1. **State the problem:** Simplify the expression $$\frac{\frac{4}{9} \left(4c^3 d \frac{1}{2}\right)^2}{3c^5 \frac{3}{2} d \frac{5}{2}}$$. 2. **Rewrite the expression clearly:** $$\frac{\frac{4}{9} \left(4c^3 d \cdot \frac{1}{2}\right)^2}{3c^5 \cdot \frac{3}{2} \cdot d \cdot \frac{5}{2}}$$ 3. **Simplify inside the parentheses:** $$4c^3 d \cdot \frac{1}{2} = 2c^3 d$$ 4. **Square the term:** $$\left(2c^3 d\right)^2 = 2^2 \cdot (c^3)^2 \cdot d^2 = 4c^6 d^2$$ 5. **Substitute back:** $$\frac{\frac{4}{9} \cdot 4c^6 d^2}{3c^5 \cdot \frac{3}{2} \cdot d \cdot \frac{5}{2}} = \frac{\frac{16}{9} c^6 d^2}{3c^5 \cdot \frac{3}{2} \cdot d \cdot \frac{5}{2}}$$ 6. **Multiply the denominator constants:** $$3 \times \frac{3}{2} \times \frac{5}{2} = 3 \times \frac{15}{4} = \frac{45}{4}$$ 7. **Rewrite denominator:** $$\frac{45}{4} c^5 d$$ 8. **Rewrite the whole fraction:** $$\frac{\frac{16}{9} c^6 d^2}{\frac{45}{4} c^5 d} = \frac{16}{9} c^6 d^2 \times \frac{4}{45} \times \frac{1}{c^5 d}$$ 9. **Simplify variables:** $$c^{6} \div c^{5} = c^{6-5} = c$$ $$d^{2} \div d = d^{2-1} = d$$ 10. **Simplify constants:** $$\frac{16}{9} \times \frac{4}{45} = \frac{64}{405}$$ 11. **Final simplified expression:** $$\frac{64}{405} c d$$ **Answer:** $$\boxed{\frac{64}{405} c d}$$