# Precalculus : Find the Roots of Complex Numbers

## Example Questions

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

Evaluate , where Â is a natural number and Â is the complex numberÂ .

Explanation:

Note that,

Â

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

What is theÂ  length ofÂ

?

Explanation:

We have

.

Hence,

.

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

Solve forÂ Â (there may be more than one solution).

Explanation:

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

Clearly, one of roots is 1.

Thus, we can factor the polynomial asÂ

so that we solve for the roots ofÂ .

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

Â

This means the solutions toÂ Â are

Â

### Example Question #4 : Find The Roots Of Complex Numbers

Recall thatÂ Â is just shorthand forÂ Â when dealing with complex numbersÂ in polar form.Â

### ExpressÂ Â Â in polarÂ form.

Explanation:

FirstÂ we recognize that we are trying to solveÂ Â whereÂ .

ThenÂ we want to convert Â into polar form using,

Â Â andÂ .

Then since De Moivre's theorem states,

Â Â if Â is an integer,Â we can say

.

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

Solve for Â (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 () : Â and now setting both Â andÂ Â equal to zero we end up with the answers Â and Â

Â

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

Compute

Â

Explanation:

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

Â

Now plug back into the original equation and solve: Â Â

Â

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

Determine the length ofÂ

Explanation:

, so Â

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

Solve for all possible solutions to the quadratic expression:

Â

Explanation:

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

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

Which of the following listsÂ all possible solutions to the quadratic expression:Â

Explanation:

SolveÂ for complex values ofÂ Â using the quadratic formula:Â

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

Determine the length ofÂ .