Let's analyze each of the given choices and determine if they are equivalent to the expression [tex]\((x^5)^{1/8}\)[/tex].
The given expression is [tex]\((x^5)^{1/8}\)[/tex]. This expression represents the eighth root of [tex]\(x^5\)[/tex] or equivalently [tex]\(x\)[/tex] raised to the power of [tex]\(\frac{5}{8}\)[/tex].
Now, let's examine each choice:
### Choice A: [tex]\((x^5)^{1/8}\)[/tex]
This is exactly the given expression. Thus, it is equivalent to [tex]\((x^5)^{1/8}\)[/tex].
### Choice B: [tex]\((x^8)^{1/5}\)[/tex]
This expression simplifies as follows:
[tex]\[
(x^8)^{1/5} = x^{8/5}
\][/tex]
Clearly, this is not equivalent to [tex]\((x^5)^{1/8}\)[/tex], because [tex]\(\frac{8}{5}\)[/tex] is not the same as [tex]\(\frac{5}{8}\)[/tex].
### Choice C: [tex]\(\sqrt[5]{x^8}\)[/tex]
This expression can be simplified using fractional exponents:
[tex]\[
\sqrt[5]{x^8} = x^{8/5}
\][/tex]
Again, [tex]\(\frac{8}{5}\)[/tex] is not equivalent to [tex]\(\frac{5}{8}\)[/tex], so this is not equivalent to [tex]\((x^5)^{1/8}\)[/tex].
### Choice D: [tex]\((\sqrt[5]{x})^8\)[/tex]
This expression can be simplified as follows:
[tex]\[
(\sqrt[5]{x})^8 = (x^{1/5})^8 = x^{8/5}
\][/tex]
Since [tex]\(\frac{8}{5}\)[/tex] is not equivalent to [tex]\(\frac{5}{8}\)[/tex], this is also not equivalent to [tex]\((x^5)^{1/8}\)[/tex].
### Choice E: [tex]\((\sqrt[8]{x})^5\)[/tex]
Simplifying this expression:
[tex]\[
(\sqrt[8]{x})^5 = (x^{1/8})^5 = x^{5/8}
\][/tex]
This is equivalent to [tex]\((x^5)^{1/8}\)[/tex], as [tex]\(\frac{5}{8}\)[/tex] matches [tex]\(\frac{5}{8}\)[/tex].
### Choice F: [tex]\(\sqrt[8]{x^5}\)[/tex]
This expression "eighth root of [tex]\(x^5\)[/tex]" can be rewritten as:
[tex]\[
\sqrt[8]{x^5} = (x^5)^{1/8} = x^{5/8}
\][/tex]
Thus, this is also equivalent to [tex]\((x^5)^{1/8}\)[/tex], since [tex]\(\frac{5}{8}\)[/tex] equals [tex]\(\frac{5}{8}\)[/tex].
Given the analysis, the choices that are equivalent to [tex]\((x^5)^{1/8}\)[/tex] are:
- A. [tex]\((x^5)^{1/8}\)[/tex]
- E. [tex]\((\sqrt[8]{x})^5\)[/tex]
- F. [tex]\(\sqrt[8]{x^5}\)[/tex]
Therefore, the correct choices are: [tex]\[ \boxed{1, 5, 6} \][/tex]
