We can simplify the expression as follows:
((mn^3)^2) / (((2^-1)(m^-1)(n))^-3) = (m^2n^6) / (2^3m^3n^3) = m^2n^6 / 2^3n^3 = m^2 / 2^3 = \boxed{\frac{m^2}{8}}.
Here are the steps in detail:
We start by simplifying the numerator and the denominator independently. In the numerator, we use the distributive property of exponents to get m2n6. In the denominator, we use the distributive property of exponents and the fact that x−a=xa1 to get 23n3.
Then, we simplify the expression by dividing the numerator and denominator by n3. We can do this because n3=0. This gives us m2/23.
Finally, we simplify the expression by dividing the numerator and denominator by 2. This gives us m^2/8.
