1. **State the problem:** Multiply the expression $3u \cdot 3u^3 v^3 \cdot 4v^5$ and simplify it. 2. **Recall the product rule for exponents:** When multiplying terms with the same base, add their exponents: $$a^m \cdot a^n = a^{m+n}$$ 3. **Multiply the coefficients:** Multiply the numbers $3$, $3$, and $4$: $$3 \cdot 3 \cdot 4 = 36$$ 4. **Multiply the $u$ terms:** $$u^{1} \cdot u^{3} = u^{1+3} = u^{4}$$ 5. **Multiply the $v$ terms:** $$v^{3} \cdot v^{5} = v^{3+5} = v^{8}$$ 6. **Combine all parts:** $$36 \cdot u^{4} \cdot v^{8} = 36u^{4}v^{8}$$ **Final answer:** $$36u^{4}v^{8}$$