To determine the rational exponent expression of [tex]\(\sqrt[3]{4n}\)[/tex], let's carefully analyze the situation.
### Understanding the Problem
The cube root of any expression [tex]\(x\)[/tex] is written as [tex]\(x^{1/3}\)[/tex]. Therefore, we need to express [tex]\(\sqrt[3]{4n}\)[/tex] in the form of [tex]\((4n)^{1/3}\)[/tex].
### Step-by-Step Solution
1. Identify the Expression:
- The given expression is [tex]\(\sqrt[3]{4n}\)[/tex], which signifies taking the cube root of the product [tex]\(4n\)[/tex].
2. Convert the Cube Root to an Exponent:
- The cube root of [tex]\(4n\)[/tex] can be written using a rational exponent:
[tex]\[
\sqrt[3]{4n} = (4n)^{1/3}
\][/tex]
3. Check Correspondence with Given Options:
- Let's match [tex]\((4n)^{1/3}\)[/tex] with the given choices:
1. [tex]\(3n^4\)[/tex] - This is not correct as it does not match our expression.
2. [tex]\((4n)^3\)[/tex] - This is incorrect because it represents the cube of [tex]\(4n\)[/tex], not the cube root.
3. [tex]\(4n^{1/3}\)[/tex] - This is incorrect because it only takes the cube root of [tex]\(n\)[/tex] and not the entirety of [tex]\(4n\)[/tex].
4. [tex]\((4n)^{1/3}\)[/tex] - This matches our resulting expression, representing the cube root of the product [tex]\(4n\)[/tex].
### Conclusion
Thus, the correct rational exponent expression for [tex]\(\sqrt[3]{4n}\)[/tex] is:
[tex]\[
(4n)^{1/3}
\][/tex]
