0\; ,\; b=-2<0\; \; \Rightarrow \; \; \varphi \in (-\frac{\pi}{2},0)\\\\|z|=\sqrt{2^2+(-2)^2}=\sqrt8\\\\argz=\varphi =arctg(\frac{2}{-2})=arctg(-1)=-arctg1=-\frac{\pi }{4}\\\\z=\sqrt8\cdot (cos(-\frac{\pi }{4})+i\, sin(-\frac{\pi }{4}))\\\\z^5=\sqrt{8^5}\cdot (cos(-\frac{5\pi }{4})+i\, sin(-\frac{5\pi }{4}))" alt="z^5=(2-2i)^5\\\\z=2-2i\; \; ,\; \; a=2>0\; ,\; b=-2<0\; \; \Rightarrow \; \; \varphi \in (-\frac{\pi}{2},0)\\\\|z|=\sqrt{2^2+(-2)^2}=\sqrt8\\\\argz=\varphi =arctg(\frac{2}{-2})=arctg(-1)=-arctg1=-\frac{\pi }{4}\\\\z=\sqrt8\cdot (cos(-\frac{\pi }{4})+i\, sin(-\frac{\pi }{4}))\\\\z^5=\sqrt{8^5}\cdot (cos(-\frac{5\pi }{4})+i\, sin(-\frac{5\pi }{4}))" align="absmiddle" class="latex-formula">
