To determine which of the given options is equivalent to [tex]\(\sqrt[5]{13^3}\)[/tex], let's walk through the steps:
1. Understanding the notation:
[tex]\(\sqrt[5]{13^3}\)[/tex] means we want the fifth root of [tex]\(13^3\)[/tex].
2. Using exponent rules:
The expression [tex]\(\sqrt[5]{13^3}\)[/tex] can be rewritten using exponents. Specifically, the [tex]\(n\)[/tex]-th root of [tex]\(a^m\)[/tex] can be expressed as [tex]\(a^{m/n}\)[/tex]. Therefore,
[tex]\[
\sqrt[5]{13^3} = 13^{\frac{3}{5}}.
\][/tex]
3. Comparing with provided options:
- [tex]\(13^2\)[/tex] is written in traditional exponential form and does not simplify to [tex]\(\sqrt[5]{13^3}\)[/tex].
- [tex]\({ }_{13} 15\)[/tex] appears to be a typographical or notation error, and it does not align with any known mathematical expression relevant here.
- [tex]\(13^{\frac{5}{3}}\)[/tex] involves a different exponent which does not match [tex]\(\frac{3}{5}\)[/tex].
- [tex]\(13^{\frac{3}{5}}\)[/tex] matches our rewritten expression exactly.
Hence, the expression equivalent to [tex]\(\sqrt[5]{13^3}\)[/tex] is:
[tex]\[ \boxed{13^{\frac{3}{5}}} \][/tex]
