To find the radical expression for [tex]\(a^{\frac{5}{7}}\)[/tex], we need to convert this expression to a form that uses a root.
In general, an expression of the form [tex]\(a^{\frac{m}{n}}\)[/tex] can be rewritten as the [tex]\(n\)[/tex]-th root of [tex]\(a\)[/tex] raised to the [tex]\(m\)[/tex]-th power:
[tex]\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \][/tex]
Here, [tex]\(m = 5\)[/tex] and [tex]\(n = 7\)[/tex], so:
[tex]\[ a^{\frac{5}{7}} = \sqrt[7]{a^5} \][/tex]
From the given options:
1. [tex]\(\sqrt[5]{a^7}\)[/tex] - This represents the fifth root of [tex]\(a^7\)[/tex], which is not the correct representation.
2. [tex]\(\sqrt[7]{a^5}\)[/tex] - This is the correct representation of [tex]\(a^{\frac{5}{7}}\)[/tex].
3. [tex]\(5 a^7\)[/tex] - This is a misinterpretation involving multiplication, not roots or exponents.
4. [tex]\(7 a^5\)[/tex] - This also represents multiplication, not the correct radical expression.
Thus, the correct radical expression for [tex]\(a^{\frac{5}{7}}\)[/tex] is:
[tex]\[ \sqrt[7]{a^5} \][/tex]
Therefore, the correct answer is the second option:
[tex]\[ \boxed{2} \][/tex]
