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4th Edition

James Stewart

Publisher: Cengage Learning

ISBN: 9781337687805

Chapter I, Problem 39E

To determine

**To find:** The roots of the cube root of *i* and sketch the roots in the complex plane.

Expert Solution

The roots of cube root of *i* are

**Theorem used:**

Roots of a complex number:

Let
*n* be a positive integer. Then
*n*th roots

**Calculation:**

Rewrite the complex number *i* in polar form.

The polar form of the complex number

Consider the complex number *i*. Here,

Obtain the argument of the complex number *i*.

Thus, the argument of the complex number *i* is

Obtain the modulus of the complex number *i*.

Thus, the value of

Therefore, the polar form of the complex number *i* is

By the above theorem, the roots of cube root of *i* are

Use online calculator to sketch the roots in the complex plane as shown below in Figure 1.

From figure 1, it is observed that all cube root of *i* form a triangle on complex plane.