Questions: Perform the indicated operation simplify. Express the answer in terms of i (as a complex number). sqrt(-8) * sqrt(-200) =

Solution Steps

To solve the problem, we need to express the square roots of negative numbers in terms of the imaginary unit \(i\), where \(i = \sqrt{-1}\). We can rewrite \(\sqrt{-8}\) as \(\sqrt{8} \cdot i\) and \(\sqrt{-200}\) as \(\sqrt{200} \cdot i\). Then, multiply these expressions together and simplify the result.

We start by expressing \(\sqrt{-8}\) in terms of \(i\): \[ \sqrt{-8} = \sqrt{8} \cdot i = 2\sqrt{2} \cdot i \approx 2.8284i \]

Next, we express \(\sqrt{-200}\) in terms of \(i\): \[ \sqrt{-200} = \sqrt{200} \cdot i = 10\sqrt{2} \cdot i \approx 14.1421i \]

Now, we multiply the two results: \[ \sqrt{-8} \cdot \sqrt{-200} = (2\sqrt{2} \cdot i) \cdot (10\sqrt{2} \cdot i) = 20 \cdot 2 \cdot i^2 = 40(-1) = -40 \]

Final Answer