The problem is to simplify the expression $\frac{4r^0 \cdot r}{3r^{-4}}$.

1. Problem Description

The problem is to simplify the expression $\frac{4r^0 \cdot r}{3r^{-4}}$.

2. Solution Steps

First, recall that any nonzero number raised to the power of 0 is

1. Thus, $r^0 = 1$.

Then, the expression becomes $\frac{4 \cdot 1 \cdot r}{3r^{-4}} = \frac{4r}{3r^{-4}}$.

Next, recall the rule for dividing exponential terms with the same base: $\frac{x^a}{x^b} = x^{a-b}$.

Applying this rule, we have $\frac{r}{r^{-4}} = r^{1 - (-4)} = r^{1+4} = r^5$.

Therefore, the expression becomes $\frac{4}{3}r^5$.

3. Final Answer

$\frac{4}{3}r^5$