The problem is to simplify the expression $\frac{3r^3}{2r^3 \cdot 3r^3}$.

1. Problem Description

The problem is to simplify the expression $\frac{3r^3}{2r^3 \cdot 3r^3}$.

2. Solution Steps

First, we simplify the denominator by multiplying the terms.

$2r^3 \cdot 3r^3 = (2 \cdot 3) \cdot (r^3 \cdot r^3)$

Using the rule $a^m \cdot a^n = a^{m+n}$, we get

$r^3 \cdot r^3 = r^{3+3} = r^6$

So the denominator becomes $6r^6$.

Thus, the expression becomes $\frac{3r^3}{6r^6}$.

Next, we simplify the fraction $\frac{3}{6}$ to $\frac{1}{2}$.

Also, using the rule $\frac{a^m}{a^n} = a^{m-n}$, we get

$\frac{r^3}{r^6} = r^{3-6} = r^{-3} = \frac{1}{r^3}$.

Therefore, we have

$\frac{3r^3}{6r^6} = \frac{1}{2} \cdot \frac{1}{r^3} = \frac{1}{2r^3}$.

3. Final Answer

The simplified expression is $\frac{1}{2r^3}$.