What is de Moivre's theorem?

de Moivre worked out a brilliant and beautiful way to solve complex equations.

If you for example have z³= 1 and you want to find all real and complex z that satisfy this equation. i is just a complex number written in rectangular form; z = 1 + i * 0 = cos(µ) + i sin(µ) . If you remember your specific angles for sine and cosine you want an angle that gets cosine(µ) = 0 and sine(µ)=1 so µ = 0+ 2nπ. Now, the + 2nπ is especially important for reasons you should see soon.

Rewrite i in euler form 1 = e^{i * (0 + 2nπ)} = z³.  Now, if we take the cube root of both sides, that will be the same as taking it to the power of 1/3.

Remember your power rules: (a^b)^c = a^{b * c}. This will give you z = e^i(0+n2π)/3. This is where the + 2nπ gets really important. If not, the answer would just be one. Put in the different values for n (n=0, n=1, n=2, etc) gives you your different angles. Plug them on your Argand diagram, and you get three different solutions: 

z₀ = 1 ; z₁ = e^{i 2π/3} and z₂ = e^{i 4π/3}. 

Hope it helped. If not, well take it in the session