To convert the exponential expression [tex]\( a^{\frac{5}{7}} \)[/tex] into a radical form, we need to understand the relationship between exponents and radicals. Specifically, the notation [tex]\( a^{\frac{m}{n}} \)[/tex] can be interpreted as the [tex]\( n \)[/tex]-th root of [tex]\( a \)[/tex] raised to the [tex]\( m \)[/tex]-th power, which can be written as [tex]\( \sqrt[n]{a^m} \)[/tex].
Let's break down the expression [tex]\( a^{\frac{5}{7}} \)[/tex]:
1. Exponent Explanation: Here, the base is [tex]\( a \)[/tex], the numerator of the exponent [tex]\( \frac{5}{7} \)[/tex] is 5, and the denominator is 7.
2. Radical Form: The denominator [tex]\( 7 \)[/tex] indicates the type of root, which is the 7th root, and the numerator [tex]\( 5 \)[/tex] indicates the power to which the base [tex]\( a \)[/tex] is raised inside the radical.
Putting this together, the radical expression corresponding to [tex]\( a^{\frac{5}{7}} \)[/tex] is:
[tex]\[ \sqrt[7]{a^5} \][/tex]
Among the provided choices:
1. [tex]\( 7 a^5 \)[/tex]: This represents 7 times [tex]\( a \)[/tex] raised to the 5th power, which is incorrect.
2. [tex]\( \sqrt[5]{a^7} \)[/tex]: This represents the 5th root of [tex]\( a \)[/tex] raised to the 7th power, which is incorrect.
3. [tex]\( \sqrt[7]{a^5} \)[/tex]: This correctly represents the 7th root of [tex]\( a \)[/tex] raised to the 5th power.
4. [tex]\( 5 a^7 \)[/tex]: This represents 5 times [tex]\( a \)[/tex] raised to the 7th power, which is incorrect.
Thus, the correct choice that matches the radical expression of [tex]\( a^{\frac{5}{7}} \)[/tex] is:
[tex]\[ \boxed{\sqrt[7]{a^5}} \][/tex]
Therefore, the answer corresponds to the third choice.
