Roots and Powers of Complex Numbers

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It is a known fact thatand in general a square root always has two roots, one positive and one negative. Alsohas four possible answers, since

Briefly, if with take the nth root of a number we obtain n answers: This is illustrated on the Argand diagram below which shows ten tenth roots of 1.

To find the nth roots of a complex number we write it in polar form if it is not already in polar form. This done we apply the following:

where

Example Find the cube roots of 27

so the cube roots are given by:

within order

ie the cube roots areThe last simplifies so the roots are

Example Find the 4th roots of 16i

so the cube roots are given by:

within order

ie the fourth roots are

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