To solve the problem of expressing the cube root of [tex]\(4n\)[/tex] using rational exponents, we need to understand the relationship between roots and exponents.
In general, the [tex]\(n\)[/tex]-th root of a number [tex]\(a\)[/tex] can be expressed using a rational exponent as [tex]\(a^{\frac{1}{n}}\)[/tex].
For example:
- The square root of [tex]\(a\)[/tex], [tex]\(\sqrt{a}\)[/tex], can be written as [tex]\(a^{\frac{1}{2}}\)[/tex].
- The cube root of [tex]\(a\)[/tex], [tex]\(\sqrt[3]{a}\)[/tex], can be written as [tex]\(a^{\frac{1}{3}}\)[/tex].
Given the expression [tex]\(\sqrt[3]{4n}\)[/tex], we need to apply the same principle. The cube root of [tex]\(4n\)[/tex] can be expressed as:
[tex]\[
(4n)^{\frac{1}{3}}
\][/tex]
Therefore, the rational exponent expression for [tex]\(\sqrt[3]{4n}\)[/tex] is [tex]\((4n)^{\frac{1}{3}}\)[/tex].
Among the given options:
- [tex]\((4n)^3\)[/tex]
- [tex]\(3n^4\)[/tex]
- [tex]\((4n)^{\frac{1}{3}}\)[/tex]
- [tex]\(4n^{\frac{1}{3}}\)[/tex]
The correct choice is:
[tex]\[
(4n)^{\frac{1}{3}}
\][/tex]
