To determine which rational exponent represents a cube root, it's important to understand the relationship between exponents and roots in general.
A root can be expressed as an exponent. Specifically, the [tex]\(n\)[/tex]th root of a number [tex]\(a\)[/tex] is written as [tex]\(a^{\frac{1}{n}}\)[/tex].
Here, we are interested in the cube root, which is the 3rd root of a number.
To express the cube root of a number [tex]\(a\)[/tex] using exponents, we use the rational exponent [tex]\(\frac{1}{3}\)[/tex]:
[tex]\[
\sqrt[3]{a} = a^{\frac{1}{3}}
\][/tex]
Thus, the correct rational exponent that represents a cube root is [tex]\(\frac{1}{3}\)[/tex].
This corresponds to option B, [tex]\(\frac{1}{3}\)[/tex].
So, the answer is B.
