Call Now to Set Up Tutoring:

(888) 888-0446

SAT II Math I : Number Theory

Study concepts, example questions & explanations for SAT II Math I

Create An Account

All SAT II Math I Resources

6 Diagnostic Tests 113 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #11 : Real And Complex Numbers

Which of the following choices gives a sixth root of sixty-four?

Possible Answers:

All of these

$1 - i \sqrt{3 }$

$1 + i \sqrt{3 }$

$-1 - i \sqrt{3 }$

$-1 + i \sqrt{3 }$

Correct answer:

All of these

Explanation:

Let be a sixth root of 64. The question is to find a solution of the equation

$x^{6} = 64$ .

Subtracting 64 from both sides, this equation becomes

$x^{6} - 64= 0$

64 is a perfect square (of 8) The binomial at left can be factored first as the difference of two squares:

$(x^{3} ) ^{2}- 8 ^{2}= 0$

$( x^{3} + 8 )( x^{3} - 8 )= 0$

8 is a perfect cube (of 2), so the two binomials can be factored as the sum and difference, respectively, of two cubes:

$x^{3} + 8 = x^{3}+ 2 ^{3} = (x+2) (x^{2}- x \cdot 2 + 2 ^{2}) = (x+2) (x^{2}-2x+4)$

$x^{3} - 8 = x^{3}+ 2 ^{3} = (x-2) (x^{2}+x \cdot 2 + 2 ^{2}) = (x-2) (x^{2}+2x+4)$

The equation therefore becomes

$(x+2) (x^{2}-2x+4) (x-2) (x^{2}+2x+4) = 0$ .

By the Zero Product Principle, one of these factors must be equal to 0.

If x+ 2 = 0 , then x = -2 ; if x-2 = 0 , then x = 2 . Therefore, and 2 are sixth roots of 64. However, these are not choices, so we examine the other polynomials for their zeroes.

If $x^{2}-2x+4 = 0$ , then, setting a= 1, b= -2, c= 4 in the following quadratic formula:

$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

$= \frac{- (-2) \pm \sqrt{ (-2)^{2}-4(1)(4)}}{2(1)}$

$= \frac{2 \pm \sqrt{ 4 - 16 }}{2}$

$= \frac{2 \pm \sqrt{ -12 }}{2}$

$= \frac{2 \pm\sqrt{4 } \cdot \sqrt{ -1} \cdot \sqrt{3 }}{2}$

$= \frac{2 \pm 2i \sqrt{3 }}{2}$

$=1 \pm i \sqrt{3 }$

If $x^{2}+2x+4= 0$ , then, setting a= 1, b=2, c= 4 in the quadratic formula:

$x = \frac{-b \pm \sqrt{b^{2}-4ac}}{2a}$

$= \frac{- 2 \pm \sqrt{ 2^{2}-4(1)(4)}}{2(1)}$

$= \frac{-2 \pm \sqrt{ 4 - 16 }}{2}$

$= \frac{-2 \pm \sqrt{ -12 }}{2}$

$= \frac{-2 \pm\sqrt{4 } \cdot \sqrt{ -1} \cdot \sqrt{3 }}{2}$

$= \frac{-2 \pm 2i \sqrt{3 }}{2}$

$=-1 \pm i \sqrt{3 }$

Therefore, the set of sixth roots of 64 is

$\left \{ -4, 4,-1 - i \sqrt{3 }, -1 + i \sqrt{3 } ,1 - i \sqrt{3 }, 1 + i \sqrt{3 } \right \}$ .

All four choices appear in this set.

Report an Error

Example Question #41 : Number Theory

Let and be complex numbers. $\overline{x}$ and $\overline{y}$ denote their complex conjugates.

xy = 13+ 7i

x= 4 - 7 i

Evaluate $\overline{x} \cdot \overline{y}$ .

Possible Answers:

7 - 13i

7 + 13i

None of these

-13 - 7i

-13 + 7i

Correct answer:

None of these

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,

$\overline{x} \cdot \overline{y} = \overline{xy}$

xy = 13+ 7i , so $\overline{xy }= 13- 7i$ , and

$\overline{x} \cdot \overline{y} =13- 7i$ ,

which is not among the choices.

Report an Error

Example Question #41 : Number Theory

$\overline{x}$ denotes the complex conjugate of .

