Which of the following is the rational exponent expression of [tex][tex]$\sqrt[3]{4n}$[/tex][/tex]?

A. [tex](4n)^3[/tex]
B. [tex]3n^4[/tex]
C. [tex](4n)^{\frac{1}{3}}[/tex]
D. [tex]4n^{\frac{1}{3}}[/tex]



Answer :

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}