Simplify the given expression: $\frac{mn^2 \cdot m^4}{2m^2}$.

1. Problem Description

Simplify the given expression: $\frac{mn^2 \cdot m^4}{2m^2}$.

2. Solution Steps

First, simplify the numerator by combining the $m$ terms using the rule $a^m \cdot a^n = a^{m+n}$:

$mn^2 \cdot m^4 = m^1 \cdot m^4 \cdot n^2 = m^{1+4}n^2 = m^5n^2$

So the expression becomes $\frac{m^5n^2}{2m^2}$.

Next, simplify the expression by dividing $m^5$ by $m^2$ using the rule $\frac{a^m}{a^n} = a^{m-n}$:

$\frac{m^5}{m^2} = m^{5-2} = m^3$

Thus, the simplified expression is $\frac{m^3n^2}{2}$.

3. Final Answer

$\frac{m^3n^2}{2}$