To determine which expression is equivalent to [tex]\(\sqrt{-80}\)[/tex], we need to understand how to handle the square root of a negative number.
The square root of a negative number involves imaginary numbers. Specifically, the square root of [tex]\(-1\)[/tex] is represented by the imaginary unit [tex]\(i\)[/tex]. Therefore:
[tex]\[
\sqrt{-80} = \sqrt{80 \times -1} = \sqrt{80} \times \sqrt{-1} = \sqrt{80} \times i
\][/tex]
Next, we need to simplify [tex]\(\sqrt{80}\)[/tex]. The number 80 can be factored into [tex]\(16 \times 5\)[/tex]. Therefore:
[tex]\[
\sqrt{80} = \sqrt{16 \times 5} = \sqrt{16} \times \sqrt{5} = 4 \times \sqrt{5}
\][/tex]
Substituting this back into our expression, we get:
[tex]\[
\sqrt{-80} = 4 \sqrt{5} \times i = 4i \sqrt{5}
\][/tex]
Thus, the expression equivalent to [tex]\(\sqrt{-80}\)[/tex] is [tex]\(4i \sqrt{5}\)[/tex]. So, the correct choice from the given options is:
4i [tex]\(\sqrt{5}\)[/tex]
