1. **Problem:** Simplify the expression $$\left(\frac{12x^3 y^2}{9xy}\right) \left(\frac{18x^5 y^3}{4x^2 y^2}\right)$$. 2. **Formula and rules:** - When multiplying fractions, multiply numerators together and denominators together. - For powers with the same base, use the rule $$a^m \times a^n = a^{m+n}$$. - When dividing powers with the same base, use $$\frac{a^m}{a^n} = a^{m-n}$$. - Simplify coefficients by dividing. 3. **Step-by-step simplification:** - Simplify each fraction first: $$\frac{12x^3 y^2}{9xy} = \frac{12}{9} \times \frac{x^3}{x} \times \frac{y^2}{y} = \frac{4}{3} x^{3-1} y^{2-1} = \frac{4}{3} x^2 y$$ - Simplify the second fraction: $$\frac{18x^5 y^3}{4x^2 y^2} = \frac{18}{4} \times x^{5-2} \times y^{3-2} = \frac{9}{2} x^3 y$$ - Multiply the simplified fractions: $$\left(\frac{4}{3} x^2 y\right) \times \left(\frac{9}{2} x^3 y\right) = \frac{4}{3} \times \frac{9}{2} \times x^{2+3} \times y^{1+1} = \frac{36}{6} x^5 y^2 = 6 x^5 y^2$$ 4. **Final answer:** $$6 x^5 y^2$$