Question

What is (-i)^3
A. -i
B. i
C. -1
D.1

Answer 1

B: i

Explanation:

Algebric explanation: as long as you remember power rules you should be good:

`(-i)^3 = (-1)^3* i^3 = -1 * i^2* i= (-1)*(-1)* i=1i`

Geometric explanation: In the complex plane you can think "multiplying by i" as "rotate 1/4 counterclockwise".
`(-1)^3` is a real number, and it's simply
`-1`. Rotate that ccw once, you're at
`-i`. Again, and you're at 1. third time (is really the charm!) you end up at
`i`.

Polar explanation Courtesy of euler's identity.

`(-i)^3 = (e^{-\frac{\pi}2i})^3=e^(-\frac32\pi i)= cos (-\frac32\pi)+isin(-\frac32\pi) = 0+i(1)= i`

Answer 2

i

Explanation:

=> -i³

=> (-1 . i )³

=> -1³ . i³

=> -1 [ i² . i ]

=> -1 [ (√-1)² . i ]

=> -1 [ -1 . i ]

=> i