The French Solution

The simplified solution to the equation $z^3 = 1 + i,$ where $i = \sqrt{-1}$ can be written in the form: $\large{a}^{\frac{b}{c}}\cdot e^{\frac{i(d+fk)\pi}{g}},$ where $a, b, c, d, f, g \in \mathbb{Z}$. Find $a+b+c+d+f+g$.