### All SAT Math Resources

## Example Questions

### Example Question #12 : Complex Numbers

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

**Possible Answers:**

**Correct answer:**

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 #13 : Complex Numbers

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

Evaluate .

**Possible Answers:**

**Correct answer:**

, so

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

### Example Question #14 : Complex Numbers

Define an operation such that, for any complex number ,

If , then evaluate .

**Possible Answers:**

**Correct answer:**

, so

, so

, and

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 as follows:

For any two complex numbers and ,

Evaluate .

**Possible Answers:**

**Correct answer:**

, so

We can simplify each expression separately by rationalizing the denominators.

Therefore,

### Example Question #16 : Complex Numbers

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

Evaluate

**Possible Answers:**

**Correct answer:**

, so

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

### Example Question #17 : Complex Numbers

Define an operation such that for any complex number ,

If , evaluate .

**Possible Answers:**

**Correct answer:**

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 #18 : Complex Numbers

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

**Possible Answers:**

**Correct answer:**

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 #244 : New Sat

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

**Possible Answers:**

**Correct answer:**

Multiply both numerator and denominator by the complex conjugate of the denominator, which is :