Question

which of the following is equivalent to $sqrt5{13^3}$?
○ $13^2$
○ $13^{15}$
○ $13^{\frac{5}{3}}$
○ $13^{\frac{3}{5}}$

Explanation:

Step1: Recall the radical to exponent rule

The rule for converting a radical to an exponent is $\sqrt[n]{a^m} = a^{\frac{m}{n}}$. Here, $n = 5$ (the index of the fifth root) and $m = 3$ (the exponent of 13 inside the root).

Step2: Apply the rule to $\sqrt[5]{13^3}$

Using the formula $\sqrt[n]{a^m}=a^{\frac{m}{n}}$, substitute $a = 13$, $m = 3$, and $n = 5$. So we get $13^{\frac{3}{5}}$.

Answer:

$13^{\frac{3}{5}}$ (the last option, which is $13^{\frac{3}{5}}$)