We will use these rules for exponents to simplify this expression:

#a = a^color(red)(1)# and #(x^color(red)(a))^color(blue)(b) = x^(color(red)(a) xx color(blue)(b))#

#(x^9y)^(1/3) + (xy^(1/9))^3 = (x^color(red)(9)y^color(red)(1))^color(blue)(1/3) + (x^color(red)(1)y^color(red)(1/9))^color(blue)(3) =#

#(x^(color(red)(9)xxcolor(blue)(1/3))y^(color(red)(1)xxcolor(blue)(1/3))) + (x^(color(red)(1)xxcolor(blue)(3))y^(color(red)(1/9)xxcolor(blue)(3))) = (x^(9/3)y^(1/3)) + (x^3y^(3/9)) = #

#x^3y^(1/3) + x^3y^(1/3) = 1x^3y^(1/3) + 1x^3y^(1/3) = (1 + 1)x^3y^(1/3) = #

#2x^3y^(1/3)#