# GRE Subject Test: Math : Complex Conjugates

## Example Questions

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### Example Question #1 : Complex Imaginary Numbers

Evaluate

You cannot divide by complex numbers

Explanation:

To divide by a complex number, we must transform the expression by multiplying it by the complex conjugate of the denominator over itself. In the problem,  is our denominator, so we will multiply the expression by  to obtain:

.

We can then combine like terms and rewrite all  terms as . Therefore, the expression becomes:

### Example Question #14 : Imaginary Numbers

Simplify:

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

Now, multiply and simplify.

Remember that

### Example Question #1 : Complex Conjugates

Simplify:

Explanation:

To get rid of the fraction, multiply the numerator and denominator by the conjugate of the denominator.

Now, multiply and simplify.

Remember that

### Example Question #1 : Complex Numbers

Divide:

The answer must be in standard form.

Explanation:

Multiply both the numerator and the denominator by the conjugate of the denominator which is  which results in

The numerator after simplification give us

The denominator is equal to

Hence, the final answer in standard form =

### Example Question #101 : Algebra

Explanation:

The definition of a complex conjugate is each of two complex numbers with the same real part and complex portions of opposite sign.

### Example Question #2 : Complex Conjugates

Which of the following is the complex conjugate of ?

Explanation:

The complex conjugate of a complex equation  is .

The complex conjugate when multiplied by the original expression will also give me a real answer.

The complex conjugate of  is

### Example Question #3 : Complex Conjugates

Simplify

Explanation:

In problems like this, you are expected to simplify by removing i from the denominator. To do this, multiply the numerator and denominator by the conjugate of the denominator (switch the sign between the two terms from either a plus to a minus or vice versa) over itself. The conjugate over itself equals 1 and does not change the value of the expression (any number multiplied by 1 is still that number). Multiplying by the conjugate is the only way to eliminate i since there will be no middle term when we foil.

Simplify i squared to -1 and then combine like terms

### Example Question #4 : Complex Conjugates

Simplify

Explanation:

In problems like this, you are expected to simplify by removing i from the denominator. To do this, multiply the numerator and denominator by the conjugate of the denominator (switch the sign between the two terms from either a plus to a minus or vice versa) over itself. The conjugate over itself equals 1 and does not change the value of the expression (any number multiplied by 1 is still that number). Multiplying by the conjugate is the only way to eliminate i since there will be no middle term when we foil.

Simplify i squared to be -1 and then combine like terms

### Example Question #5 : Complex Conjugates

Simplify

Explanation:

In problems like this, you are expected to simplify by removing i from the denominator. To do this, multiply the numerator and denominator by the conjugate of the denominator (switch the sign between the two terms from either a plus to a minus or vice versa) over itself. The conjugate over itself equals 1 and does not change the value of the expression (any number multiplied by 1 is still that number). Multiplying by the conjugate is the only way to eliminate i since there will be no middle term when we foil.

Simplify i squared to be -1 and then combine like terms

Simplify