To simplify [tex]\(\sqrt{-384}\)[/tex], we need to deal with the negative sign under the square root, which involves imaginary numbers. Here's how we can simplify it step-by-step:
1. Identify the negative sign:
The square root of a negative number introduces the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
2. Separate the negative component:
We can express [tex]\(\sqrt{-384}\)[/tex] as [tex]\(\sqrt{-1 \times 384}\)[/tex].
3. Simplify using the imaginary unit [tex]\(i\)[/tex]:
Using the property that [tex]\(\sqrt{-1} = i\)[/tex], we have:
[tex]\[
\sqrt{-384} = \sqrt{384} \times \sqrt{-1} = \sqrt{384} \times i
\][/tex]
4. Simplify [tex]\(\sqrt{384}\)[/tex]:
We first find the square root of 384. The square root of 384 is approximately 19.595917942265423.
5. Combine the simplified parts:
We now have:
[tex]\[
\sqrt{-384} = 19.595917942265423i
\][/tex]
Therefore, the simplified form of [tex]\(\sqrt{-384}\)[/tex] is:
[tex]\[
\boxed{19.595917942265423i}
\][/tex]
