Note that, \(\displaystyle{z}={0}\) is not a branch point of \(\displaystyle{f{{\left({z}\right)}}}\). To find the branch points of \(\displaystyle{f{{\left({z}\right)}}}\), solve the equation

\(\displaystyle{z}^{{3}}-{2}={0}\Rightarrow{z}^{{3}}={2}{e}^{{{2}{k}\pi{i}}}\Rightarrow{z}={2}^{{\frac{{{1}}}{{{3}}}}}{e}^{{{\frac{{{2}{k}\pi{i}}}{{{3}}}}}},{k}={0},{1},{2}\)

\(\displaystyle{z}^{{3}}-{2}={0}\Rightarrow{z}^{{3}}={2}{e}^{{{2}{k}\pi{i}}}\Rightarrow{z}={2}^{{\frac{{{1}}}{{{3}}}}}{e}^{{{\frac{{{2}{k}\pi{i}}}{{{3}}}}}},{k}={0},{1},{2}\)