Simplify $ \frac{789 \times 789 \times 789 + 211 \times 211 \times 211}{789 \times 789 - 789 \times 211 + 211 \times 211} $

### Concept & Formula This large fraction is a classic substitution problem based on the algebraic identity for the sum of two cubes. By mapping the numbers to variables, the complex expression perfectly simplifies: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ ### Step-by-Step Solution * Let $a = 789$ and $b = 211$. * The numerator consists of $(789)^3 + (211)^3$, which is $a^3 + b^3$. * The denominator consists of $(789)^2 - (789 \times 211) + (211)^2$, which is $a^2 - ab + b^2$. * Substitute the variables into the fraction: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} $$ * Expand the numerator using the sum of cubes identity: $$ \frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2} $$ * The quadratic term $(a^2 - ab + b^2)$ cancels out completely from the top and bottom. * We are left with simply $(a + b)$. * Substitute the original numbers back: $$ 789 + 211 = 1000 $$ ### Exam Strategy & Shortcut Whenever you see a numerator with a sum of cubes ($x \times x \times x + y \times y \times y$) and a denominator with the corresponding quadratic, bypass the algebra entirely. The answer will always just be the sum of the two base numbers: $x + y$. In this case, $789 + 211 = 1000$. This should take you 3 seconds to solve. ### Common Pitfall A massive trap is attempting to multiply these 3-digit numbers manually. Examiners specifically design these questions to penalize students who rely on brute-force calculation rather than algebraic pattern recognition. ### Final Answer Therefore, the correct answer is **1000**.