To find the radical form of the expression [tex]\(a^{\frac{5}{7}}\)[/tex], we need to understand how to convert an expression with a fractional exponent into a radical expression.
The general rule for converting a fractional exponent [tex]\(a^{\frac{m}{n}}\)[/tex] into a radical expression is:
[tex]\[ a^{\frac{m}{n}} = \sqrt[n]{a^m} \][/tex]
In this specific problem, the exponent is [tex]\(\frac{5}{7}\)[/tex], meaning we have [tex]\(a^{\frac{5}{7}}\)[/tex]. According to the rule stated above:
[tex]\[ a^{\frac{5}{7}} = \sqrt[7]{a^5} \][/tex]
This tells us that [tex]\(a^{\frac{5}{7}}\)[/tex] is equivalent to the 7th root of [tex]\(a\)[/tex] raised to the power of 5.
Now, let's match this expression to the options provided:
1. [tex]\(\sqrt[5]{a^7}\)[/tex]
2. [tex]\(\sqrt[7]{a^5}\)[/tex]
3. [tex]\(5 a^7\)[/tex]
4. [tex]\(7 a^5\)[/tex]
The correct radical expression is [tex]\(\sqrt[7]{a^5}\)[/tex], which corresponds to the second option.
Therefore, the answer is:
[tex]\[
2
\][/tex]
