# Algebra II : Complex Imaginary Numbers

## Example Questions

### Example Question #51 : Imaginary Numbers

Simplify:

Explanation:

We know that  and . Therefore . We know . Therefore we will have just . Our final answer is .

### Example Question #51 : Complex Imaginary Numbers

Simplify:

Explanation:

We know that  and  simplifies to . Since  our final answer is .

### Example Question #53 : Imaginary Numbers

Simplify:

Explanation:

To get rid of a complex number, we multiply by the conjugate which is opposite the sign.

### Example Question #54 : Complex Imaginary Numbers

Simplify:

Explanation:

To get rid of a complex number, we multiply by the conjugate which is opposite the sign.

### Example Question #52 : Complex Imaginary Numbers

Simplify:

Explanation:

Write out some of the imaginary terms.

The powers of the imaginary number can be rewritten using the product of exponents.

Replace all the terms back into the expression.

### Example Question #55 : Imaginary Numbers

Evaluate:

Explanation:

The imaginary term  is equivalent to .

This means that:

Substitute this term back into the numerator.

There is no need to use extra steps such as multiplying by the conjugate of the denominator to simplify.

### Example Question #53 : Complex Imaginary Numbers

Solve:

Explanation:

Evaluate by first evaluating the imaginary terms.

We can also use the product rule of exponents to simplify the higher powered imaginary numbers.

Note that evaluating the denominator will give us a zero denominator.

This will indicate that the expression will approach to infinity.

### Example Question #51 : Imaginary Numbers

Simplify:

Explanation:

Write the first few terms of the imaginary term.

Notice that these terms will be in a pattern for higher order imaginary terms.

Rewrite the numerator using the product of exponents.

### Example Question #54 : Complex Imaginary Numbers

Evaluate:

Explanation:

Using the exponential property, we can expand the term in parentheses by multiplying the inner and outer powers together.

Evaluate  by breaking this up as a product of imaginary powers.

This means that:

### Example Question #60 : Complex Imaginary Numbers

Evaluate:

Explanation:

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

Simplify the bottom using the FOIL method.

Simplify the expression by distributing all terms.  Recall that  and .  Replace the term.

Simplify the top.

Divide this by forty.