$$\frac{753 \times 753 + 247 \times 247 - 753 \times 247}{753 \times 753 \times 753 + 247 \times 247 \times 247} = x$$

### Concept & Formula This expression is the exact reciprocal (inverted form) of the standard sum of cubes algebraic identity. $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ ### Step-by-Step Solution * Substitute variables for the recurring numerical values: * Let $a = 753$ * Let $b = 247$ * Rewrite the fraction using these defined variables: $$\frac{a^2 + b^2 - ab}{a^3 + b^3}$$ * Rearrange the numerator slightly for clarity so it matches the standard identity format: $$\frac{a^2 - ab + b^2}{a^3 + b^3}$$ * Expand the denominator using the sum of cubes factorization rule: $$\frac{a^2 - ab + b^2}{(a + b)(a^2 - ab + b^2)}$$ * Cancel the trinomial factor $(a^2 - ab + b^2)$ from both the numerator and denominator. Since the numerator completely cancels out, a 1 remains on top: $$\frac{1}{a + b}$$ * Insert the original numbers and complete the simple addition: $$\frac{1}{753 + 247} = \frac{1}{1000}$$ ### Exam Strategy & Shortcut Look closely at where the cubed terms ($a^3$ and $b^3$) are located. If they are in the numerator, the answer is just $a + b$. If the cubed terms are in the denominator (as they are in this specific problem), the answer is the reciprocal: $\frac{1}{a + b}$. Simply add $753 + 247 = 1000$ and place it under a 1. ### Common Pitfall The primary mistake here is failing to notice that the fraction is inverted compared to standard questions, leading students to incorrectly select "1000" instead of "1/1000". Always verify whether the $a^3 \pm b^3$ sequence is on the top or bottom of the fraction line. ### Final Answer Therefore, the correct answer is **$\frac{1}{1000}$**.