Apply the laws of indices to simplify variables and coefficients: Compute 12 x^2 y^4 z^3 ÷ (2 x y^2 × 3 y z^2) and express the answer in simplest monomial form.

Introduction / Context:
This is a basic practice question on simplifying products and quotients of monomials. The idea is to reduce the numerical coefficient and then subtract exponents of like bases appearing in numerator and denominator.

Given Data / Assumptions:

• Numerator: 12 x^2 y^4 z^3.
• Denominator: (2 x y^2) × (3 y z^2) = 6 x y^3 z^2.
• All variables x, y, z are nonzero real numbers.

Concept / Approach:
Divide coefficients: 12 ÷ 6 = 2. For variables, subtract exponents according to a^m / a^n = a^(m−n). Do this separately for x, y, and z. The result should be a single monomial with positive exponents.

Step-by-Step Solution:

Verification / Alternative check:
Substitute simple nonzero values (e.g., x = y = z = 1) to see that both sides equal 2, supporting the algebra.

Why Other Options Are Wrong:
xyz and 1/(xyz) ignore the coefficient change and/or exponent subtraction; “2 xyz” is equivalent in spacing to 2xyz (correct), but other distractors like x^2 y z reflect incorrect exponent arithmetic.

Common Pitfalls:
Forgetting to combine the two factors in the denominator first and mishandling exponent subtraction order when dividing.

Final Answer:
2xyz