To determine which expression is equivalent to [tex]\(\left(\frac{1}{z^{3/5}}\right)^{-1/5}\)[/tex], we need to simplify it step-by-step.
1. Let's start by rewriting the initial expression:
[tex]\[
\left(\frac{1}{z^{3/5}}\right)^{-1/5}
\][/tex]
2. Recall that [tex]\(\frac{1}{z^{3/5}}\)[/tex] can be rewritten as [tex]\(z^{-3/5}\)[/tex]. So the expression becomes:
[tex]\[
\left(z^{-3/5}\right)^{-1/5}
\][/tex]
3. Next, apply the exponent rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex]. Here, [tex]\(a = z\)[/tex], [tex]\(m = -3/5\)[/tex], and [tex]\(n = -1/5\)[/tex].
[tex]\[
\left(z^{-3/5}\right)^{-1/5} = z^{(-3/5) \cdot (-1/5)}
\][/tex]
4. Now, we need to multiply the exponents [tex]\(-3/5\)[/tex] and [tex]\(-1/5\)[/tex]:
[tex]\[
(-3/5) \times (-1/5) = 3/25
\][/tex]
5. Therefore, the expression simplifies to:
[tex]\[
z^{3/25}
\][/tex]
So, the expression equivalent to [tex]\(\left(\frac{1}{z^{3/5}}\right)^{-1/5}\)[/tex] is:
[tex]\[
\boxed{z^{3/25}}
\][/tex]
Thus, the correct answer is:
A. [tex]\(z^{3/25}\)[/tex]
