Simplify $ \frac{658 \times 658 \times 658 - 328 \times 328 \times 328}{658 \times 658 + 658 \times 328 + 328 \times 328} $

### Concept & Formula This expression is structured around the algebraic identity for the difference of two cubes. By replacing the large numbers with variables, the expression collapses into a simple subtraction: $$ a^3 - b^3 = (a - b)(a^2 + ab + b^2) $$ ### Step-by-Step Solution * Let $a = 658$ and $b = 328$. * The numerator is the difference of cubes: $(658)^3 - (328)^3 \rightarrow a^3 - b^3$. * The denominator is a quadratic sum: $(658)^2 + (658 \times 328) + (328)^2 \rightarrow a^2 + ab + b^2$. * Rewrite the full expression algebraically: $$ \frac{a^3 - b^3}{a^2 + ab + b^2} $$ * Expand the numerator using the identity formula: $$ \frac{(a - b)(a^2 + ab + b^2)}{a^2 + ab + b^2} $$ * Cancel out the common quadratic factor $(a^2 + ab + b^2)$ from the numerator and denominator. * This leaves exactly: $(a - b)$. * Substitute the original numbers back in: $$ 658 - 328 = 330 $$ ### Exam Strategy & Shortcut Recognize the pattern instantly: if the numerator has a minus sign connecting the cubed terms ($x^3 - y^3$), the entire expression will always simplify to exactly $(x - y)$. You can skip all intermediate steps and just subtract the base numbers: $658 - 328 = 330$. ### Common Pitfall A frequent error is confusing the sum and difference formulas, accidentally adding the two numbers ($658 + 328 = 986$) instead of subtracting them. Always check the sign linking the cubic terms in the numerator to determine your final operation. ### Final Answer Therefore, the correct answer is **330**.