section_path: Definition and basic operations/Powers and roots
section_name: Powers and roots
paragraph_text: Powers and roots
  
  The n -th power of a complex number can be computed using de Moivre's formula, which is obtained by repeatedly applying the above formula for the product: z n = z ⋅ ⋯ ⋅ z ⏟ n factors = (r (cos φ + i sin φ)) n = r n (cos n φ + i sin n φ). {\displaystyle z^{n}=\underbrace {z\cdot \dots \cdot z} _{n{\text{ factors}}}=(r(\cos \varphi +i\sin \varphi))^{n}=r^{n}\,(\cos n\varphi +i\sin n\varphi).} For example, the first few powers of the imaginary unit i are i, i 2 = − 1, i 3 = − i, i 4 = 1, i 5 = i, … {\displaystyle i,i^{2}=-1,i^{3}=-i,i^{4}=1,i^{5}=i,\dots }.
  
  The n n th roots of a complex number z are given by z 1 / n = r n (cos (φ + 2 k π n) + i sin (φ + 2 k π n)) {\displaystyle z^{1/n}={\sqrt{r}}\left(\cos \left({\frac {\varphi +2k\pi }{n}}\right)+i\sin \left({\frac {\varphi +2k\pi }{n}}\right)\right)} for 0 ≤ k ≤ n − 1. (Here r n {\displaystyle {\sqrt{r}}} is the usual (positive) n th root of the positive real number r.) Because sine and cosine are periodic, other integer values of k do not give other values. For any z ≠ 0 {\displaystyle z\neq 0}, there are, in particular n distinct complex n -th roots. For example, there are 4 fourth roots of 1, namely
  
  In general there is no natural way of distinguishing one particular complex n th root of a complex number. (This is in contrast to the roots of a positive real number x, which has a unique positive real n -th root, which is therefore commonly referred to as the n -th root of x.) One refers to this situation by saying that the n th root is a n -valued function of z.