((16x^2y)/(81xy^2))/((24x^2y)/(54x^3y^3))
=color(red)(((16x^2y)/(81xy^2)))*color(blue)(((54x^3y^3)/(24x^2y)))
=(color(red)((8)(2)(x^2y))color(blue)((6)(9)(xy^2)(x^2y)))/(color(red)((9)(9)(xy^2))color(blue)((4)(6)(x^2y)))
=((8)(2)cancel((x^2y))(6)(9)(xy^2)(x^2y))/((9)(9)(xy^2)(4)(6)cancel((x^2y)))
=((8)(2)(6)(9)cancel((xy^2))(x^2y))/((9)(9)cancel((xy^2))(4)(6))
=((8)(2)cancel((6))cancel((9))(x^2y))/((9)cancel((9))(4)cancel((6)))
=(cancel((8))^2(2)(x^2y))/((9)cancel((4)))
=(4x^2y)/9