# SAT II Math II : Real and Complex Numbers

## Example Questions

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### Example Question #11 : Real And Complex Numbers

Evaluate:

Explanation:

Use the square of a sum pattern

where :

### Example Question #11 : Real And Complex Numbers

is a complex number;  denotes the complex conjugate of .

Which of the following could be the value of ?

Any of the numbers in the other four choices could be equal to .

Any of the numbers in the other four choices could be equal to .

Explanation:

The product of a complex number  and its complex conjugate  is

Setting  and  accordingly for each of the four choices, we want to find the choice for which :

For each given value of .

### Example Question #12 : Real And Complex Numbers

denotes the complex conjugate of .

If , then evaluate .

Explanation:

Applying the Power of a Product Rule:

The complex conjugate of an imaginary number  is ; the product of the two is

, so, setting  in the above pattern:

Consequently,

### Example Question #11 : Real And Complex Numbers

Let  and  be complex numbers.  and  denote their complex conjugates.

Evaluate .

Explanation:

Knowing the actual values of  and   is not necessary to solve this problem. The product of the complex conjugates of two numbers is equal to the complex conjugate of the product of the numbers; that is,

We are given that  is therefore the conjugate of , or .

### Example Question #13 : Real And Complex Numbers

denotes the complex conjugate of .

If , then evaluate .

Explanation:

By the difference of squares pattern,

If , then .

Consequently:

Therefore,

### Example Question #13 : Real And Complex Numbers

The fraction  is equivalent to which of the following?

Undefined

Explanation:

Start by multiplying the fraction by .

Since , we can then simplify the fraction:

Thus, the fraction is equivalent to .

### Example Question #14 : Real And Complex Numbers

The fraction  is equivalent to which of the following?

Explanation:

Start by multiplying both the denominator and the numerator by the conjugate of , which is .

Next, recall , and combine like terms.

Finally, simplify the fraction.

### Example Question #11 : Real And Complex Numbers

Evaluate

Explanation:

To evaluate a power of , divide the exponent by 4 and note the remainder.

The remainder is 3, so

Consequently, using the Product of Powers Rule:

### Example Question #12 : Real And Complex Numbers

Let  be a complex number.  denotes the complex conjugate of

and .

How many of the following expressions could be equal to ?

(a)

(b)

(c)

(d)

None

Three

Two

One

Four

Two

Explanation:

is a complex number, so  for some real ; also, .

Therefore,

Substituting:

Therefore, we can eliminate choices (c) and (d).

Also, the product

Setting  and substituting 10 for , we get

Therefore, either  or  - making two the correct response.

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