1. **State the problem:** Simplify the expression $$(3a^3b)^2(4ab^2)^3$$. 2. **Recall the rules:** - When raising a power to another power, multiply the exponents: $$(x^m)^n = x^{mn}$$. - When multiplying like bases, add the exponents: $$x^m \cdot x^n = x^{m+n}$$. - Coefficients (numbers) multiply normally. 3. **Apply the power to each factor inside the parentheses:** $$(3a^3b)^2 = 3^2 \cdot (a^3)^2 \cdot b^2 = 9a^{6}b^{2}$$ $$(4ab^2)^3 = 4^3 \cdot a^3 \cdot (b^2)^3 = 64a^{3}b^{6}$$ 4. **Multiply the two results:** $$9a^{6}b^{2} \cdot 64a^{3}b^{6} = (9 \cdot 64) \cdot a^{6+3} \cdot b^{2+6} = 576a^{9}b^{8}$$ 5. **Final simplified expression:** $$\boxed{576a^{9}b^{8}}$$