To simplify [tex]\(\sqrt{-48}\)[/tex], we follow a step-by-step approach as detailed below:
1. Recognize that we are dealing with the square root of a negative number. To handle this, we use the imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
[tex]\[
\sqrt{-48} = \sqrt{(-1) \times 48}
\][/tex]
2. Break this down using [tex]\(\sqrt{-1} = i\)[/tex]:
[tex]\[
\sqrt{-48} = \sqrt{-1} \times \sqrt{48} = i \times \sqrt{48}
\][/tex]
3. Next, simplify [tex]\(\sqrt{48}\)[/tex]. To do this, factor 48 into its prime factors and look for perfect squares:
[tex]\[
48 = 16 \times 3
\][/tex]
Therefore,
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3}
\][/tex]
4. Since [tex]\(\sqrt{16} = 4\)[/tex]:
[tex]\[
\sqrt{48} = 4 \times \sqrt{3}
\][/tex]
5. Combine this with the previously included [tex]\(i\)[/tex]:
[tex]\[
i \times \sqrt{48} = i \times (4 \times \sqrt{3}) = 4i \times \sqrt{3}
\][/tex]
6. Thus, the simplified form of [tex]\(\sqrt{-48}\)[/tex] is:
[tex]\[
\sqrt{-48} = 4i \sqrt{3}
\][/tex]
Therefore, the correct option is:
[tex]\[
\boxed{4i \sqrt{3}}
\][/tex]
