# Precalculus : Find the Roots of Complex Numbers

## Example Questions

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

What is the  length of

?

Explanation:

We have

.

Hence,

.

### Example Question #1 : 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

.

### Example Question #5 : Find The Roots Of Complex Numbers

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

Explanation:

To solve for the roots, just set equal to zero and solve for z using the quadratic formula () :  and now setting both  and  equal to zero we end up with the answers  and

### Example Question #6 : Find The Roots Of Complex Numbers

Compute

Explanation:

To solve this question, you must first derive a few values and convert the equation into exponential form: :

Now plug back into the original equation and solve:

### Example Question #7 : Find The Roots Of Complex Numbers

Determine the length of

Explanation:

, so

### Example Question #8 : Find The Roots Of Complex Numbers

Solve for all possible solutions to the quadratic expression:

Explanation:

Solve for complex values of m using the aforementioned quadratic formula:

### Example Question #9 : Find The Roots Of Complex Numbers

Which of the following lists all possible solutions to the quadratic expression:

Explanation:

Solve for complex values of  using the quadratic formula:

### Example Question #10 : Find The Roots Of Complex Numbers

Determine the length of .