Question #dd5ca

1 Answer
Eddie
May 1, 2017

Start with the Euler's Formula: #1 = cos (2n) pi + i sin (2n) pi = e^(i 2n pi)#

So #1^(1/3) = (e^(i2n pi))^(1/3) = e^((i 2n pi)/3)#

#n = 0: = cis 0 = 1# This is the principal (real) root.

#n = 1: = cis (2pi)/3 =-1/2 +sqrt3/2 i#

#n = 2: = cis (4pi)/3 =-1/2 - sqrt3/2 i#

Thereafter the roots repeat. In the complex plane, each number has n distinct nth roots.

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