$$\frac{256 \times 256 - 144 \times 144}{112}$$ is equal to

### Concept & Formula This problem utilizes the fundamental difference of two squares identity, revealing a hidden relationship between the numerator and the denominator. $$a^2 - b^2 = (a - b)(a + b)$$ ### Step-by-Step Solution * Identify the structure of the numerator, which is a difference of squares: $(256)^2 - (144)^2$. * Let $a = 256$ * Let $b = 144$ * Expand the numerator using the algebraic identity: $$\frac{(256 - 144)(256 + 144)}{112}$$ * Perform the subtraction operation within the first set of parentheses: $$256 - 144 = 112$$ * Substitute this result back into the numerator: $$\frac{112 \times (256 + 144)}{112}$$ * Cancel out the matching number 112 from both the top and bottom of the fraction: $$256 + 144$$ * Complete the remaining addition operation to find the final value: $$256 + 144 = 400$$ ### Exam Strategy & Shortcut When presented with $\frac{a^2 - b^2}{c}$, immediately check if the denominator $c$ is equal to either $(a - b)$ or $(a + b)$. In this case, a quick glance shows that $256 - 144 = 112$. Since the denominator is exactly $(a - b)$, it cancels out, leaving only $(a + b)$ as the answer. You can mentally add $256 + 144 = 400$ without putting pen to paper. ### Common Pitfall Squaring 256 and 144 manually to get 65536 and 20736 respectively before subtracting and dividing by 112. While this technically works, it is extremely tedious, highly susceptible to arithmetic errors, and wastes an unacceptable amount of time on a competitive test. ### Final Answer Therefore, the correct answer is **400**.