$ \frac{(13)^3 + 7^3}{(13)^2 + 7^2 - x} = 20 $

### Concept & Formula This problem tests your structural understanding of the "Sum of Cubes" formula. Instead of asking for the final result, it provides the result and asks you to identify a missing structural component. The underlying identity is: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} = a + b $$ ### Step-by-Step Solution Let's define our variables based on the obvious base numbers: * Let $ a = 13 $ * Let $ b = 7 $ Notice the result given on the right side of the equation is 20. Observe that $ 20 = 13 + 7 = a + b $. So, the equation provided is essentially stating: $$ \frac{a^3 + b^3}{a^2 + b^2 - x} = a + b $$ We know from our standard algebraic identity that for this relationship to hold true, the denominator MUST be exactly $ a^2 - ab + b^2 $. Let's rearrange our formula's denominator to match the visual layout of the problem's denominator: $ a^2 + b^2 - ab $ Comparing this perfect formula denominator ($ a^2 + b^2 - ab $) to the given problem's denominator ($ 13^2 + 7^2 - x $), it becomes clear that $ x $ is simply the missing $ ab $ term. Therefore: $$ x = a \times b $$ $$ x = 13 \times 7 $$ $$ x = 91 $$ ### Exam Strategy & Shortcut When you see a sum of cubes fraction that equals the sum of the two base numbers ($ 13^3 + 7^3 $ leading to $ 20 $), you instantly know the equation is utilizing the standard formula. The missing piece subtracting from the squares in the denominator is always the product of the two numbers. Multiply $ 13 \times 7 $ mentally to get 91. ### Common Pitfall Students who rely on brute-forcing arithmetic might attempt to cube 13 (2197) and cube 7 (343), add them together (2540), and then solve $ \frac{2540}{218 - x} = 20 $. While this eventually yields 91, it takes a massive amount of unnecessary time and invites calculation errors. Use algebra to bypass arithmetic. ### Final Answer **Therefore, the correct answer is 91.**