We need to determine which of the given expressions is equivalent to [tex]\( 5^{\frac{7}{3}} \)[/tex].
Firstly, consider the properties of exponents that we'll use for this solution:
- [tex]\( a^{m/n} = \sqrt[n]{a^m} \)[/tex]
- [tex]\( a^{m/n} = (a^m)^{1/n} \)[/tex]
Given the expression [tex]\( 5^{\frac{7}{3}} \)[/tex], let's analyze the options one by one.
1. [tex]\( 5^{-4} \)[/tex]:
[tex]\[
5^{-4} = \frac{1}{5^4}
\][/tex]
This is clearly not equivalent to [tex]\( 5^{\frac{7}{3}} \)[/tex], since the exponent is negative and does not match.
2. [tex]\( 5^4 \)[/tex]:
[tex]\[
5^4
\][/tex]
The exponent here is 4, which is not the same as [tex]\( \frac{7}{3} \)[/tex].
3. [tex]\( \sqrt[7]{5^3} \)[/tex]:
[tex]\[
\sqrt[7]{5^3} = 5^{3/7}
\][/tex]
This again is not equivalent to [tex]\( 5^{\frac{7}{3}} \)[/tex].
4. [tex]\( \sqrt[3]{5^7} \)[/tex]:
[tex]\[
\sqrt[3]{5^7} = (5^7)^{1/3} = 5^{7 \cdot \frac{1}{3}} = 5^{\frac{7}{3}}
\][/tex]
This is equivalent to [tex]\( 5^{\frac{7}{3}} \)[/tex].
Therefore, the expression [tex]\( \sqrt[3]{5^7} \)[/tex] is equivalent to [tex]\( 5^{\frac{7}{3}} \)[/tex]. The correct answer is:
[tex]\[
\boxed{\sqrt[3]{5^7}}
\][/tex]
