Question

how to find the 3 third roots of I=e^(ipi/2) I know you have to add 2pi to each one...

Answer 1

nah, just divide the unit circle in two three equal parts

Answer 2

one of them is at \[\large e^{\frac{\pi}{6}i}\]

Answer 3

Yah if you want to add 2pi, you would write it like this:\[\Large\rm \left(e^{i(\pi/2+2k \pi)}\right)^{1/3}\]So full rotations get us to the same location right?
That's what the 2kpi is all about.

Since we're taking the third root, we only need k to take on the first 3 values ( starting from 0 ) to get the 3 unique roots.\[\Large\rm \left(e^{i(\pi/2+2k \pi)}\right)^{1/3},\qquad k=0,1,2\]And simplify from there! :)

Answer 4

of course you can do ti the donkey way and add \(2\pi\) and then once again divide by 3
you would get \(2\pi+\frac{\pi}{2}=\frac{5\pi}{2}\) and then \(\frac{5\pi}{6}\)for another root

Answer 5

but really what you are doing is dividing the unit circle in to three parts, where one of them is at \(\frac{\pi}{6}\)

Answer 6

so if your first polar coordinate is 5pi/2 then add 2pi for the second coordinate don't you get 9pi/2?

Answer 7

please help! :(

Answer 8

Wops the first one should have been pi/6! :O

You took your pi/2 and multiplied it by 1/3.

Now what's tricky is, you're not adding 2pi to find the other roots.
You're adding 2pi/3 ( you divide the 2pi by 3 as well ).

So the secon root should give you an argument like... pi/6 + 2pi/3 = 5pi/6

Answer 9

ahh that makes so much moe sense thank you! so the last root is 9pi/6=3pi/2?