To determine which expression is equivalent to [tex]\(\sqrt{-80}\)[/tex], let's break it down step by step:
1. Understand the complex number property:
The square root of a negative number involves an imaginary unit [tex]\(i\)[/tex], where [tex]\(i = \sqrt{-1}\)[/tex].
2. Express [tex]\(\sqrt{-80}\)[/tex] using the imaginary unit:
We know that [tex]\(\sqrt{-80}\)[/tex] can be written as:
[tex]\[
\sqrt{-80} = \sqrt{-1 \times 80} = \sqrt{-1} \times \sqrt{80}
\][/tex]
Since [tex]\(\sqrt{-1} = i\)[/tex], we can write:
[tex]\[
\sqrt{-80} = i \times \sqrt{80}
\][/tex]
3. Simplify [tex]\(\sqrt{80}\)[/tex]:
To simplify [tex]\(\sqrt{80}\)[/tex], we factorize 80 into its prime factors:
[tex]\[
80 = 16 \times 5
\][/tex]
Therefore, we can write:
[tex]\[
\sqrt{80} = \sqrt{16 \times 5}
\][/tex]
Using the property of square roots, [tex]\(\sqrt{ab} = \sqrt{a} \times \sqrt{b}\)[/tex], we get:
[tex]\[
\sqrt{80} = \sqrt{16} \times \sqrt{5}
\][/tex]
Since [tex]\(\sqrt{16} = 4\)[/tex], this becomes:
[tex]\[
\sqrt{80} = 4 \times \sqrt{5}
\][/tex]
4. Combine the results:
Putting it all together, we have:
[tex]\[
\sqrt{-80} = i \times \sqrt{80} = i \times (4 \times \sqrt{5}) = 4i \times \sqrt{5}
\][/tex]
Therefore, the expression that is equivalent to [tex]\(\sqrt{-80}\)[/tex] is:
[tex]\[
\boxed{4i \sqrt{5}}
\][/tex]
