Calculus: Powers and Roots of Complex Numbers

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Professor Burger explains how to find the powers and roots of complex numbers. The equation of a complex number is z= r(cosx + isinx). To raise the complex number to a power, n, the equation is z^n = r^n[cos(nx) + isin(nx)]. In general, if you are raising a complex number to the power of n or 1/n (taking the nth root), you will come up with n solutions, as you will always have one solution for each of the degrees of power. When taking the root of a complex number, you will find one solution for each degree of power. To find the nth root of a complex number the equation is n root of z = (n root r) *[cos ((x + 2 Pi K)/n) + 1 sin ((x + 2 Pi K)/n)] where k = 0, 1, 2,...n-1.

Taught by Professor Edward Burger, this lesson was selected from a broader, comprehensive course.

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[lostguitarartist]

how about if it not a perfect square