1. **State the problem:** Simplify the expression $$\frac{(3m^{-2} n)^{-3}}{6mn^{-2}}$$ assuming $$m \neq 0$$ and $$n \neq 0$$. 2. **Recall the rules:** - Power of a product: $$(ab)^c = a^c b^c$$ - Negative exponent: $$a^{-b} = \frac{1}{a^b}$$ - Division of powers with the same base: $$\frac{a^m}{a^n} = a^{m-n}$$ 3. **Simplify the numerator:** $$(3m^{-2} n)^{-3} = 3^{-3} (m^{-2})^{-3} n^{-3} = \frac{1}{3^3} m^{6} n^{-3} = \frac{m^{6}}{27 n^{3}}$$ 4. **Rewrite the entire expression:** $$\frac{\frac{m^{6}}{27 n^{3}}}{6 m n^{-2}} = \frac{m^{6}}{27 n^{3}} \times \frac{1}{6 m n^{-2}} = \frac{m^{6}}{27 n^{3}} \times \frac{1}{6 m} \times n^{2}$$ 5. **Combine terms:** $$= \frac{m^{6} n^{2}}{27 \times 6 \times m \times n^{3}} = \frac{m^{6} n^{2}}{162 m n^{3}}$$ 6. **Cancel common factors:** $$= \frac{\cancel{m^{6}}^{m^{5}} \cancel{n^{2}}}{162 \cancel{m} \cancel{n^{2}} n} = \frac{m^{5}}{162 n}$$ **Final answer:** $$\frac{m^{5}}{162 n}$$ This matches the first option.