# Precalculus : Find the Roots of Complex Numbers

## Example Questions

### Example Question #1 : Find The Roots Of Complex Numbers

Evaluate , where is a natural number and is the complex number .      Explanation:

Note that, ### Example Question #2 : Find The Roots Of Complex Numbers

What is the  length of ?      Explanation:

We have .

Hence, .

### Example Question #3 : Find The Roots Of Complex Numbers

Solve for (there may be more than one solution).     Explanation:

Solving that equation is equivalent to solving the roots of the polynomial .

Clearly, one of roots is 1.

Thus, we can factor the polynomial as so that we solve for the roots of .

Using the quadratic equation, we solve for roots, which are .

This means the solutions to are ### Example Question #4 : Find The Roots Of Complex Numbers

Recall that is just shorthand for when dealing with complex numbers in polar form.

### Express in polar form.      Explanation:

First we recognize that we are trying to solve where .

Then we want to convert into polar form using, and .

Then since De Moivre's theorem states, if is an integer, we can say .

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