To determine which of the given options is equivalent to the expression [tex]\(\sqrt[5]{13^3}\)[/tex], we can use properties of exponents and radicals. The notation [tex]\(\sqrt[5]{13^3}\)[/tex] represents the fifth root of [tex]\(13^3\)[/tex].
Recall the property of exponents and radicals:
[tex]\[
\sqrt[n]{a^m} = a^{\frac{m}{n}}
\][/tex]
Thus, [tex]\(\sqrt[5]{13^3}\)[/tex] can be rewritten using this property:
[tex]\[
\sqrt[5]{13^3} = 13^{\frac{3}{5}}
\][/tex]
Now, let's compare this to the options provided:
1. [tex]\(13^2\)[/tex]
2. [tex]\(13^{15}\)[/tex]
3. [tex]\(13^{\frac{5}{3}}\)[/tex]
4. [tex]\(13^{\frac{3}{5}}\)[/tex]
Clearly, none of the first three options have the same exponent as [tex]\(\sqrt[5]{13^3}\)[/tex]. Only the fourth option,
[tex]\[ 13^{\frac{3}{5}}, \][/tex]
matches the form we derived.
Therefore, the expression [tex]\(\sqrt[5]{13^3}\)[/tex] is equivalent to [tex]\(13^{\frac{3}{5}}\)[/tex].
