Let's simplify the given expression:

$\frac{\sqrt{3 \cdot 5^{-3}}}{\sqrt{3^{-1}}} \times \sqrt{5} \times \sqrt[5]{3 \cdot 5^4}$

Step-by-step simplification:

1. Simplify the square roots:

   • $\sqrt{3 \cdot 5^{-3}}$ can be written as $\sqrt{3} \times \sqrt{5^{-3}} = \sqrt{3} \times 5^{-\frac{3}{2}}$.
   • $\sqrt{3^{-1}} = 3^{-\frac{1}{2}}$.

   So the first part of the expression becomes:

   $\frac{\sqrt{3} \times 5^{-\frac{3}{2}}}{3^{-\frac{1}{2}}}$

2. Simplify this fraction: Using the properties of exponents:

   $\frac{\sqrt{3}}{3^{-\frac{1}{2}}} = 3^{\frac{1}{2} + \frac{1}{2}} = 3^1 = 3$

   So the expression simplifies to:

   $3 \times 5^{-\frac{3}{2}}$

3. Include the rest of the expression: Now include the rest of the terms: $\sqrt{5} \times \sqrt[5]{3 \cdot 5^4}$.

   Let's simplify $\sqrt[5]{3 \cdot 5^4}$:

   • (\sqrt[5]{