1. **State the problem:** Multiply and simplify the expression assuming $u \geq 0$: $$\sqrt{6u^5} \cdot \sqrt{3u^3} \div \sqrt{3^2} \cdot 2 \cdot (u^2)^4$$ 2. **Use the property of square roots:** $$\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}$$ and $$(a^m)^n = a^{mn}$$ 3. **Multiply the square roots in the numerator:** $$\sqrt{6u^5} \cdot \sqrt{3u^3} = \sqrt{(6u^5)(3u^3)} = \sqrt{18u^{8}}$$ 4. **Simplify the denominator:** $$\sqrt{3^2} \cdot 2 \cdot (u^2)^4 = \sqrt{9} \cdot 2 \cdot u^{8} = 3 \cdot 2 \cdot u^{8} = 6u^{8}$$ 5. **Rewrite the expression:** $$\frac{\sqrt{18u^{8}}}{6u^{8}}$$ 6. **Simplify the square root:** $$\sqrt{18u^{8}} = \sqrt{9 \cdot 2 \cdot u^{8}} = \sqrt{9} \cdot \sqrt{2} \cdot \sqrt{u^{8}} = 3 \cdot \sqrt{2} \cdot u^{4}$$ 7. **Substitute back:** $$\frac{3 \cdot \sqrt{2} \cdot u^{4}}{6u^{8}}$$ 8. **Simplify the fraction:** $$\frac{3}{6} = \frac{\cancel{3}}{\cancel{6}} = \frac{1}{2}$$ and $$\frac{u^{4}}{u^{8}} = u^{4-8} = u^{-4} = \frac{1}{u^{4}}$$ 9. **Final simplified expression:** $$\frac{\sqrt{2}}{2u^{4}}$$ **Answer:** $\boxed{\frac{\sqrt{2}}{2u^{4}}}$