Simplify by using the imaginary unit \( i \). \[ \sqrt{-2} \cdot \sqrt{-8} \]

To simplify \( \sqrt{-2} \cdot \sqrt{-8} \), we first express the square roots in terms of the imaginary unit \( i \). We have: \[ \sqrt{-2} = \sqrt{2} \cdot i \] and \[ \sqrt{-8} = \sqrt{8} \cdot i = \sqrt{4 \cdot 2} \cdot i = 2\sqrt{2} \cdot i. \] Now, substitute these values into the original expression: \[ \sqrt{-2} \cdot \sqrt{-8} = (\sqrt{2} \cdot i) \cdot (2\sqrt{2} \cdot i). \] Combining the terms gives: \[ \sqrt{2} \cdot 2\sqrt{2} \cdot i^2. \] Since \( i^2 = -1 \), we can simplify: \[ \sqrt{2} \cdot 2\sqrt{2} \cdot (-1) = -2 \cdot 2 = -4. \] Thus, the simplified expression is: \[ \sqrt{-2} \cdot \sqrt{-8} = -4. \]