Simplify an expression with cube roots and negative powers: Compute ∛(x^6) ÷ ∛(x^12) × x^(−3) × ∛(x^9).

Introduction / Context:
This problem combines cube roots with integer and negative exponents. Converting all radicals to fractional exponents allows straightforward combination via exponent addition and subtraction rules.

Given Data / Assumptions:

• Expression: x^(6/3) ÷ x^(12/3) × x^(−3) × x^(9/3).
• Assume x ≠ 0 to avoid division by zero.

Concept / Approach:
Use ∛(x^k) = x^(k/3). Then combine exponents linearly: when multiplying, add exponents; when dividing, subtract exponents. Keep track of the signs carefully, especially for the negative power term x^(−3).

Step-by-Step Solution:

Verification / Alternative check:
Pick x = 2: ∛(64)/∛(4096) × 2^(−3) × ∛(512) = 4/16 × 1/8 × 8 = (1/4) × (1/8) × 8 = 1/4 = 1/2^2, confirming 1/x^2.

Why Other Options Are Wrong:

• 1/x, x, 1: Each misses at least one exponent step or cancels incorrectly.

Common Pitfalls:
Forgetting that dividing by x^4 reduces exponent by 4, or cancelling x^(−3) incorrectly with x^3.

Final Answer: