Question

What does the graph of the seventh root of unity look like on a unit circle like the one in the pictured attached?

Answer 1

There are seven seventh roots of unity, `e^{ {2pi k i }/7}`, all on the unit circle, `r=1` above. The first one is at `theta={2pi}/7 = 360^circ/7 = 51 3/7 \ ^circ`, and there are others at `{4pi}/7, {6pi}/7, {8pi}/7, {10 pi}/7, {12 pi}/7` and of course at `0` radians, i.e. unity itself.

Explanation:

Euler's Identity to an even integer power of `2k` tell us

`(e^{i pi})^{2 k} = (-1)^{2k}`

`e^{2pi k i} = 1`

Now we see

`1^(1/7) = (e^{2pi k i})^{1/7} = e^{ {2pi k i }/7}`

That's seven distinct seventh roots, given by any seven consecutive `k`s. (After that they repeat.)

Answer 2

Another view
See below

Explanation:

It is interesting to note that the `n^(th)` roots of unity lie on the unit circle on the complex plane and form a regular polygon of with n sides.

The 7 roots of `z^7 =1: z in CC` are shown below.