Question

which of the following is equivalent to ( (5)^{\frac{7}{3}} )?
( \bigcirc 5\frac{4}{4} )
( \bigcirc 5^4 )
( \bigcirc sqrt7{5^3} )
( \bigcirc sqrt3{5^7} )

Explanation:

Step1: Recall the exponent rule

The rule for converting a rational exponent to a radical is \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\) (or \((\sqrt[n]{a})^{m}\)). Here, we have \((5)^{\frac{7}{3}}\).

Step2: Apply the exponent rule

Using the rule \(a^{\frac{m}{n}}=\sqrt[n]{a^{m}}\), for \(a = 5\), \(m=7\), and \(n = 3\), we get \(\sqrt[3]{5^{7}}\).

Answer:

The option with \(\sqrt[3]{5^{7}}\) (the fourth option, assuming the options are in order from left to right as \(5_{\frac{4}{}}\) (maybe a typo), \(5^{4}\), \(\sqrt[7]{5^{3}}\), \(\sqrt[3]{5^{7}}\))