1. **State the problem:** Simplify the expression $$\left( \frac{-vu^3}{2u^4v^{-3} \cdot -u^4v^{-4}} \right)^3$$. 2. **Rewrite the denominator:** Multiply the terms inside the denominator: $$2u^4v^{-3} \cdot -u^4v^{-4} = 2 \cdot (-1) \cdot u^{4+4} \cdot v^{-3 + (-4)} = -2u^8v^{-7}$$. 3. **Rewrite the entire fraction:** $$\frac{-vu^3}{-2u^8v^{-7}}$$ 4. **Simplify the fraction:** The negatives cancel out: $$\frac{-vu^3}{-2u^8v^{-7}} = \frac{vu^3}{2u^8v^{-7}}$$ 5. **Divide powers with the same base:** $$\frac{v}{v^{-7}} = v^{1 - (-7)} = v^{8}$$ $$\frac{u^3}{u^8} = u^{3 - 8} = u^{-5}$$ 6. **Rewrite the simplified fraction:** $$\frac{vu^3}{2u^8v^{-7}} = \frac{v^{8}u^{-5}}{2} = \frac{v^{8}}{2u^{5}}$$ 7. **Apply the cube to the entire fraction:** $$\left( \frac{v^{8}}{2u^{5}} \right)^3 = \frac{v^{24}}{2^3 u^{15}} = \frac{v^{24}}{8u^{15}}$$ **Final answer:** $$\boxed{\frac{v^{24}}{8u^{15}}}$$