Question

Find all seventh roots of unity and sketch them on the axes below

Answer 1

You have to find all \(z\) such that \(z^7-1=0\). Let \(\large z^7=re^{i\theta}=re^{i(\theta+2n\pi)}\), where \(n=0,1,2,\cdots,6\); then these roots are \[\large r^{1/7}e^{i(\theta+2n\pi)/7}=r^{1/7}\left(\cos\left(\frac{\theta+2n\pi}{7}\right)+i\sin\left(\frac{\theta+2n\pi}{7}\right)\right)\].
You can get all the details here: http://tutorial.math.lamar.edu/Extras/ComplexPrimer/Roots.aspx

Answer 2

I have a graph, but I can't post it for some reason http://www.google.com/imgres?imgurl=http%3A%2F%2Fuser-content.enotes.com%2F7c4540d5f581a45d753fc2cff4180d9b3cb69630_thumb.png&imgrefurl=http%3A%2F%2Fwww.enotes.com%2Fhomework-help%2Ffind-all-seventh-roots-unity-sketch-them-axes-436533&h=287&w=282&tbnid=aGzfxmKJbQW5JM%3A&zoom=1&docid=WUxe1gxLnyuV-M&ei=pg55U7atMtOSqAbAgYGIAQ&tbm=isch&client=safari&ved=0CFUQMygBMAE&iact=rc&uact=3&dur=849&page=1&start=0&ndsp=32

Answer 3

@ranga can you help?

Answer 4

The first reply is right on the mark.

Here is a Wolfram link: https://www.wolframalpha.com/input/?i=x^7-1%3D0

Answer 5

Under complex roots in Wolfram, click on More roots to get 6 complex roots and combined with one real root of 1 you will have 7 roots.

Answer 6

so are all 7 of the complex roots my answer and the graph at the bottom is what I sketch?

Answer 7

complex solutions*

Answer 8

@helpme1.2 can you answer my last question? Just view the link

Answer 9

sorry not my link. the wolfram link

Answer 10

i would assume that they should be your 7 complex numbers

Answer 11

and the graph? that;s what I think, but I'm just verifying

Answer 12

yes, Wolfram is accurate