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

1 Answer
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