{ 2(cos2 pi/5 + i sin 2pi/5)} ^8, express result in polar form?

Question

in this pi= 180 degree

the answer is 255 (cos pi /5 + i sin pi/5)

pls. explain the answer

Answer 1

cos 2pi/5 + isin 2pi/5 = exp(i2pi/5) Euler's
Now just raise the right side to the exponent. Thus you multiply the exponent:
[exp(i2pi/5)^8 = exp(i8*2*pi/5) = exp(i16pi/5)
This is periodic with a period of 2pi, so you can subtract multples of 2pi
16pi/5 = 3pi + pi/5 = 2pi + 6pi/5 , right?
So the result is
exp(i6pi/5)

Oop! I forgot the constant out front. Just raise it to the 8 power:
2^8 = 256.

Note also that you can get to the same vector by going -4pi/5

Answer 2

256 * Exp(-2.51327)