The problem as given: $$\left(4m^3n\right)^\frac{1}{4} \times \left(8mn^3\right)^\frac{1}{2}$$
distributed the exponent through.
$$\left(4^{1/4}m^{3/4}n^{1/4}\right)\left(8^{1/2}m^{1/2}n^{3/2}\right)$$
Then because $a^na^m = a^{n+m}$ and $a^{n}b^{n} = (ab)^n$
$$\left(2^2\right)^{1/4} \left(8^{1/2}\right)\left(m^{3/4 + 1/2}\right) \left(n^{1/4+6/4}\right) = 16^{1/2}m^{5/4}n^{7/4}$$
Am I correct?
Edited to fix mistake pointed out by "Old John"