Chapter 6.5, Problem 93E

(a)

To determine

To find the indicated roots of the complex number.

Explanation of Solution

Given information:

The given expression is

cube roots of

Using the formula for nth roots of a complex number which states that,

For a positive integer the complex number has exactly distinct roots given by,

  

Where.

  

Hence, its modulus is

  

Its argument is

  

But the complex number lies in the second quadrant, therefore

  

Thus, the trigonometric form comes out to be

  

Using the above formula-

  

So, for roots are as follows

For 0,

  

For 1,

  

For 2,

  

(b)

To determine

To represent the roots graphically.

Explanation of Solution

Representing the solution graphically as follows-

  

(c)

To determine

To represent the roots in standard form.

Explanation of Solution

Writing the roots in standard form as follows-

  

For,

  

For,