Many scientists and researchers have used multilayered neural networks to approximate multivariate functions for several years [see Lin and Cao 2015, Li and Xu 2007, Suzuki 1998 & Wang and Xu 2010]. They have established both upper and lower bounds of simultaneous approximations for 1st and 2nd orders, spaces of function to approximate and approximators as well. That work solves many applicant issues in science and engineer. Our goal was to achieve that both bounds of modulus of smoothness of order m for a pth Lebegue integrable multivariate function that is approximated by a multi-layered feedforwrd neural network.