To determine which choices are equivalent to the expression [tex]\(\sqrt{-16}\)[/tex], we need to understand how to simplify square roots of negative numbers. In mathematics, the square root of a negative number can be expressed using the imaginary unit [tex]\(i\)[/tex], where [tex]\(i\)[/tex] is defined as [tex]\(\sqrt{-1}\)[/tex].
Step-by-Step Solution:
1. Simplify the given expression:
[tex]\[
\sqrt{-16} = \sqrt{16 \cdot (-1)}
\][/tex]
We can separate this into two parts:
[tex]\[
\sqrt{16} \cdot \sqrt{-1}
\][/tex]
Now we know:
[tex]\[
\sqrt{16} = 4
\][/tex]
and
[tex]\[
\sqrt{-1} = i
\][/tex]
Therefore:
[tex]\[
\sqrt{-16} = 4i
\][/tex]
2. Analyze each choice:
- Choice A: [tex]\(4i\)[/tex]
[tex]\[
4i = 4i
\][/tex]
This is equivalent to [tex]\(\sqrt{-16}\)[/tex].
- Choice B: -4
[tex]\[
-4
\][/tex]
This is not a complex number and is not equivalent to [tex]\(\sqrt{-16}\)[/tex].
- Choice C: [tex]\(i \sqrt{16}\)[/tex]
[tex]\[
i \sqrt{16} = i \cdot 4 = 4i
\][/tex]
This is equivalent to [tex]\(\sqrt{-16}\)[/tex].
- Choice D: [tex]\(-\sqrt{16}\)[/tex]
[tex]\[
-\sqrt{16} = -4
\][/tex]
This is not equivalent to [tex]\(\sqrt{-16}\)[/tex].
3. Conclusion:
The choices that match the expression [tex]\(\sqrt{-16}\)[/tex] are:
- [tex]\(4i\)[/tex] which is choice A.
- [tex]\(i \sqrt{16}\)[/tex] which is choice C.
Therefore, the correct choices are A and C.
