We are asked to simplify the expression $\frac{(2m^2n)^3}{(mn^3)^2 \times (4m)^2}$.

1. Problem Description

We are asked to simplify the expression $\frac{(2m^2n)^3}{(mn^3)^2 \times (4m)^2}$.

2. Solution Steps

First, let's simplify the numerator.

$(2m^2n)^3 = 2^3 \times (m^2)^3 \times n^3 = 8m^6n^3$

Next, let's simplify the denominator.

$(mn^3)^2 = m^2 \times (n^3)^2 = m^2n^6$

$(4m)^2 = 4^2 \times m^2 = 16m^2$

$(mn^3)^2 \times (4m)^2 = (m^2n^6) \times (16m^2) = 16m^4n^6$

Now, we can rewrite the expression as:

$\frac{8m^6n^3}{16m^4n^6}$

We can simplify the fraction by dividing the coefficients and using the quotient rule for exponents.

$\frac{8}{16} = \frac{1}{2}$

$\frac{m^6}{m^4} = m^{6-4} = m^2$

$\frac{n^3}{n^6} = n^{3-6} = n^{-3} = \frac{1}{n^3}$

Combining these, we get:

$\frac{1}{2} \times m^2 \times \frac{1}{n^3} = \frac{m^2}{2n^3}$

3. Final Answer

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