Nth root
n th roots

Every complex number has n different n th roots in the complex plane. These are

η, η ω, η ω 2, …, η ω n − 1, {\displaystyle \eta,\;\eta \omega,\;\eta \omega ^{2},\;\ldots,\;\eta \omega ^{n-1},}

where η is a single n th root, and 1, ω, ω,... ω are the n th roots of unity. For example, the four different fourth roots of 2 are

2 4, i 2 4, − 2 4, and − i 2 4. {\displaystyle {\sqrt{2}},\quad i{\sqrt{2}},\quad -{\sqrt{2}},\quad {\text{and}}\quad -i{\sqrt{2}}.}

In polar form, a single n th root may be found by the formula

r e i θ n = r n ⋅ e i θ / n. {\displaystyle {\sqrt{re^{i\theta }}}={\sqrt{r}}\cdot e^{i\theta /n}.}

Here r is the magnitude (the modulus, also called the absolute value) of the number whose root is to be taken; if the number can be written as a+bi then r = a 2 + b 2 {\displaystyle r={\sqrt {a^{2}+b^{2}}}}. Also, θ {\displaystyle \theta } is the angle formed as one pivots on the origin counterclockwise from the positive horizontal axis to a ray going from the origin to the number; it has the properties that cos θ = a / r, {\displaystyle \cos \theta =a/r,} sin θ = b / r, {\displaystyle \sin \theta =b/r,} and tan θ = b / a. {\displaystyle \tan \theta =b/a.}

Thus finding n th roots in the complex plane can be segmented into two steps. First, the magnitude of all the n th roots is the n th root of the magnitude of the original number. Second, the angle between the positive horizontal axis and a ray from the origin to one of the n th roots is θ / n {\displaystyle \theta /n}, where θ {\displaystyle \theta } is the angle defined in the same way for the number whose root is being taken. Furthermore, all n of the n th roots are at equally spaced angles from each other.

If n is even, a complex number's n th roots, of which there are an even number, come in additive inverse pairs, so that if a number r is one of the n th roots then r = – r is another. This is because raising the latter's coefficient –1 to the n th power for even n yields 1: that is, (– r) = (–1) × r = r.

As with square roots, the formula above does not define a continuous function over the entire complex plane, but instead has a branch cut at points where θ / n is discontinuous.
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