00:01
Okay, this question wants us to find all fourth roots negative 16i.
00:07
So we're looking for any number such that z to the fourth is equal to negative 16 i.
00:16
And our equation in the book needs our complex number in polar form.
00:22
And this is quite easy to write in polar form, because we're looking for where z to the fourth is equal to, well it's just pointing straight down on the imaginary axis so our magnitude is 16 and our angle is 270 and now that we have our polar form we can use the book equation which tells us that z sub k is just equal to the nth root of the magnitude times cosine of theta 0 over n plus 2k pi over n and the same for sign.
01:37
Sorry, there should be theta not.
01:47
So we always have as many roots as the power.
01:53
So since we have a fourth root, we're looking for four of them.
01:57
So our first root, when we plug in zero, we get, well, this nth root of ours, just the fourth root of 16, so two.
02:09
And then cosine of our initial angle divided by the root.
02:21
So it's 270 divided by 4, which is 67 .5 degrees.
02:38
And then we don't have to worry about adding any angle yet...