### All Precalculus Resources

## 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:**

Note that,

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

What is the length of

?

**Possible Answers:**

**Correct answer:**

We have

.

Hence,

.

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

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

**Possible Answers:**

**Correct answer:**

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 #1 : 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:**

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 #1 : Find The Roots Of Complex Numbers

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

**Possible Answers:**

**Correct answer:**

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 #1 : Find The Roots Of Complex Numbers

Compute

**Possible Answers:**

**Correct answer:**

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 #1 : Find The Roots Of Complex Numbers

Determine the length of

**Possible Answers:**

**Correct answer:**

, so

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

Solve for all possible solutions to the quadratic expression:

**Possible Answers:**

**Correct answer:**

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

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

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

**Possible Answers:**

**Correct answer:**

Solve for complex values of using the quadratic formula:

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

Determine the length of .

**Possible Answers:**

**Correct answer:**

To begin, we must recall that . Plug this in to get . Length must be a positive value, so we'll take the absolute value: . Therefore the length is 3.

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