Question

How do you use De Moivre's Theorem to simplify `z=(2[cos(5pi/4)+isin(5pi/4)])^5` to `16sqrt2 + 16sqrt2 i`?

Answer 1

`z=(2[cos(5pi/4)+isin(5pi/4)])^5`

`=>z=2^5[cos((5xx5pi)/4)+isin((5xx5pi)/4)]`

`=>z=2^5[cos(6pi+pi/4)+isin(6pi+pi/4)]`

`=>z=2^5[cos(pi/4)+isin(pi/4)]`

`=>z=32[1/sqrt2+i/sqrt2]`

`=>z=16sqrt2+16sqrt2i`