1. **State the problem:** Simplify the expression $$\left( \frac{3a^2 b^3}{cd} \right)^3 \div \left( \frac{3ab^4}{c^2 d^3} \right)^2$$. 2. **Recall the rules:** - When raising a fraction to a power, raise both numerator and denominator to that power: $$\left( \frac{x}{y} \right)^n = \frac{x^n}{y^n}$$. - Division of fractions is multiplication by the reciprocal: $$\frac{A}{B} \div \frac{C}{D} = \frac{A}{B} \times \frac{D}{C}$$. - When multiplying powers with the same base, add exponents: $$x^m \times x^n = x^{m+n}$$. 3. **Apply the powers:** $$\left( \frac{3a^2 b^3}{cd} \right)^3 = \frac{3^3 (a^2)^3 (b^3)^3}{c^3 d^3} = \frac{27 a^{6} b^{9}}{c^{3} d^{3}}$$ $$\left( \frac{3ab^4}{c^2 d^3} \right)^2 = \frac{3^2 (a)^2 (b^4)^2}{(c^2)^2 (d^3)^2} = \frac{9 a^{2} b^{8}}{c^{4} d^{6}}$$ 4. **Rewrite the division as multiplication by reciprocal:** $$\frac{27 a^{6} b^{9}}{c^{3} d^{3}} \div \frac{9 a^{2} b^{8}}{c^{4} d^{6}} = \frac{27 a^{6} b^{9}}{c^{3} d^{3}} \times \frac{c^{4} d^{6}}{9 a^{2} b^{8}}$$ 5. **Multiply numerators and denominators:** $$= \frac{27 a^{6} b^{9} c^{4} d^{6}}{9 a^{2} b^{8} c^{3} d^{3}}$$ 6. **Simplify coefficients:** $$\frac{\cancel{27}^{3 \times 9} \cdot \cancel{a^{6}} \cdot \cancel{b^{9}} \cdot c^{4} \cdot d^{6}}{\cancel{9}^{3 \times 3} \cdot \cancel{a^{2}} \cdot \cancel{b^{8}} \cdot c^{3} \cdot d^{3}} = 3 \cdot a^{6-2} \cdot b^{9-8} \cdot c^{4-3} \cdot d^{6-3}$$ 7. **Calculate exponents:** $$= 3 a^{4} b^{1} c^{1} d^{3} = 3 a^{4} b c d^{3}$$ **Final answer:** $$\boxed{3 a^{4} b c d^{3}}$$