Chapter 9.3, Problem 9AYU

To determine

Tofind how many distinct cube roots will a nonzero complex number have.

Answer to Problem 9AYU

There will be exactly three cube roots for the complex number.

Explanation of Solution

Let us consider a non-zero complex numberfor which the cube root to be found. So, by using the theorem for finding complex roots, the cube root of this complex number can be found-

  

where

For cube root the value of is 3 and hence, the values ofare

As takes three values, there will be exactly three cube roots for the complex number.

Thus, every nonzero complex number will have exactly three cube roots.