To determine the radical expression of [tex]\( a^{\frac{5}{7}} \)[/tex], we need to convert this exponent form into a radical form.
In general, for any real number [tex]\( a \)[/tex] and rational exponent [tex]\( \frac{m}{n} \)[/tex],
[tex]\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\][/tex]
Here, [tex]\( a \)[/tex] is raised to the power [tex]\( m \)[/tex] and then we take the [tex]\( n \)[/tex]-th root of the result.
For the given expression [tex]\( a^{\frac{5}{7}} \)[/tex]:
- [tex]\( m \)[/tex] is 5.
- [tex]\( n \)[/tex] is 7.
Using the general formula for converting an exponent to a radical expression we get:
[tex]\[
a^{\frac{5}{7}} = \sqrt[7]{a^5}
\][/tex]
Now let's consider the provided options:
1. [tex]\(\sqrt[5]{a^7}\)[/tex] - This represents the 5th root of [tex]\( a^7 \)[/tex], which is not the expression we need.
2. [tex]\(\sqrt[7]{a^5}\)[/tex] - This represents the 7th root of [tex]\( a^5 \)[/tex], which is exactly the expression we're looking for.
3. [tex]\(5 a^7\)[/tex] - This represents 5 times [tex]\( a^7 \)[/tex], which is a different type of expression.
4. [tex]\(7 a^5\)[/tex] - This represents 7 times [tex]\( a^5 \)[/tex], which is also a different type of expression.
So, the correct radical expression matching [tex]\( a^{\frac{5}{7}} \)[/tex] is:
[tex]\[
\sqrt[7]{a^5}
\][/tex]
Hence, the correct answer is:
[tex]\(\sqrt[7]{a^5}\)[/tex]
