# College Algebra : Complex Numbers

## Example Questions

### Example Question #1 : Complex Numbers

Consider the following definitions of imaginary numbers:

Then,

Explanation:

### Example Question #2 : Complex Numbers

What is the value of ?

Explanation:

When dealing with imaginary numbers, we multiply by foiling as we do with binomials. When we do this we get the expression below:

Since we know that  we get  which gives us

### Example Question #2 : Complex Numbers

What is the value of  ?

Explanation:

Recall that the definition of imaginary numbers gives that  and thus that . Therefore, we can use Exponent Rules to write

### Example Question #3 : Complex Numbers

Explanation:

When adding complex numbers, add the real parts and the imaginary parts separately to get another complex number in standard form.

Adding the real parts gives , and adding the imaginary parts gives .

### Example Question #4 : Complex Numbers

Divide:

The answer must be in standard form.

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator which is  which results in

The numerator after simplification give us

The denominator is equal to

Hence, the final answer in standard form =

### Example Question #1081 : Linear Equations

Divide:

Answer must be in standard form.

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator which is  resulting in

This is equal to

Since  you can make that substitution of  in place of  in both numerator and denominator, leaving:

When you then cancel the negatives in both numerator and denominator (remember that , simplifying each term), you're left with a denominator of  and a numerator of , which equals .

### Example Question #6 : Complex Numbers

Evaluate:

Explanation:

Use the FOIL method to simplify. FOIL means to mulitply the first terms together, then multiply the outer terms together, then multiply the inner terms togethers, and lastly, mulitply the last terms together.

The imaginary  is equal to:

Write the terms for .

Replace  with the appropiate values and simplify.

### Example Question #7 : Complex Numbers

Explanation:

Combine like terms:

Distribute:

Combine like terms:

### Example Question #3 : Complex Numbers

Rationalize the complex fraction:

Explanation:

To rationalize a complex fraction, multiply numerator and denominator by the conjugate of the denominator.

Multiply: