# Algebra II : Complex Imaginary Numbers

## Example Questions

### Example Question #41 : Complex Imaginary Numbers

Simplify, if possible:

Explanation:

Multiply the top and the bottom of the fraction by the conjugate of the denominator.

Expand the top and bottom.

Recall that:

Replace the imaginary term for .

Simplify this fraction.

### Example Question #42 : Complex Imaginary Numbers

Evaluate:

Explanation:

Identify the least common denominator of the imaginary terms by using the FOIL method.

Simplify the terms.  Recall that:

and

The expression becomes:

Convert the fractions of the original problem.

Simplify the numerators.

Replace the  term.

Replace the denominator with the value of the LCD.

Factor out a negative one.

Simplify by multiplying both the top and bottom by the conjugate of the denominator.  Factor both the top and bottom using the FOIL method.

Replace the  term.

Simplify the fraction.

### Example Question #41 : Imaginary Numbers

Simplify:

Explanation:

Multiply the top and the bottom by the conjugate of the denominator.

Since the i-terms are imaginary numbers, write out the actual value of  and .

Simplify both top and bottom by the FOIL method.

Divide the two terms.

### Example Question #43 : Complex Imaginary Numbers

Simplify:

Explanation:

Multiply the top and bottom by the conjugate of the denominator.

Simplify the top and bottom.  The bottom can be simplified by the FOIL method.

The expression becomes:

Recall that  and .  Replace the values.

### Example Question #45 : Complex Imaginary Numbers

Simplify, if possible:

Explanation:

In order to simplify this, we will need to multiply both the top and bottom by the conjugate of the denominator.

Simplify the top and bottom by FOIL method.  Recall that:   and .

Solve the denominator.

Divide the numerator and denominator.

### Example Question #41 : Complex Imaginary Numbers

Simplify:

Explanation:

Write out the power terms of .

Then,

Replace the terms.

Multiply the top and bottom by the conjugate of the denominator.

Simplify both terms by the FOIL method.

Divide the numerator by the denominator.

### Example Question #42 : Complex Imaginary Numbers

Evaluate:

Explanation:

Simplify the numerator.  Recall that  and .

Replace the value of .

Rewrite the fraction and split it into two terms.

Simplify both terms.

To simplify , multiply  on the top and bottom.

This means that:

### Example Question #41 : Imaginary Numbers

Solve:

Explanation:

Multiply the numerator and denominator by the conjugate of the denominator.

Simplify the top and bottom.

The fraction becomes:

Recall that  since .  Replace the value.

### Example Question #41 : Imaginary Numbers

None of these.

Explanation:

FOIL:

Simplify like terms:

Simplify :

### Example Question #41 : Imaginary Numbers

Simplify:

Explanation:

Use the FOIL method to simplify the denominator.

The imaginary term is defined as:

This means that:

Replace the term.