# SAT Math : How to multiply complex numbers

## Example Questions

### Example Question #21 : Complex Numbers

Raise  to the power of 3.

None of the other choices gives the correct response.

Explanation:

To raise any expression  to the third power, use the pattern

Setting :

Taking advantage of the Power of a Product Rule:

Since ,

and

:

Collecting real and imaginary terms:

### Example Question #201 : Exponents

Evaluate .

None of the other choices gives the correct response.

Explanation:

Apply the Power of a Product Rule:

,

and

so, substituting and evaluating:

### Example Question #41 : Squaring / Square Roots / Radicals

Raise  to the power of 4.

Explanation:

The easiest way to find  is to note that

.

Therefore, we can find the fourth power of  by squaring , then squaring the result.

Using the binomial square pattern to square :

Applying the Power of a Product Property:

Since  by definition:

Square this using the same steps:

,

the correct response.

### Example Question #42 : Squaring / Square Roots / Radicals

Evaluate

None of the other choices gives the correct response.

None of the other choices gives the correct response.

Explanation:

Apply the Power of a Product Rule:

Applying the Product of Powers Rule:

raised to any multiple of 4 is equal to 1, and , so, substituting and evaluating:

This is not among the given choices.

### Example Question #43 : Squaring / Square Roots / Radicals

is the complex conjugate of .

Evaluate

.

Explanation:

conforms to the perfect square trinomial pattern

.

The easiest way to solve this problem is to add  and , then square the sum.

The complex conjugate of a complex number  is .

,

so  is the complex conjugate of this;

and

Substitute 14 for :

.

### Example Question #35 : Complex Numbers

is the complex conjugate of .

Evaluate

.

Explanation:

conforms to the perfect square trinomial pattern

.

The easiest way to solve this problem is to add  and , then square the sum.

The complex conjugate of a complex number  is .

,

so  is the complex conjugate of this;

and

Substitute 8 for :

.

### Example Question #36 : Complex Numbers

is the complex conjugate of .

Evaluate

.

Explanation:

conforms to the perfect square trinomial pattern

.

The easiest way to solve this problem is to subtract  and , then square the difference.

The complex conjugate of a complex number  is .

,

so  is the complex conjugate of this;

Substitute  for :

By definition, , so, substituting,

,

the correct choice.

### Example Question #211 : Exponents

Remember that .

Simplify:

Explanation:

Use FOIL to multiply complex numbers as follows:

Since , it follows that , so then:

Combining like terms gives:

### Example Question #42 : Squaring / Square Roots / Radicals

Simplify:

Explanation:

Use FOIL:

Combine like terms:

But since , we know

### Example Question #41 : Squaring / Square Roots / Radicals

is the complex conjugate of .

Evaluate

.

Explanation:

conforms to the perfect square trinomial pattern

.

The easiest way to solve this problem is to subtract  and , then square the difference.

The complex conjugate of a complex number  is .

,

so  is the complex conjugate of this;

Taking advantage of the Power of a Product Rule and the fact that :