Question

How do you simplify `(2+2i)^6`?

Answer 1

`(2+2i)^6 = -512i`

Explanation:

`color(white)()`
Using polar form

`2+2i = 2sqrt(2)(cos (pi/4) + i sin (pi/4))`

So:

`(2+2i)^6 = (2sqrt(2))^6(cos (pi/4) + i sin (pi/4))^6`

`=(2^(3/2))^6 (cos 6(pi/4) + i sin 6(pi/4))`

`=2^9 (cos ((3pi)/2) + i sin ((3pi)/2))`

`=2^9 * (-i)`

`=-512i`

`color(white)()`
Using rectangular form

`(2+2i)^2 = (2(1+i))^2 = 4(1+i)(1+i) = 4(1+2i+i^2) = 4(2i) = 8i`

So:

`(2+2i)^6 = ((2+2i)^2)^3 = (8i)^3 = 8^3i^3 = -512i`