algebra complex number solve z7 =1 Solution According to the fundamental theorem of algebra, the equation z7=1 has 7 roots, called the roots of unity. Only one of them is obvious: z=1. The other six require polar form. First write 1 in polar form: 1 = ePi*in, where n is any integer. Then set z7=ePi*in and take the seventh root: z=ePi*in/7 Now substitute in values for n. You only need the values n=0,1,2,3,4,5, and 6, because the answer will repeat itself for all other n (due to the periodicity of complex exponentials). For n=0, you get the obvious solution, z0=e0=1. For n=1, you get z1=ePi*i/7, which you can write in standard form using Euler\'s identity: z1=cos(Pi/7)+i sin(Pi/7) In this way, you get the seven seventh roots of unity. If you graph them in the complex plane, you will find they all lie equally spaced on the unit circle..