To simplify the expression [tex]\(\sqrt{-18}\)[/tex], follow these steps:
1. Identify the Negative Under the Square Root:
- The square root of a negative number involves the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
2. Rewrite the Expression:
- Rewrite [tex]\(\sqrt{-18}\)[/tex] as [tex]\(\sqrt{18} \cdot \sqrt{-1}\)[/tex].
- Since [tex]\(\sqrt{-1} = i\)[/tex], the expression becomes [tex]\(\sqrt{18} \cdot i\)[/tex].
3. Simplify [tex]\(\sqrt{18}\)[/tex]:
- Factorize 18 inside the square root: [tex]\(18 = 9 \times 2\)[/tex].
- Then, [tex]\(\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2}\)[/tex].
- Since [tex]\(\sqrt{9} = 3\)[/tex], it simplifies to [tex]\(3\sqrt{2}\)[/tex].
4. Combine the Results:
- Combine the results from step 2 and step 3: [tex]\(\sqrt{18} \cdot i = 3\sqrt{2} \cdot i\)[/tex].
Therefore, the simplified form of [tex]\(\sqrt{-18}\)[/tex] is [tex]\(3i\sqrt{2}\)[/tex].
Among the given choices, the correct answer is:
[tex]\[
\boxed{3 i \sqrt{2}}
\][/tex]
