To determine the correct radical expression for [tex]\( a^{\frac{5}{7}} \)[/tex], let's examine the options provided.
### Step-by-Step Analysis:
1. Understanding [tex]\( a^{\frac{5}{7}} \)[/tex]:
The expression [tex]\( a^{\frac{5}{7}} \)[/tex] represents a number raised to the power of a fraction. This fraction suggests that we could write it as a radical (or root) expression, generally represented as [tex]\( \sqrt[n]{a^m} \)[/tex], where [tex]\( n \)[/tex] is the denominator and [tex]\( m \)[/tex] is the numerator of the fraction.
2. Converting [tex]\( a^{\frac{5}{7}} \)[/tex] to Radical Form:
The general rule for converting a fractional exponent [tex]\( a^{\frac{m}{n}} \)[/tex] into radical form is:
[tex]\[
a^{\frac{m}{n}} = \sqrt[n]{a^m}
\][/tex]
Therefore:
[tex]\[
a^{\frac{5}{7}} = \sqrt[7]{a^5}
\][/tex]
3. Comparing with Given Options:
- Option 1: [tex]\( \sqrt[5]{a^7} \)[/tex]
- This corresponds to [tex]\( a^{\frac{7}{5}} \)[/tex], not [tex]\( a^{\frac{5}{7}} \)[/tex].
- Option 2: [tex]\( \sqrt[7]{a^5} \)[/tex]
- This indeed matches our derived radical form of [tex]\( a^{\frac{5}{7}} \)[/tex].
- Option 3: [tex]\( 5 a^7 \)[/tex]
- This is a linear combination of multiplying 5 and [tex]\( a^7 \)[/tex], and does not involve transforming the exponent in fractions.
- Option 4: [tex]\( 7 a^5 \)[/tex]
- This also is a linear combination and does not represent the radical form of [tex]\( a^{\frac{5}{7}} \)[/tex].
### Conclusion:
The correct radical expression for [tex]\( a^{\frac{5}{7}} \)[/tex] is indeed:
[tex]\[
\sqrt[7]{a^5}
\][/tex]
Therefore, the correct option is:
Option 2: [tex]\( \sqrt[7]{a^5} \)[/tex]
