# Algebra II : Complex Imaginary Numbers

## Example Questions

### Example Question #31 : Complex Imaginary Numbers

Simplify:

Explanation:

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

Multiply the numerator with the numerator.  Use the FOIL method.

Recall that , which indicates that .

Multiply the denominator with the denominator.

Divide the numerator with the denominator.

### Example Question #32 : Complex Imaginary Numbers

Solve:

Explanation:

In order to simplify this expression, we will need to multiply the numerator and denominator with the conjugate of the denominator.

Simplify the top and bottom.  The value of .

Divide the numerator with the denominator.

### Example Question #33 : Complex Imaginary Numbers

Simplify:

Explanation:

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

Simplify the top and the bottom.

Recall that , and .

This means that:

Re-substitute the actual values back into the fraction.

Reduce and split this fraction.

### Example Question #34 : Complex Imaginary Numbers

Simplify:

Explanation:

The values are imaginary.  Recall that:

This means that:

Rewrite the numerator.

Multiply the top and bottom by the conjugate of .

Rewrite the terms using the value of .

### Example Question #35 : Complex Imaginary Numbers

Simplify:

Explanation:

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

Multiply the numerator.

The term  since .  Replace the term and simplify.

Multiply the denominator.

The expression becomes:

Dividing the double negatives will result into a positive value.

### Example Question #36 : Complex Imaginary Numbers

Simplify:

Explanation:

Simplify the denominator.  Recall that , and .

Multiply the top and bottom by the conjugate of the denominator.  Use the FOIL method to simplify the denominator.

Replace the values of the known imaginary terms.

Simplify this fraction.

### Example Question #37 : Complex Imaginary Numbers

Simplify:

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.  The bottom can be simplified by the FOIL method.

The value of  .  Replace the term.

Reduce this fraction.

### Example Question #38 : Complex Imaginary Numbers

Simplify:

Explanation:

In order to simplify this expression, we will need to multiply the numerator and denominator by the conjugate of the denominator.

Simplify both the top and bottom.  Recall that:

Divide the numerator with the denominator.

Simplify this term and split the fraction.

### Example Question #39 : Complex Imaginary Numbers

Solve:

Explanation:

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

Simplify the top and bottom using the FOIL method.

Note that:

Replace the term.  The numerator becomes:

Simplify the denominator.

Divide the numerator with the denominator.

### Example Question #40 : Complex Imaginary Numbers

Simplify:

Explanation:

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

Simplify the numerator.  Recall that , and .

Simplify the denominator by FOIL method.

Divide the numerator with the denominator.