Simplify $\frac{(359 + 256)^2 + (359 - 256)^2}{359 \times 359 + 256 \times 256}$

### Concept & Formula This problem uses standard algebraic identities to eliminate the large numbers completely. The relevant identities are: $$(a + b)^2 = a^2 + 2ab + b^2$$ $$(a - b)^2 = a^2 - 2ab + b^2$$ Adding them gives: $$(a + b)^2 + (a - b)^2 = 2(a^2 + b^2)$$ ### Step-by-Step Solution * **Step 1:** Let $a = 359$ and $b = 256$. * **Step 2:** Substitute these variables into the given expression: $$\frac{(a + b)^2 + (a - b)^2}{a^2 + b^2}$$ * **Step 3:** Apply the algebraic identity to the numerator: $$\frac{2(a^2 + b^2)}{a^2 + b^2}$$ * **Step 4:** Cancel out the common term $(a^2 + b^2)$ from the numerator and denominator: $$2$$ ### Exam Strategy & Shortcut **Pattern Recognition:** Notice the structural template $\frac{(a+b)^2 + (a-b)^2}{a^2 + b^2}$. In this specific format, the actual values of $a$ and $b$ are completely irrelevant because they will always cancel out, leaving a constant answer of $2$. ### Common Pitfall * **Actual Calculation:** Attempting to add, subtract, and square the actual numbers $359$ and $256$ is an immediate time-sink that leads to massive numbers and high error probability. Recognize the algebraic form first. ### Final Answer **Therefore, the correct answer is 2.**