To determine which of the given expressions is an imaginary number, we need to evaluate each expression and check if the result is imaginary. Let's analyze each one step-by-step.
1. [tex]\(\sqrt{-8}\)[/tex]
This represents the square root of a negative number. In mathematics, the square root of a negative number is an imaginary number. Specifically:
[tex]\[
\sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = \sqrt{8} \cdot i
\][/tex]
where [tex]\(i\)[/tex] is the imaginary unit. Therefore, [tex]\(\sqrt{-8}\)[/tex] is indeed an imaginary number.
2. [tex]\(-\sqrt[3]{-8}\)[/tex]
This represents the negative cube root of [tex]\(-8\)[/tex]. The cube root of a negative number is a real number. For instance:
[tex]\[
\sqrt[3]{-8} = -2
\][/tex]
Therefore:
[tex]\[
-\sqrt[3]{-8} = -(-2) = 2
\][/tex]
Hence, [tex]\(-\sqrt[3]{-8}\)[/tex] is a real number, not an imaginary one.
3. [tex]\(\sqrt[3]{-8}\)[/tex]
This represents the cube root of [tex]\(-8\)[/tex]. The cube root of a negative number is indeed a real number, since:
[tex]\[
\sqrt[3]{-8} = -2
\][/tex]
Therefore, [tex]\(\sqrt[3]{-8}\)[/tex] is a real number.
4. [tex]\(-\sqrt{8}\)[/tex]
This represents the negative square root of 8. The square root of a positive number is a real number, and taking the negative of a real number remains real:
[tex]\[
\sqrt{8} \approx 2.828 \quad \text{so} \quad -\sqrt{8} \approx -2.828
\][/tex]
Hence, [tex]\(-\sqrt{8}\)[/tex] is also a real number.
Among the given expressions, only [tex]\(\sqrt{-8}\)[/tex] results in an imaginary number.
Thus, the expression that is an imaginary number is:
[tex]\[
\boxed{\sqrt{-8}}
\][/tex]
