1. **State the problem:** Simplify the expression $$\frac{(12 m^5 n^{-2})(5 m^{-11} n^6)}{15 m^3 n^{-4}}$$. 2. **Multiply the numerator terms:** $$12 \times 5 = 60$$ $$m^{5} \times m^{-11} = m^{5 + (-11)} = m^{-6}$$ $$n^{-2} \times n^{6} = n^{-2 + 6} = n^{4}$$ So numerator becomes $$60 m^{-6} n^{4}$$. 3. **Rewrite the expression:** $$\frac{60 m^{-6} n^{4}}{15 m^{3} n^{-4}}$$ 4. **Divide coefficients:** $$\frac{60}{15} = 4$$ 5. **Divide variables using exponent subtraction:** $$m^{-6} \div m^{3} = m^{-6 - 3} = m^{-9}$$ $$n^{4} \div n^{-4} = n^{4 - (-4)} = n^{8}$$ 6. **Combine all parts:** $$4 m^{-9} n^{8}$$ 7. **Express negative exponent as reciprocal:** $$4 \frac{n^{8}}{m^{9}}$$ **Final simplified expression:** $$\boxed{\frac{4 n^{8}}{m^{9}}}$$