1. The problem is to simplify the expression $\sqrt{-8} \cdot \sqrt{-200}$.\n\n2. Recall the property of square roots: $\sqrt{a} \cdot \sqrt{b} = \sqrt{a \cdot b}$, which holds for non-negative $a$ and $b$. For negative numbers, we use imaginary numbers where $\sqrt{-1} = i$.\n\n3. Rewrite each square root using $i$: $\sqrt{-8} = \sqrt{8} \cdot i$ and $\sqrt{-200} = \sqrt{200} \cdot i$.\n\n4. Substitute back: $\sqrt{-8} \cdot \sqrt{-200} = (\sqrt{8} \cdot i) \cdot (\sqrt{200} \cdot i) = \sqrt{8} \cdot \sqrt{200} \cdot i^2$.\n\n5. Since $i^2 = -1$, the expression becomes $\sqrt{8 \cdot 200} \cdot (-1) = -\sqrt{1600}$.\n\n6. Simplify $\sqrt{1600}$: $\sqrt{1600} = 40$.\n\n7. Therefore, the final answer is $-40$.