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## Example Questions

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

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

**Possible Answers:**

**Correct answer:**

Explanation:

Note that,

### Example Question #2 : Find The Roots Of Complex Numbers

What is the length of

?

**Possible Answers:**

**Correct answer:**

Explanation:

We have

.

Hence,

.

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

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

**Possible Answers:**

**Correct answer:**

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.

**Possible Answers:**

**Correct answer:**

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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### All Precalculus Resources

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