To determine which expression is equivalent to [tex]\(\sqrt{-80}\)[/tex], let's analyze the value step by step.
Firstly, recall that we can rewrite [tex]\(\sqrt{-80}\)[/tex] as:
[tex]\[
\sqrt{-80} = \sqrt{-1 \cdot 80}
\][/tex]
We know that [tex]\(\sqrt{-1}\)[/tex] is denoted as [tex]\(i\)[/tex], where [tex]\(i\)[/tex] is the imaginary unit. Therefore, the expression becomes:
[tex]\[
\sqrt{-80} = \sqrt{80} \cdot i
\][/tex]
Next, simplify [tex]\(\sqrt{80}\)[/tex]:
[tex]\[
\sqrt{80} = \sqrt{16 \cdot 5} = \sqrt{16} \cdot \sqrt{5}
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex], we get:
[tex]\[
\sqrt{80} = 4 \cdot \sqrt{5}
\][/tex]
Now substitute this back into the expression for [tex]\(\sqrt{-80}\)[/tex]:
[tex]\[
\sqrt{-80} = 4 \cdot \sqrt{5} \cdot i = 4i \cdot \sqrt{5}
\][/tex]
Having found that [tex]\(\sqrt{-80} = 4i \sqrt{5}\)[/tex], we can match this with the given choices:
[tex]\[
-4 \sqrt{5}
\][/tex]
[tex]\[
-4 i \sqrt{5}
\][/tex]
[tex]\[
4i \sqrt{5}
\][/tex]
[tex]\[
4 \sqrt{5}
\][/tex]
Clearly, the expression 4i [tex]\(\sqrt{5}\)[/tex] matches our derived equivalent expression for [tex]\(\sqrt{-80}\)[/tex].
Therefore, the correct answer is:
4i [tex]\(\sqrt{5}\)[/tex]
