SAT Math : Complex Numbers

Study concepts, example questions & explanations for SAT Math

varsity tutors app store varsity tutors android store varsity tutors amazon store varsity tutors ibooks store

Example Questions

Example Question #201 : Exponents

Evaluate .

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

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.

Possible Answers:

Correct answer:

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 #31 : Complex Numbers

Evaluate 

Possible Answers:

None of the other choices gives the correct response.

Correct answer:

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 

.

Possible Answers:

Correct answer:

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 

.

Possible Answers:

Correct answer:

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 

.

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

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: 

Possible Answers:

Correct answer:

Explanation:

Use FOIL:

Combine like terms:

But since , we know

Example Question #32 : Complex Numbers

 is the complex conjugate of .

Evaluate 

.

Possible Answers:

Correct answer:

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 :

Example Question #40 : Complex Numbers

Raise  to the fourth power.

Possible Answers:

None of these

Correct answer:

Explanation:

By the Power of a Power Rule, the fourth power of any number is equal to the square of the square of that number:

Therefore, one way to raise  to the fourth power is to square it, then to square 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:

Learning Tools by Varsity Tutors