$\displaystyle \left(\sum_{i=1}^n \left(D_i^3-x^3\right)N_i\right)=m$

$\displaystyle \sum_{i=1}^n D_i^3 N_i - \sum_{i=1}^n N_i x^3 =m$

rewrite this a bit as

$\displaystyle C_1 = C_2 x^3 $ where

$C_1 = \displaystyle \left(\left(\sum_{i=1}^n D_i^3 N_i\right)-m\right)$ and $\displaystyle C_2=\sum_{i-1}^n N_i$

finally

$x=\sqrt[3]{\dfrac{C_1}{C_2}}$