If n is a positive integer greater than unity and z is a complex number satisfying the equation $z^n=(z+1{)}^n$, then

$\frac{z}{1+z}=1^{1/n}\text{, i.e., nth }$root of unity

$\begin{array}{l}\therefore \left|\frac{z}{1+z}\right|=1\text{ or }|z{|}^2=|1+z{|}^2\\ x^2+y^2=(x+1{)}^2+y^2\to 2x+1=0\end{array}$

$x=-\frac{1}{2},y$ may be any number $\in R\Rightarrow \left(\mathrm{d}\right)$

$\therefore \mathrm{Re}\left(z\right)=x=-\frac{1}{2}<0$.