Simplify the expression $\frac{(2m^2n)^3}{(mn^3)^2 \times (4m)^2}$.

1. Problem Description

Simplify the expression $\frac{(2m^2n)^3}{(mn^3)^2 \times (4m)^2}$.

2. Solution Steps

First, simplify the numerator:

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

Next, simplify the denominator:

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

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

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

Then, rewrite the expression:

$\frac{(2m^2n)^3}{(mn^3)^2 \times (4m)^2} = \frac{8m^6n^3}{16m^4n^6}$

Now, simplify the fraction by dividing the coefficients and subtracting the exponents of like variables:

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

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

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

Combining these results:

$\frac{8m^6n^3}{16m^4n^6} = \frac{1}{2} m^2 \frac{1}{n^3} = \frac{m^2}{2n^3}$

3. Final Answer

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