# SAT Math : How to divide complex numbers

## Example Questions

### Example Question #11 : Complex Numbers

For which of the following values of  is the value of  least?

Explanation:

is the same as  , which means that the bigger the answer to  is, the smaller the fraction will be.

Therefore,  is the correct answer because

.

### Example Question #2 : How To Divide Complex Numbers

Define an operation  so that for any two complex numbers  and :

Evaluate .

Explanation:

, so

Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is :

### Example Question #12 : Complex Numbers

Define an operation  such that, for any complex number

If , then evaluate .

Explanation:

, so

, so

, and

Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is :

### Example Question #11 : Complex Numbers

Define an operation  as follows:

For any two complex numbers  and ,

Evaluate .

Explanation:

, so

We can simplify each expression separately by rationalizing the denominators.

Therefore,

### Example Question #11 : Complex Numbers

Define an operation  so that for any two complex numbers  and :

Evaluate

Explanation:

, so

Rationalize the denominator by multiplying both numerator and denominator by the complex conjugate of the latter, which is :

### Example Question #15 : Complex Numbers

Define an operation  such that for any complex number ,

If , evaluate .

Explanation:

First substitute our variable N in where ever there is an a.

Thus, , becomes .

Since , substitute:

In order to solve for the variable we will need to isolate the variable on one side with all other constants on the other side. To do this, apply the oppisite operation to the function.

First subtract i from both sides.

Now divide by 2i on both sides.

From here multiply the numerator and denominator by i to further solve.

Recall that  by definition. Therefore,

.

### Example Question #1 : How To Divide Complex Numbers

Let . What is the following equivalent to, in terms of :

Explanation:

Solve for x first in terms of y, and plug back into the equation.

Then go back to the equation you are solving for:

substitute in

### Example Question #11 : Complex Numbers

Simplify the expression by rationalizing the denominator, and write the result in standard form: