To determine the rational exponent expression of the given expression [tex]$\sqrt[5]{7n}$[/tex], we can follow a clear step-by-step method.
1. Understand the Problem:
We need to convert the fifth root of [tex]\( 7n \)[/tex] into an expression with rational exponents.
2. Recall the Rule for Converting Roots to Rational Exponents:
The [tex]\( n \)[/tex]-th root of a number [tex]\( a \)[/tex] is expressed with rational exponents as:
[tex]\[
\sqrt[n]{a} = a^{\frac{1}{n}}
\][/tex]
3. Apply the Rule:
- Here, we have the fifth root of [tex]\( 7n \)[/tex].
- According to the rule, this can be written as:
[tex]\[
\sqrt[5]{7n} = (7n)^{\frac{1}{5}}
\][/tex]
4. Check the Options Given:
- Option 1: [tex]\( 5 n^7 \)[/tex] — This is [tex]\( 5 \times n^7 \)[/tex], not related to our expression.
- Option 2: [tex]\( (7 n)^5 \)[/tex] — This represents [tex]\( 7n \)[/tex] raised to the power of 5, not [tex]\((7n)^{\frac{1}{5}}\)[/tex].
- Option 3: [tex]\( 7 n^{\frac{1}{5}} \)[/tex] — This means [tex]\( 7 \times n^{\frac{1}{5}} \)[/tex], which only raises [tex]\( n \)[/tex] to the power [tex]\( \frac{1}{5} \)[/tex].
- Option 4: [tex]\( (7 n)^{\frac{1}{5}} \)[/tex] — This correctly represents the entire expression [tex]\( 7n \)[/tex] raised to the power [tex]\( \frac{1}{5} \)[/tex].
5. Select the Correct Option:
Among the given options, the correct representation of [tex]\( \sqrt[5]{7n} \)[/tex] using rational exponents is:
[tex]\[
(7 n)^{\frac{1}{5}}
\][/tex]
Therefore, the correct answer is:
[tex]\[
\boxed{(7 n)^{\frac{1}{5}}}
\][/tex]
