Question

(cos(pi/5)+i Sin(pi/5))^20

show work?

Answer 1

raising to the power of 20 is the same as multiplying the angle by 20 and taking the absolute value to the 20th power. the absolute value here is 1, so it remains 1.
20*pi/5=4pi
sin(4pi)=0, cos(4pi)=1 so answer is 1

Answer 2

\[(cos(\frac{\pi}{5}) + i\ sin(\frac{\pi}{5}))^{20} = (e^{i\frac{\pi}{5}})^{20} = e^{i4\pi}\]
\[=cos(4\pi) + isin(4\pi) = cos(0) + isin(0) = 1 + 0i = 1\]