How do you simplify #(-1)^(1/3)#?

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Answer 1 · 2016-01-07T19:54:27

It depends.

In Real arithmetic you don't have any choice:

#(-1)^(1/3) = -1#

In Complex arithmetic you might prefer to define:

#(-1)^(1/3) = cos(pi/3) + i sin(pi/3) = 1/2 + sqrt(3)/2 i#

or you might prefer to stick with #(-1)^(1/3) = -1#.
It rather depends on the context.

#-1# has three cube roots:

#-1#

#1/2+sqrt(3)/2 i#

#1/2-sqrt(3)/2 i#

If you are dealing with Complex arithmetic then you commonly want to be aware of all three.