Question

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

Explanation:

Step1: Recall the exponent rule for radicals

The rule states that \( a^{\frac{m}{n}}=\sqrt[n]{a^{m}} \), where \( a \) is the base, \( m \) is the numerator of the exponent, and \( n \) is the denominator of the exponent.

Step2: Apply the rule to \( 5^{\frac{7}{3}} \)

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

Step3: Eliminate other options

• For \( 5^{-4} \), it is \( \frac{1}{5^{4}} \), which is not equal to \( 5^{\frac{7}{3}} \).
• For \( \sqrt[7]{5^{3}} \), by the exponent rule, it is \( 5^{\frac{3}{7}} \), not \( 5^{\frac{7}{3}} \).
• For \( 5^{4} \), it is \( 5\times5\times5\times5 = 625 \), while \( 5^{\frac{7}{3}}=\sqrt[3]{5^{7}}=\sqrt[3]{78125}\approx42.7 \), so they are not equal.

Answer:

B. \(\sqrt[3]{5^{7}}\)