$ \left( \frac{785 \times 785 \times 785 + 435 \times 435 \times 435}{785 \times 785 + 435 \times 435 - 785 \times 435} \right) $ simplifies to

### Concept & Formula This problem is a classic aptitude question designed to test your recognition of the "Sum of Cubes" algebraic identity. The formula is: $$ a^3 + b^3 = (a + b)(a^2 - ab + b^2) $$ When rearranged as a fraction, it becomes a powerful simplification tool: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} = a + b $$ ### Step-by-Step Solution To make the algebraic structure obvious, let's replace the large numbers with variables: * Let $ a = 785 $ * Let $ b = 435 $ Now, substitute these variables into the expression given in the problem: $$ \text{Numerator} = a \cdot a \cdot a + b \cdot b \cdot b = a^3 + b^3 $$ $$ \text{Denominator} = a \cdot a + b \cdot b - a \cdot b = a^2 + b^2 - ab $$ Putting it all together, the expression is exactly: $$ \frac{a^3 + b^3}{a^2 - ab + b^2} $$ Using the algebraic identity, we can expand the numerator: $$ = \frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2} $$ The entire trinomial block $ (a^2 - ab + b^2) $ cancels out from both the top and the bottom: $$ = a + b $$ Now, simply add the original numerical values: $$ = 785 + 435 $$ $$ = 1220 $$ ### Exam Strategy & Shortcut You should instantly recognize this structure. Whenever you see $ \frac{\text{Cube} + \text{Cube}}{\text{Square} + \text{Square} - \text{Product}} $, the answer is **ALWAYS** just the sum of the two base numbers ($a + b$). You don't need to write down any formulas—just confidently add $785 + 435$ in seconds. ### Common Pitfall The trap here is attempting to calculate the actual products (e.g., $ 785 \times 785 \times 785 $). This will take several minutes, waste massive amounts of scratch paper, and practically guarantee an arithmetic error. Always look for algebraic patterns in massive numbers. ### Final Answer **Therefore, the correct answer is 1220.**