Let's work on simplifying the expression [tex]\(-\sqrt{-200}\)[/tex] step by step.
1. The given expression is [tex]\(-\sqrt{-200}\)[/tex].
2. First, address the negative sign inside the square root. We know that [tex]\(\sqrt{-1} = i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit. So,
[tex]\[
\sqrt{-200} = \sqrt{-1 \times 200} = \sqrt{-1} \times \sqrt{200} = i \sqrt{200}
\][/tex]
3. Next, simplify [tex]\(\sqrt{200}\)[/tex]. Notice that [tex]\(200\)[/tex] can be factored into [tex]\(100 \times 2\)[/tex], thus:
[tex]\[
\sqrt{200} = \sqrt{100 \times 2} = \sqrt{100} \times \sqrt{2}
\][/tex]
4. We know that [tex]\(\sqrt{100} = 10\)[/tex], so:
[tex]\[
\sqrt{200} = 10 \times \sqrt{2}
\][/tex]
5. Now substitute back into the expression we were simplifying:
[tex]\[
\sqrt{-200} = i \times 10 \times \sqrt{2} = 10i \sqrt{2}
\][/tex]
6. Finally, apply the negative sign from the original problem:
[tex]\[
-\sqrt{-200} = -10i \sqrt{2}
\][/tex]
Therefore, the simplified expression is:
[tex]\[
-\sqrt{-200} = -10i \sqrt{2}
\][/tex]
The correct answer is [tex]\(D. -10i \sqrt{2}\)[/tex].
