1. **State the problem:** Simplify the expression $$\frac{(-3mn)^2 \cdot 64(m^2 n)^3}{16m^2 n^4 (mn^2)^3} \times \frac{24(m^2 n^2)^4}{3(m^2 n^3)^2}$$ 2. **Recall exponent rules:** - $(ab)^n = a^n b^n$ - $a^m \cdot a^n = a^{m+n}$ - $\frac{a^m}{a^n} = a^{m-n}$ - $(a^m)^n = a^{mn}$ 3. **Simplify each part:** - $(-3mn)^2 = (-3)^2 m^2 n^2 = 9 m^2 n^2$ - $64(m^2 n)^3 = 64 m^{2\cdot3} n^3 = 64 m^6 n^3$ - Numerator top: $9 m^2 n^2 \times 64 m^6 n^3 = 9 \times 64 m^{2+6} n^{2+3} = 576 m^8 n^5$ - Denominator top: $16 m^2 n^4 (mn^2)^3 = 16 m^2 n^4 \times m^3 n^{2\cdot3} = 16 m^{2+3} n^{4+6} = 16 m^5 n^{10}$ - First fraction simplified: $$\frac{576 m^8 n^5}{16 m^5 n^{10}}$$ 4. **Simplify the first fraction:** - Divide coefficients: $\frac{576}{16} = 36$ - Apply exponent subtraction: $$m^{8-5} = m^3$$ $$n^{5-10} = n^{-5} = \frac{1}{n^5}$$ - So first fraction is $$36 m^3 n^{-5} = \frac{36 m^3}{n^5}$$ 5. **Simplify the second fraction:** - Numerator: $24 (m^2 n^2)^4 = 24 m^{2\cdot4} n^{2\cdot4} = 24 m^8 n^8$ - Denominator: $3 (m^2 n^3)^2 = 3 m^{2\cdot2} n^{3\cdot2} = 3 m^4 n^6$ - Fraction: $$\frac{24 m^8 n^8}{3 m^4 n^6}$$ 6. **Simplify the second fraction:** - Divide coefficients: $\frac{24}{3} = 8$ - Apply exponent subtraction: $$m^{8-4} = m^4$$ $$n^{8-6} = n^2$$ - So second fraction is $$8 m^4 n^2$$ 7. **Multiply the two simplified results:** $$\frac{36 m^3}{n^5} \times 8 m^4 n^2 = 36 \times 8 \times m^{3+4} \times \frac{n^2}{n^5} = 288 m^7 n^{-3} = \frac{288 m^7}{n^3}$$ **Final answer:** $$\boxed{\frac{288 m^7}{n^3}}$$