How do I find the fifth root of a complex number?

1 Answer
Nov 1, 2015

Answer:

Convert to polar form first, then...

Explanation:

A Complex number in the form #r(cos theta + i sin theta)# has #5#th roots:

#root(5)(r)(cos (theta/5) + i sin (theta/5))#

#root(5)(r)(cos ((theta + 2pi)/5) + i sin ((theta+2pi)/5))#

#root(5)(r)(cos ((theta + 4pi)/5) + i sin ((theta+4pi)/5))#

#root(5)(r)(cos ((theta + 6pi)/5) + i sin ((theta+6pi)/5))#

#root(5)(r)(cos ((theta + 8pi)/5) + i sin ((theta+8pi)/5))#

Conventionally your original #theta# is in the range #(-pi, pi]# or the range #[0, 2pi)# according to your definition of #Arg(z)# and the first of these five roots is the Principal Complex fifth root.