$ \frac{38 \times 38 \times 38 + 34 \times 34 \times 34 + 28 \times 28 \times 28 - 38 \times 34 \times 84}{38 \times 38 + 34 \times 34 + 28 \times 28 - 38 \times 34 - 34 \times 28 - 38 \times 28} $ is equal to

### Concept & Formula This massive calculation is built entirely around one of the most complex, yet standard, algebraic identities involving the sum of three cubes. The identity is: $$ a^3 + b^3 + c^3 - 3abc = (a + b + c)(a^2 + b^2 + c^2 - ab - bc - ca) $$ When restructured as a fraction, it provides an instant simplification: $$ \frac{a^3 + b^3 + c^3 - 3abc}{a^2 + b^2 + c^2 - ab - bc - ca} = a + b + c $$ ### Step-by-Step Solution Let's establish our variables: * Let $ a = 38 $ * Let $ b = 34 $ * Let $ c = 28 $ The denominator perfectly matches the formula: $ a^2 + b^2 + c^2 - ab - bc - ca $. The numerator looks like $ a^3 + b^3 + c^3 $, but the final term is $ - 38 \times 34 \times 84 $. Where is the $ -3abc $? Let's calculate what $ 3abc $ should be: $$ 3abc = 3 \times 38 \times 34 \times 28 $$ Notice that $ 3 \times 28 = 84 $. Therefore, $ 3abc $ can be rewritten as $ 38 \times 34 \times 84 $. This means the numerator is indeed exactly $ a^3 + b^3 + c^3 - 3abc $. Substitute the entire structure into our formula fraction: $$ \frac{a^3 + b^3 + c^3 - 3abc}{a^2 + b^2 + c^2 - ab - bc - ca} $$ According to the identity, the massive polynomial cancels out, leaving only: $$ = a + b + c $$ Substitute the numbers back in and sum them up: $$ = 38 + 34 + 28 $$ $$ = 100 $$ ### Exam Strategy & Shortcut Whenever you see a fraction containing the sum of three cubes ($ a^3 + b^3 + c^3 $) in the numerator and the sum of three squares in the denominator, you can confidently assume it is testing this specific identity. Don't even worry about verifying the $ 3abc $ term; test makers design these specifically to cancel out. Jump straight to adding the three base numbers ($ 38 + 34 + 28 = 100 $). ### Common Pitfall The main point of confusion is the $ 84 $ at the end of the numerator. Students expect to see a $ 3 $ and all three variables explicitly written out. When they see $ 84 $, they assume the formula doesn't apply and either abandon the question or attempt impossible brute-force arithmetic. Always check if constants have been multiplied together to hide the formula. ### Final Answer **Therefore, the correct answer is 100.**