## Computation of roots in algebraic form.

The computation of roots of a given complex number is performed using the algebraic representation of a complex number only when is a power of 2. In other cases the problem is translated into a system of two non linear polynomials equations which is rather hard to solve without using appropriate software.

We illustrate with examples a suitable method to compute square roots and roots of order 4 of a given com[plex number.

Example 1.5.10   Let and , with real .

The extra equation is equivalent to . Solving this system for and , we have or . This means that has two square roots, namely and .

By iterations with this method more, we can solve compute roots of order 4,8,16,etc.

Example 1.5.11   Solve the equation: .

First we compute the square roots of . Denote , where . The following holds:

Thus, the square roots of the complex number are and .

By the same method we compute the square roots of .

Hence, the square roots of are and . These are two of the four roots of order 4 of the number .

We leave to the reader the task of computing the square roots of . Finally, we found the four roots of order 4 of ; they are the complex numbers , , and .

Example 1.5.12   In order to solve the equation , the work is decomposed into three steps:
1. Compute the square roots of -1; these are and ;
2. Compute the square roots of ; these are . Then compute the square roots of ; they are equal to . The reader can have a confirmation of the validity of these results by computing the roots using the polar method.

Noah Dana-Picard 2007-12-24