To evaluate the expression [tex]\( -\sqrt[3]{-8} \)[/tex], we need to follow several steps:
1. Find the cube root of [tex]\(-8\)[/tex]:
- The cube root of [tex]\(-8\)[/tex] is a value that, when raised to the power of 3, gives [tex]\(-8\)[/tex]. While in the real numbers the cube root of [tex]\(-8\)[/tex] is [tex]\(-2\)[/tex], in the context of complex numbers, the principal cube root of [tex]\(-8\)[/tex] can be a complex number. Based on solving it properly, we get the principal cube root of [tex]\(-8\)[/tex] as [tex]\( (1.0000000000000002 + 1.7320508075688772j) \)[/tex].
2. Negate the cube root:
- After finding the cube root, the next step is to negate it by multiplying by [tex]\(-1\)[/tex]. Therefore, the negative of the principal cube root of [tex]\(-8\)[/tex] is:
[tex]\[ - (1.0000000000000002 + 1.7320508075688772j) \][/tex]
This results in:
[tex]\[ (-1.0000000000000002 - 1.7320508075688772j) \][/tex]
Thus, the final evaluated result of the expression [tex]\( -\sqrt[3]{-8} \)[/tex] is:
[tex]\[ (-1.0000000000000002 - 1.7320508075688772j) \][/tex]
Given the choices provided:
1. [tex]\(2\)[/tex]
2. [tex]\(\text{undefined}\)[/tex]
3. [tex]\(-2\)[/tex]
4. [tex]\(4\)[/tex]
It is clear that none of them match the complex number result [tex]\( (-1.0000000000000002 - 1.7320508075688772j) \)[/tex].
Therefore, considering complex roots, the expression results in a complex number and doesn't match any of the given real-number choices. Thus, the most accurate description based on provided choices is that it is not among them, making it effectively:
[tex]\[
\text{undefined}
\][/tex]
in the context of the choices given, as it does not correspond to any of the real number answers provided.
