If  $z+z^{-1}=1,then \text{\hspace{0.17em}}{z}^{100}+z^{-100} isequal to$

$$\begin{gathered} z+z^{-1}=1\Rightarrow z^2-z+1=0\Rightarrow z=-\omega or-{\omega }^2 \\ Forz=-\omega ,z^{100}+z^{-100}={(-\omega )}^{100}+{(-\omega )}^{-100} \\ =\omega +\frac{1}{\omega }=\omega +{\omega }^2=-1 \\ z=-{\omega }^2,z^{100}+z^{-100}={(-{\omega }^2)}^{100}+{(-{\omega }^2)}^{-100} \\ For {\omega }^{200}+\frac{1}{{\omega }^{200}}={\omega }^2+\frac{1}{{\omega }^2}={\omega }^2+\omega =-1 \end{gathered}$$