[tex]1=e^{i0}=e^{i2\pi}=e^{i4\pi}=\cdots[/tex]
[tex]\implies1^{1/7}=e^{i(0+2\pi k)/7}[/tex]
This means the seventh roots of unity are
[tex]e^{i0},e^{i2\pi/7},e^{i4\pi/7},e^{i6\pi/7},e^{i8\pi/7},e^{i10\pi/7},e^{i12\pi/7},e^{i14\pi/7},\ldots[/tex]
but notice that [tex]\dfrac{14\pi}7=2\pi\equiv0\mod{2\pi}[/tex], which means the seventh roots begin to repeat, so we only need to worry about [tex]k=0,1,\ldots,6[/tex].
