Question

I need help plz. Its DeMoivres Theorem

Answer 1

`\bf \qquad \textit{power of two complex numbers} \\\\\ [\quad r[cos(\theta)+isin(\theta)]\quad ]^n\implies r^n[cos(n\cdot \theta)+isin(n\cdot \theta)] \\\\[-0.35em] \rule{34em}{0.25pt}\\\\ z=\stackrel{a}{-1}\stackrel{b}{-√(3)~i}~~ \begin{cases} r=√(a^2+b^2)\\ \theta =tan^(-1)\left( (b)/(a) \right)\\[-0.5em] \hrulefill\\ r=\sqrt{(-1)^2+(√(3))^2}\\ \qquad √(4)\\ \qquad 2\\ \theta =tan^(-1)\left( (-√(3))/(-1) \right)\\\\ \qquad (4\pi )/(3) \end{cases}`

`\bf z=2\left[ cos\left( (4\pi )/(3) \right) -i~sin\left( (4\pi )/(3) \right)\right]\implies z^6=\left[ 2\left[ cos\left( (4\pi )/(3) \right) -i~sin\left( (4\pi )/(3) \right)\right] \right]^6 \\\\\\ z^6=2^6\left[ cos\left( 6\cdot (4\pi )/(3) \right) -i~sin\left( 6\cdot (4\pi )/(3) \right)\right]\implies z^6=64[cos(8\pi )-i~sin(8\pi )] \\\\\\ z^6=64[(-1)-i~(0)]\implies z^6=\stackrel{\stackrel{a}{\downarrow }}{-64}~~\stackrel{\stackrel{b}{\downarrow }}{-0i}`