Simplify the compound monomial quotient: Simplify (6 a^(-2) b c^(-3) / (4 a b^(-3) c^2)) ÷ (5 a^(-3) b^2 c^(-1) / (3 a b^(-2) c^3)) and express the final answer using positive/negative exponents as needed.

Introduction / Context:
This is a standard indices exercise involving negative exponents and division of monomials. The safest route is to simplify each fraction separately using index rules, then divide them. Carefully add or subtract exponents when multiplying or dividing like bases, and reduce numerical coefficients.

Given Data / Assumptions:

• First factor: (6 a^(-2) b c^(-3)) / (4 a b^(-3) c^2).
• Second factor: (5 a^(-3) b^2 c^(-1)) / (3 a b^(-2) c^3).
• All variables represent nonzero real numbers (so divisions are valid).

Concept / Approach:
Use three rules repeatedly: a^m / a^n = a^(m−n); (ab)^m = a^m b^m; and handle coefficients separately. Simplify each fraction to a single monomial, then compute the quotient of those monomials by subtracting exponents and dividing coefficients.

Step-by-Step Solution:

Verification / Alternative check:
Convert to positive-exponent form: 9a/(10c). Any equivalent representation matches the simplified result.

Why Other Options Are Wrong:
Options with c, c^2, or c^(−3) come from exponent slips; a^(−1) would require subtracting in reverse.

Common Pitfalls:
Mixing the direction of subtraction for exponents during division and forgetting to invert the second coefficient when dividing fractions.

Final Answer:
9/10 · a c^-1