To determine which expression is equivalent to [tex]\((5)^{\frac{7}{3}}\)[/tex], we need to recognize how fractional exponents work.
The expression [tex]\((5)^{\frac{7}{3}}\)[/tex] can be understood by breaking down the fraction in the exponent:
- The denominator (3) represents the root.
- The numerator (7) represents the power.
Thus, [tex]\((5)^{\frac{7}{3}}\)[/tex] is equivalent to taking the cube root of [tex]\(5\)[/tex] raised to the 7th power. Mathematically, this is expressed as:
[tex]\[
(5)^{\frac{7}{3}} = \sqrt[3]{5^7}
\][/tex]
Among the given options:
1. [tex]\(5^{-4}\)[/tex] - This represents raising 5 to the power of -4. This is not equivalent to our expression.
2. [tex]\(5^4\)[/tex] - This represents raising 5 to the power of 4. This is also not equivalent to our expression.
3. [tex]\(\sqrt[7]{5^3}\)[/tex] - This represents the 7th root of [tex]\(5^3\)[/tex], which does not match our expression.
4. [tex]\(\sqrt[3]{5^7}\)[/tex] - This represents the cube root of [tex]\(5^7\)[/tex], which is exactly the same as our expression [tex]\((5)^{\frac{7}{3}}\)[/tex].
Therefore, the correct equivalent expression to [tex]\((5)^{\frac{7}{3}}\)[/tex] is:
[tex]\[
\sqrt[3]{5^7}
\][/tex]
