Finding the nth Root of a Complex Number

Baldwin

Let our complex number z = 5 + 3i. Let’s find the 7^th roots of z.

First, converting to trigonometric form z = r(cos q + isin q) we get

z = 34^1/2(cos 31^o + isin 31^o)

de’Moivre’s theorem says that zⁿ = rⁿ(cosnq + isinnq), so

Each of the next six roots will have the same radius, but the angles will each have to increase by 1/7 of a full circle, which is 2p/7, or 0.898 radians, or 1/7 of 360^o = 51.43^o. That makes all seven roots

These are the 7 7^th roots of 5 + 3i.