Question

Возвести в степень n=100 комплексное число z=2i-2

Answer 1

$z=2i-2\a=-2,; ; b=2\|z|= sqrt{a^2+b^2}= sqrt{(-2)^2+2^2}= sqrt{8}=2 sqrt{2} \\ left { {{cos alpha = frac{a}{|z|} } atop {sin alpha = frac{b}{|z|} }} right. = textgreater left { {{cos alpha =frac{-2}{2 sqrt{2}} } atop {sin alpha = frac{2}{2 sqrt{2} } }} right. = textgreater left { {{cos alpha =- frac{ sqrt{2} }{2} } atop {sin alpha = frac{ sqrt{2} }{2} }} right.= textgreater alpha = frac{3 pi }{4}\\z=|z|(cos alpha +isin alpha )\z=(2 sqrt{2})(cosfrac{3 pi }{4}+isinfrac{3 pi }{4})$

$z^n=(|z|)^n(cos alpha n+isin alpha n)\\z^{100}=(2 sqrt{2})^{100}(cos frac{3 pi *100}{4}+isin frac{3 pi *100}{4})=\=2^{100}*2^{50}(cos75 pi +isin75 pi )=2^{150}(-1+i*0)=-2^{150}$