Simplify a quotient of monomials using index laws: Evaluate 6a^3 b^3 c^2 ÷ 2a b^2 c.

Introduction / Context:
This is a straightforward monomial simplification using exponent subtraction for like bases and coefficient division. Such manipulations are routine in algebraic simplification before factorization or substitution steps.

Given Data / Assumptions:

• Expression: (6a^3 b^3 c^2) / (2a b^2 c).

Concept / Approach:
For like bases, a^m / a^n = a^(m−n). Also divide numeric coefficients: 6/2 = 3.

Step-by-Step Solution:

Verification / Alternative check:
Plug small values (e.g., a=b=c=2): LHS = 6*8*8*4 / (2*2*4*2) = 1536 / 32 = 48; RHS = 3*(4)*2*2 = 48. Matches.

Why Other Options Are Wrong:

• They have incorrect exponent arithmetic or extra factors, contradicting the laws a^m / a^n = a^(m−n).

Common Pitfalls:
Adding exponents during division or miscounting one of the variables’ exponents.

Final Answer: