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

1 Answer
George C.
Sep 2, 2015

If you express your complex number in polar form as #r(cos theta + i sin theta)#, then it has fourth roots:

#alpha = root(4)(r)(cos (theta/4) + i sin (theta/4))#, #i alpha#, #-alpha# and #- i alpha#

Explanation:

Given #a+ib#, let #r = sqrt(a^2+b^2)#, #theta = "atan2"(b, a)#

Then #a + ib = r (cos theta + i sin theta)#

This has one #4th# root #alpha = root(4)(r)(cos (theta/4) + i sin (theta/4))#

There are three other #4th# roots: #i alpha#, #-alpha# and #-i alpha#

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