1. **State the problem:** Multiply the expressions $ \frac{3m^5 n^6}{8m - 6n} $ and $ \frac{24m - 18n}{27m^2 n^{10}} $ and simplify. 2. **Write the multiplication:** $$ \frac{3m^5 n^6}{8m - 6n} \times \frac{24m - 18n}{27m^2 n^{10}} $$ 3. **Factor numerators and denominators where possible:** - Factor out common terms in denominators and numerators: $$ 8m - 6n = 2(4m - 3n) $$ $$ 24m - 18n = 6(4m - 3n) $$ 4. **Rewrite the expression with factored terms:** $$ \frac{3m^5 n^6}{2(4m - 3n)} \times \frac{6(4m - 3n)}{27m^2 n^{10}} $$ 5. **Multiply the numerators and denominators:** $$ \frac{3m^5 n^6 \times 6(4m - 3n)}{2(4m - 3n) \times 27m^2 n^{10}} $$ 6. **Cancel common factors $ (4m - 3n) $:** $$ \frac{3m^5 n^6 \times \cancel{6}(\cancel{4m - 3n})}{2(\cancel{4m - 3n}) \times 27m^2 n^{10}} $$ 7. **Simplify constants:** $$ \frac{3m^5 n^6 \times 6}{2 \times 27m^2 n^{10}} = \frac{18m^5 n^6}{54m^2 n^{10}} $$ 8. **Cancel common factors in constants:** $$ \frac{\cancel{18}^3 m^5 n^6}{\cancel{54}^9 m^2 n^{10}} = \frac{3 m^5 n^6}{9 m^2 n^{10}} $$ 9. **Cancel powers of variables using $ a^m / a^n = a^{m-n} $:** $$ 3 \times m^{5-2} \times n^{6-10} / 9 = \frac{3 m^3 n^{-4}}{9} $$ 10. **Simplify constants:** $$ \frac{3}{9} = \frac{1}{3} $$ 11. **Final simplified expression:** $$ \frac{1}{3} m^3 n^{-4} = \frac{m^3}{3 n^4} $$ **Answer:** $$ \boxed{\frac{m^3}{3 n^4}} $$