To solve the problem of finding an expression equivalent to \(\sqrt[5]{13^3}\), we can use properties of exponents and roots.
1. Start with the expression \(\sqrt[5]{13^3}\).
2. Recognize that a fifth root can be rewritten as an exponent with a denominator of 5. Thus, \(\sqrt[5]{x} = x^{1/5}\).
3. Apply this property to our expression:
[tex]\[
\sqrt[5]{13^3} = (13^3)^{1/5}
\][/tex]
4. When raising a power to another power, we multiply the exponents. So:
[tex]\[
(13^3)^{1/5} = 13^{3 \cdot \frac{1}{5}}
\][/tex]
5. Simplify the exponent:
[tex]\[
3 \cdot \frac{1}{5} = \frac{3}{5}
\][/tex]
6. Combine this back into the expression with the base 13:
[tex]\[
13^{\frac{3}{5}}
\][/tex]
Therefore, \(\sqrt[5]{13^3} = 13^{\frac{3}{5}}\).
The correct equivalent expression is:
[tex]\[
\boxed{13^{\frac{3}{5}}}
\][/tex]
