# High School Math : Using Expressions with Complex Numbers

## Example Questions

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### Example Question #1 : Using Expressions With Complex Numbers

Simplify

Explanation:

Multiplying top and bottom by the complex conjugate  eliminates i from the denominator

### Example Question #2 : Using Expressions With Complex Numbers

Multiply, then simplify.

Explanation:

Use FOIL to multiply. We can treat as a variable for now, just as if it were an .

Combine like terms.

Now we can substitute for .

Simplify.

### Example Question #3 : Using Expressions With Complex Numbers

Which of the following is equivalent to , where ?

Explanation:

We will need to simplify the rational expression by removing imaginary terms from the denominator. Often in such problems, we want to multiply the numerator and denominator by the conjugate of the denominator, which will usually eliminate the imaginary term from the denominator.

In this problem, the denominator is . Remember that, in general, the conjugate of the complex number  is equal to , where a and b are both nonzero constants. Thus, the conjugate of  is equal to .

We need to multiply both the numerator and denominator of the fraction  by .

Next, we use the FOIL method to simplify both the numerator and denominator. The FOIL method of binomial multiplication requires us to mutiply together the first, outside, inner, and last terms of each binomial and then sum them.

Now, we can start simplifying.

Use the fact that .

### Example Question #4 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

Explanation:

Begin by factoring the radical expression using imaginary numbers:

Now, multiply the factors:

Simplify. Remember that  is equivalent to

### Example Question #5 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

Explanation:

Begin by completing the square:

Now, multiply the factors. Remember that  is equivalent to

### Example Question #6 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

Explanation:

Begin by multiplying the numerator and denominator by the complement of the denominator:

Combine like terms:

Simplify. Remember that  is equivalent to

### Example Question #7 : Using Expressions With Complex Numbers

Simplify the following complex number expression:

Explanation:

Begin by factoring the radical expression using imaginary numbers:

Now, multiply the factors:

Simplify. Remember that  is equivalent to

### Example Question #8 : Using Expressions With Complex Numbers

Solve the following complex number equation:

Explanation:

Begin by simplifying the equation. Divide both sides by  and take the square root.

### Example Question #9 : Using Expressions With Complex Numbers

Solve the following complex number equation:

Explanation:

Begin by simplifying the equation. Divide both sides by  and take the square root.

### Example Question #10 : Using Expressions With Complex Numbers

Solvethe following complex number expression for :