To determine which expression is equivalent to [tex]\(\sqrt[3]{8}^x\)[/tex], let's start by rewriting [tex]\(\sqrt[3]{8}\)[/tex] in exponential form.
Firstly, we know that:
[tex]\[
\sqrt[3]{8} \text{ is equivalent to } 8^{\frac{1}{3}}
\][/tex]
Therefore:
[tex]\[
\sqrt[3]{8}^x = (8^{\frac{1}{3}})^x
\][/tex]
Now apply the power of a power property of exponents, which states [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]:
[tex]\[
(8^{\frac{1}{3}})^x = 8^{(\frac{1}{3} \cdot x)} = 8^{\frac{x}{3}}
\][/tex]
Thus, the expression [tex]\(8^{\frac{x}{3}}\)[/tex] is equivalent to [tex]\(\sqrt[3]{8}^x\)[/tex].
So, the correct answer is
[tex]\[
8^{\frac{x}{3}}
\][/tex]
