The problem as given: $(4m^3n)^\frac{1}{4} \times (8mn^3)^\frac{1}{2}$
distributed the exponted threw.
$(4^{1/4}m^{3/4}n^{1/4})(8^{1/2}m^{1/2}n^{3/2})$
Then because $a^na^m = a^{n+m} \; \& \; a^{n}b^{n} = (ab)^n$
$(2^2)^{1/4} (8^{1/2})(m^{3/4 + 1/2})( n^{1/4+6/4}) = 16^{1/2}(m^{5/4})n^{7/4}$
am I correct?
edited to fix mistake pointed out by "Old John"