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

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### 1 Answer

Find the seventh roots of unity:

Solve `z^7=1` where `z=a+bi in CC`

`z^7=1`

`(r"cis" alpha)^7=7"cis"0^@` (`"cis" alpha=cos alpha + isin alpha` )

`r^7"cis"7alpha=7"cis"0^@`

`r^7=7 ==> r=root(7)(7)`

`7alpha=0 ==>alpha=0 + (k360)/7,k in 1,2,3,4,5,6`

So the seventh roots of unity are :

`root(7)(7)"cis"((k360)/7)` for k=0,1,2,3,4,5,6 or

`root(7)(7)(cos ((k360)/7)+isin((k360)/7))` which are

`root(7)(7)(cos0+isin0)`

`root(7)(7)(cos(360/7)+isin(360/7))`

`root(7)(7)(cos(720/7)+isin(720/7))`

etc...

To plot in the argand plane (the complex plane) there will be 7 vectors of length `root(7)(7)~~1.32` ; the first along the positive x-axis from the origin and the ends of each of the others lying on the vertices of an inscribed heptagon

(inscribed in a circle of radius `root(7)(7)` )