# GRE Subject Test: Math : Operations on Complex Numbers

## Example Questions

### Example Question #1 : Operations On Complex Numbers

Expand and Simplify:

Explanation:

Step 1: We will multiply the two complex conjugates:  and .

Step 2: Replace  with .

Simplify:

Step 3: Multiply the result of the complex conjugates to the other parentheses,.

The final answer after the product of all three binomials is

### Example Question #1 : Operations On Complex Numbers

Expand: .

Explanation:

Quick Way:
Step 1: Expand  .

.

Remember:

Step 2:

By this equivalence, I can just raise the answer of  to the power .

. Replace ..

Long Way:

### Example Question #1 : Operations On Complex Numbers

Multiply:

Explanation:

Step 1: FOIL:

Recall, FOIL means to multiply the first terms in both binomials together, the outer terms together, the inner terms together, and finally, the last terms together.

Step 2: Simplify:

Step 3: Recall: . Replace and simplify.

### Example Question #1 : Operations On Complex Numbers

Explanation:

When adding imaginary numbers, simply add the real parts and the imaginary parts.

Explanation:

### Example Question #11 : Imaginary Numbers & Complex Functions

What is the value of ?

Explanation:

Distribute and Multiply:

Simplify all terms...

### Example Question #1 : Operations On Complex Numbers

What is the value: ?

Explanation:

Step 1: Recall the cycle of imaginary numbers to a random power .

If , then

If , then

If , then

If , then

If , then

and so on....

The cycle repeats every  terms.

For ANY number , you can break down that term into smaller elementary powers of i.

Step 2: Distribute the  to all terms in the parentheses:

.

Step 3: Recall the rules for exponents:

Step 4: Use the rules to rewrite the expression in Step 2:

Step 5: Simplify the results in Step 4. Use the rules in Step 1.:

Step 6: Write the answer in  form, where  is the real part and  is the imaginary part:

We get

Explanation:

### Example Question #263 : Gre Subject Test: Math

Explanation:

In order to subtract complex numbers, we must first distribute the negative sign to the second complex number.

### Example Question #1 : Operations On Complex Numbers

Explanation:

First we must distribute