Simplify. \[ \left(-5 u v^{3}\right)^{3} \] Write your answer without parentheses.

To simplify \(\left(-5 u v^{3}\right)^{3}\), we apply the power of a product rule. This means we will raise each factor inside the parentheses to the third power: \[ (-5)^{3} \cdot (u)^{3} \cdot (v^{3})^{3} \] Calculating each part: \[ (-5)^{3} = -125, \] \[ (u)^{3} = u^{3}, \] \[ (v^{3})^{3} = v^{9}. \] Putting it all together, we have: \[ -125 u^{3} v^{9}. \] Thus, the simplified expression is: \[ \boxed{-125 u^{3} v^{9}}. \]