73)^3$ or $(0.27)^3$ leads to multi-digit fractional decimals, wasting critical exam runtime and introducing immediate transcription or precision errors. ### Final Answer **Therefore, the correct answer is 1.**