To determine the rational exponent expression of the cube root of [tex]\(4n\)[/tex], let us begin by understanding what the problem is asking.
Step 1: Understanding the Cube Root
The cube root of a number [tex]\(x\)[/tex] can be expressed using a rational exponent as follows:
[tex]\[
\sqrt[3]{x} = x^{\frac{1}{3}}
\][/tex]
Step 2: Applying the Cube Root to [tex]\(4n\)[/tex]
We need to apply the cube root to the expression [tex]\(4n\)[/tex]. Using the same rule mentioned above, we get:
[tex]\[
\sqrt[3]{4n} = (4n)^{\frac{1}{3}}
\][/tex]
Step 3: Identifying the Correct Option
Now, let us compare this expression with the given choices:
1. [tex]\((4n)^3\)[/tex]
2. [tex]\(3n^4\)[/tex]
3. [tex]\((4n)^{\frac{1}{3}}\)[/tex]
4. [tex]\(4n^{\frac{1}{3}}\)[/tex]
Among these options, the expression [tex]\((4n)^{\frac{1}{3}}\)[/tex] exactly matches the rational exponent form of the cube root of [tex]\(4n\)[/tex].
Step 4: Final Answer
Thus, the correct rational exponent expression of [tex]\(\sqrt[3]{4n}\)[/tex] is:
[tex]\[
(4n)^{\frac{1}{3}}
\][/tex]
Therefore, the right choice is the third option:
\boxed{3}
