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Roots of Complex Numbers

Recall that a consequence of the Fundamental Theorem of Algebra is that a polynomial equation of degree n has n solutions in the complex number system. Each solution is an nth root of the equation. The nth root of a complex number is defined as follows.

 

Definition of nth Root of a Complex Number

The complex number u = a + bi is an nth root of the complex number z if z = un = (a + bi)n.

 

STUDY TIP

The nth roots of a complex number are useful for solving some polynomial equations. For instance, you can use DeMoivre’s Theorem to solve the polynomial equation x4 + 16 = 0 by writing -16 as 16(cos π + i sin π).

 

To find a formula for an nth root of a complex number, let u be an nth root of z, where u = s(cos β + i sin β) and z = r(cos θ + i sin θ).

By DeMoivre’s Theorem and the fact that un = z  you have

sn(cos nβ + i sin nβ) = r (cos θ + i sin θ).

Taking the absolute values of both sides of this equation, it follows that sn = r. Substituting back into the previous equation and dividing by r, you get

cos nβ + i sin nβ = cos θ + i sin θ

Thus, it follows that

cos nβ = cos θ and sin nβ = sin θ.

Because both sine and cosine have a period of 2π, these last two equations have solutions if and only if the angles differ by a multiple of 2π. Consequently, there must exist an integer k such that

By substituting this value for into the polar form of u, you get the following result.

NOTE

When k exceeds n - 1 the roots begin to repeat. For instance, if k = n the angle is coterminal with θ/n which is also obtained when k = 0.

 

Theorem

nth Roots of a Complex Number

For a positive integer n, the complex number z = r(cos θ + i sin θ) has exactly n distinct nth roots given by

where k = 0, 1, 2, ..., n - 1.

This formula for the nth roots of a complex number z has a nice geometrical interpretation, as shown in the figure below.

Note that because the nth roots of z all have the same magnitude they all lie on a circle of radius with center at the origin. Furthermore, because successive nth roots have arguments that differ by 2π/n, the n roots are equally spaced along the circle.

 

Example 1

Finding the nth Roots of a Complex Number

Find the three cube roots of z = -2 + 2i.

Solution

Because z lies in Quadrant II, the polar form for z is

By the formula for nth roots, the cube roots have the form

Finally, for k = 0, 1, and 2, you obtain the roots

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