To determine which expression is equivalent to [tex]\(\sqrt[5]{13^3}\)[/tex], let's break it down step by step:
1. Understand the given expression:
[tex]\[
\sqrt[5]{13^3}
\][/tex]
2. Express using exponents:
The fifth root of a number can be written as an exponent of [tex]\(\frac{1}{5}\)[/tex].
[tex]\[
\sqrt[5]{13^3} = (13^3)^{\frac{1}{5}}
\][/tex]
3. Simplify using the power rule for exponents:
[tex]\[
(13^3)^{\frac{1}{5}} = 13^{3 \cdot \frac{1}{5}} = 13^{\frac{3}{5}}
\][/tex]
Thus, the expression [tex]\(\sqrt[5]{13^3}\)[/tex] simplifies to [tex]\(13^{\frac{3}{5}}\)[/tex].
Given the options:
- [tex]\(13^2\)[/tex]
- 1315
- [tex]\(13^{\frac{5}{3}}\)[/tex]
- [tex]\(13^{\frac{3}{5}}\)[/tex]
The correct equivalent expression is:
[tex]\[
13^{\frac{3}{5}}
\][/tex]
So, the answer is [tex]\(13^{\frac{3}{5}}\)[/tex].
