### 17.2 Powers and Roots

## INTRODUCTION

Recall from calculus that a point (*x*, *y*) in rectangular coordinates can also be expressed in terms of polar coordinates (*r*, *θ*). We shall see in this section that the ability to express a complex number *z* in terms of *r* and *θ* greatly facilitates finding powers and roots of *z*.

### Polar Form

Rectangular coordinates (*x*, *y*) and polar coordinates (*r*, *θ*) are related by the equations *x* = *r* cos *θ* and *y* = *r* sin *θ* (see Section 14.1). Thus a nonzero complex number *z* = *x* + *iy* can be written as *z* = (*r* cos *θ*) + *i*(*r* sin *θ*) or

(1)

We say that (1) is the **polar form** of the complex number *z*. We see from FIGURE 17.2.1 that the polar ...

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