First, we will use this rule of exponents to simplify the term within the parenthesis:
#x^color(red)(a)/x^color(blue)(b) = x^(color(red)(a)-color(blue)(b))#
#((3m^(1/6)n^(color(red)(1/3)))/(4n^(color(blue)(-2/3))))^2 = ((3m^(1/6)n^(color(red)(1/3)-color(blue)(-2/3)))/4)^2 = ((3m^(1/6)n^(3/3))/4)^2 =#
#((3m^(1/6)n^1)/4)^2#
Now, use these two rules for exponents to simplify the terms for the entire expression:
#a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#
#((3^color(red)(1)m^(color(red)(1/6))n^color(red)(1))/4^color(red)(1))^color(blue)(2) = (3^(color(red)(1)xxcolor(blue)(2))m^(color(red)(1/6)xxcolor(blue)(2))n^(color(red)(1)xxcolor(blue)(2)))/4^(color(red)(1)xxcolor(blue)(2)) = (3^2m^(2/6)n^2)/4^2 = (9m^(1/3)n^2)/16#
