Algebraic identity application: Evaluate (2.247)^3 + (1.730)^3 + (1.023)^3 - 3 * 2.247 * 1.730 * 1.023, divided by [(2.247)^2 + (1.730)^2 + (1.023)^2 - 2.247*1.730 - 1.730*1.023 - 2.247*1.023].
Aptitude
Simplification
Difficulty: Easy
Choose an option
Answer
Correct Answer: 5
Explanation
Introduction / Context:This problem directly uses a well-known algebraic identity. When x + y + z ≠ 0, the expression (x^3 + y^3 + z^3 - 3xyz) / (x^2 + y^2 + z^2 - xy - yz - zx) simplifies elegantly to x + y + z.
Given Data / Assumptions:
- x = 2.247, y = 1.730, z = 1.023
- We assume standard algebraic identities apply and no rounding is needed before substitution.
Concept / Approach:Use the identity: (x^3 + y^3 + z^3 - 3xyz) / (x^2 + y^2 + z^2 - xy - yz - zx) = x + y + z, provided x + y + z ≠ 0. Here, the numbers are chosen so their sum is exact and simple.
Step-by-Step Solution:
Compute sum: x + y + z = 2.247 + 1.730 + 1.023 = 5.000By identity, the entire fraction simplifies to x + y + z = 5Verification / Alternative check:
You can numerically evaluate numerator and denominator separately, but the identity guarantees the result as long as x + y + z ≠ 0.Why Other Options Are Wrong:
- 1.730, 4, 5.247: These are distractors derived from partial sums or misapplied identities. Only the exact sum 5 satisfies the identity here.
Common Pitfalls:
- Attempting to expand everything numerically, increasing chances of arithmetic error.
- Forgetting the identity or misremembering coefficients.
Final Answer:
5