$$\frac{768 \times 768 \times 768 + 232 \times 232 \times 232}{768 \times 768 - 768 \times 232 + 232 \times 232} = x$$

### Concept & Formula This mathematical expression maps perfectly to the factorization of the sum of two perfect cubes. The large numbers are merely placeholders for variables in a standard formula. $$a^3 + b^3 = (a + b)(a^2 - ab + b^2)$$ ### Step-by-Step Solution * Identify the recurring large numbers and assign them to algebraic variables: * Let $a = 768$ * Let $b = 232$ * Rewrite the provided fraction completely in terms of $a$ and $b$: $$\frac{a \times a \times a + b \times b \times b}{a \times a - a \times b + b \times b}$$ * Simplify the expression using cube and square exponents: $$\frac{a^3 + b^3}{a^2 - ab + b^2}$$ * Expand the numerator using the standard sum of cubes algebraic identity: $$\frac{(a + b)(a^2 - ab + b^2)}{a^2 - ab + b^2}$$ * The large trinomial $(a^2 - ab + b^2)$ is present in both the numerator and the denominator, so they cancel each other out completely: $$a + b$$ * Substitute the original numerical values back into the simplified expression and evaluate: $$768 + 232 = 1000$$ ### Exam Strategy & Shortcut Memorize the structural form $\frac{a^3 + b^3}{a^2 - ab + b^2}$. When you spot a sum of three multiplied identical terms in the numerator and a corresponding mix of squares in the denominator, the answer is always simply $a + b$. Mentally add the two unique numbers in the problem ($768 + 232$) to arrive at 1000 instantly. ### Common Pitfall Students frequently panic at the sight of massive multiplication operations and attempt digit-by-digit calculation, which is a guaranteed way to run out of time on an aptitude test. Always look for algebraic reductions first. ### Final Answer Therefore, the correct answer is **1000**.