$$\frac{854 \times 854 \times 854 - 276 \times 276 \times 276}{854 \times 854 + 854 \times 276 + 276 \times 276} = x$$

### Concept & Formula This problem is a direct application of the algebraic factorization for the difference of two perfect cubes, designed to eliminate complex arithmetic. $$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$$ ### Step-by-Step Solution * Assign variables to the repeating base numbers to expose the underlying algebraic equation: * Let $a = 854$ * Let $b = 276$ * Translate the numerical fraction into variable notation: $$\frac{a^3 - b^3}{a^2 + ab + b^2}$$ * Expand the numerator using the difference of cubes identity: $$\frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2}$$ * Cancel the identical trinomial factors present in both the top and bottom of the fraction: $$a - b$$ * Substitute the original numbers back in and calculate the final basic subtraction: $$854 - 276 = 578$$ ### Exam Strategy & Shortcut Recognize the blueprint of $\frac{a^3 - b^3}{a^2 + ab + b^2}$. Whenever you see this specific arrangement of multiplied terms with a subtraction sign in the cubed numerator, the entire complex expression collapses down to just $a - b$. Perform a quick mental subtraction of $854 - 276 = 578$ to solve the question in under 5 seconds. ### Common Pitfall Ignoring the signs is a frequent mistake. If the numerator has a minus sign between the cubed terms (like this problem), the operation is subtraction ($a - b$). If it has a plus sign, the operation is addition ($a + b$). Confusing the two will lead you straight to a distractor option. ### Final Answer Therefore, the correct answer is **578**.