To simplify the expression [tex]\(\sqrt{-48}\)[/tex]:
1. Recognize that the square root of a negative number can be expressed using the imaginary unit [tex]\( i \)[/tex], where [tex]\( i = \sqrt{-1} \)[/tex].
2. Therefore, [tex]\(\sqrt{-48}\)[/tex] can be broken down as [tex]\(\sqrt{-1 \times 48}\)[/tex].
[tex]\[
\sqrt{-48} = \sqrt{-1 \times 48}
\][/tex]
3. We can separate this into two parts: the square root of [tex]\(-1\)[/tex] and the square root of [tex]\(48\)[/tex].
[tex]\[
\sqrt{-48} = \sqrt{-1} \times \sqrt{48} = i \times \sqrt{48}
\][/tex]
4. Now, simplify [tex]\(\sqrt{48}\)[/tex]. Notice that [tex]\(48\)[/tex] can be factored into [tex]\(16 \times 3\)[/tex] (since [tex]\(16\)[/tex] is a perfect square).
[tex]\[
\sqrt{48} = \sqrt{16 \times 3} = \sqrt{16} \times \sqrt{3} = 4 \sqrt{3}
\][/tex]
5. Combining these results, we find:
[tex]\[
\sqrt{-48} = i \times 4 \sqrt{3} = 4i \sqrt{3}
\][/tex]
Thus, after step-by-step simplification, we get:
[tex]\[
\sqrt{-48} = 4i \sqrt{3}
\][/tex]
So, the correct answer is:
[tex]\[4i \sqrt{3}\][/tex]
Choose the correct option:
[tex]\[
\boxed{4i \sqrt{3}}
\][/tex]
