To determine the rational exponent expression of [tex]\(\sqrt[5]{7n}\)[/tex], we'll translate the radical notation into exponentiation notation.
A radical expression like [tex]\(\sqrt[5]{x}\)[/tex] can be written in exponential form as:
[tex]\[ x^{\frac{1}{5}} \][/tex]
In this case, we want to find the equivalent expression for [tex]\(\sqrt[5]{7n}\)[/tex].
First, express the radical using rational exponents:
[tex]\[ \sqrt[5]{7n} = (7n)^{\frac{1}{5}} \][/tex]
Let's now examine the options provided:
1. [tex]\( 5 n^7 \)[/tex]
2. [tex]\( (7 n)^5 \)[/tex]
3. [tex]\( 7 n^{\frac{1}{5}} \)[/tex]
4. [tex]\( (7 n)^{\frac{1}{5}} \)[/tex]
Comparing these options with our derivation:
1. [tex]\( 5 n^7 \)[/tex] is clearly not in the form of a fifth root.
2. [tex]\( (7 n)^5 \)[/tex] raises [tex]\(7n\)[/tex] to the power of 5 instead of taking the fifth root.
3. [tex]\( 7 n^{\frac{1}{5}} \)[/tex] would imply only [tex]\(n\)[/tex] is raised to the power [tex]\(\frac{1}{5}\)[/tex], and 7 remains unaffected by the exponential form, which is not correct.
4. [tex]\( (7 n)^{\frac{1}{5}} \)[/tex] correctly captures the entire fifth root expression.
Therefore, the rational exponent expression equivalent to [tex]\(\sqrt[5]{7n}\)[/tex] is:
[tex]\[ (7 n)^{\frac{1}{5}} \][/tex]
Hence, the correct option is:
[tex]\[ (7 n)^{\frac{1}{5}} \][/tex]
