Question

use trigonometric form and de moivre's theorem to simplify the expression.

(2+2i)^3

Answer 1

2+2i=
\[\sqrt{8}(\cos(45) +i*\sin(45))\]

Answer 2

\[(\sqrt{8}(\cos(45)+i*\sin(45))^3 = \sqrt{8}^3*(\cos(45)+i*\sin(45))^3\]
This is where good ol' De'Moivre comes in.

Answer 3

\[=\sqrt{8}^3(\cos(3*45)+i*\sin(3*45))\]

Answer 4

cos(8*pi/4)? I believe the angle is simple pi/4.

Answer 5

And then after De'Moivre's, you should be left with cos(3pi/4) + i*sin(3pi/4)

Answer 6

\[(2+2i) 3 =(8 √ ) 3 (\cos3×pi 4 +isin3×pi 4 )=162 √ (\cos3pi/4+isin3pi/4)=\]

Answer 7

Yes, that.

Answer 8

i think i get it, thanks smoothmath!