To determine which of the given expressions are equivalent to the expression \((x^8)^{1/5}\), we need to simplify each expression and see if they match.
Let's simplify each option step by step:
Expression A: \((x^8)^{1/5}\)
[tex]\[
(x^8)^{1/5} = x^{(8 \cdot 1/5)} = x^{8/5}
\][/tex]
Expression B: \(\sqrt[8]{x^5}\)
[tex]\[
\sqrt[8]{x^5} = x^{5/8}
\][/tex]
Expression C: \(\sqrt[5]{x^8}\)
[tex]\[
\sqrt[5]{x^8} = x^{8/5}
\][/tex]
Expression D: \((x^5)^{1/8}\)
[tex]\[
(x^5)^{1/8} = x^{(5 \cdot 1/8)} = x^{5/8}
\][/tex]
Expression E: \((\sqrt[8]{x})^5\)
[tex]\[
(\sqrt[8]{x})^5 = (x^{1/8})^5 = x^{(1/8 \cdot 5)} = x^{5/8}
\][/tex]
Expression F: \((\sqrt[5]{x})^8\)
[tex]\[
(\sqrt[5]{x})^8 = (x^{1/5})^8 = x^{(1/5 \cdot 8)} = x^{8/5}
\][/tex]
To see which of these expressions are equivalent to \((x^8)^{1/5} = x^{8/5}\):
- Expression A: \((x^8)^{1/5} = x^{8/5}\)
- Expression B: \(\sqrt[8]{x^5} = x^{5/8}\) (not equivalent)
- Expression C: \(\sqrt[5]{x^8} = x^{8/5}\)
- Expression D: \((x^5)^{1/8} = x^{5/8}\) (not equivalent)
- Expression E: \((\sqrt[8]{x})^5 = x^{5/8}\) (not equivalent)
- Expression F: \((\sqrt[5]{x})^8 = x^{8/5}\)
Thus, the expressions that are equivalent to \((x^8)^{1/5}\) are:
[tex]\[
\boxed{\text{A, C, F}}
\][/tex]