If x = 5 + 2i , then evaluate $x^{3} \cdot \overline{x} ^{3}$ .

Possible Answers:

1,000

9,261

None of these

24, 389

Correct answer:

24, 389

Explanation:

Applying the Power of a Product Rule:

$x^{3} \cdot \overline{x} ^{3}=(x \overline{x})^{3}$

The complex conjugate of an imaginary number x = a+ bi is $\overline{x} = a - bi$ ; the product of the two is

$x \overline{x} = a^{2} + b^{2}$

x = 5 + 2i , so, setting a = 5 , b = 2 in the above pattern:

$x \overline{x} = 5^{2} + 2^{2} = 25 + 4 = 29$

Consequently,

$x^{3} \cdot \overline{x} ^{3}=(x \overline{x})^{3} = 29 ^{3}= 24, 389$

Report an Error

View SAT Subject Test in Mathematics Level 1 Tutors

Sharif
Certified Tutor

NCCU, Bachelors, Physics and Mathematics.

View SAT Subject Test in Mathematics Level 1 Tutors

Tarvinder
Certified Tutor

University of Arkansas - Monticello, Bachelor of Science, Mathematics. Tulane University of Louisiana, Master of Science, Com...

View SAT Subject Test in Mathematics Level 1 Tutors

Leonard
Certified Tutor

Florida State University, Bachelor of Science, Environmental Science.

All SAT II Math I Resources

6 Diagnostic Tests 113 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

GRE Tutors in Atlanta, Physics Tutors in Boston, MCAT Tutors in Houston, English Tutors in Miami, Chemistry Tutors in Chicago, ACT Tutors in San Diego, Algebra Tutors in Boston, SSAT Tutors in Houston, ISEE Tutors in Boston, GMAT Tutors in San Francisco-Bay Area

Popular Courses & Classes

LSAT Courses & Classes in Denver, LSAT Courses & Classes in Atlanta, Spanish Courses & Classes in Seattle, MCAT Courses & Classes in Los Angeles, MCAT Courses & Classes in Miami, LSAT Courses & Classes in Seattle, Spanish Courses & Classes in San Diego, Spanish Courses & Classes in Miami, SSAT Courses & Classes in Seattle, SSAT Courses & Classes in Phoenix

Popular Test Prep

GRE Test Prep in Houston, ISEE Test Prep in Houston, SAT Test Prep in Seattle, ACT Test Prep in Washington DC, SAT Test Prep in Houston, MCAT Test Prep in San Diego, SSAT Test Prep in Denver, SAT Test Prep in Chicago, GMAT Test Prep in Phoenix, ACT Test Prep in Chicago

DMCA Complaint

If you believe that content available by means of the Website (as defined in our Terms of Service) infringes one or more of your copyrights, please notify us by providing a written notice (“Infringement Notice”) containing the information described below to the designated agent listed below. If Varsity Tutors takes action in response to an Infringement Notice, it will make a good faith attempt to contact the party that made such content available by means of the most recent email address, if any, provided by such party to Varsity Tutors.

Your Infringement Notice may be forwarded to the party that made the content available or to third parties such as ChillingEffects.org.

Please be advised that you will be liable for damages (including costs and attorneys’ fees) if you materially misrepresent that a product or activity is infringing your copyrights. Thus, if you are not sure content located on or linked-to by the Website infringes your copyright, you should consider first contacting an attorney.

Please follow these steps to file a notice:

You must include the following:

A physical or electronic signature of the copyright owner or a person authorized to act on their behalf; An identification of the copyright claimed to have been infringed; A description of the nature and exact location of the content that you claim to infringe your copyright, in \ sufficient detail to permit Varsity Tutors to find and positively identify that content; for example we require a link to the specific question (not just the name of the question) that contains the content and a description of which specific portion of the question – an image, a link, the text, etc – your complaint refers to; Your name, address, telephone number and email address; and A statement by you: (a) that you believe in good faith that the use of the content that you claim to infringe your copyright is not authorized by law, or by the copyright owner or such owner’s agent; (b) that all of the information contained in your Infringement Notice is accurate, and (c) under penalty of perjury, that you are either the copyright owner or a person authorized to act on their behalf.

Send your complaint to our designated agent at:

Charles Cohn Varsity Tutors LLC
101 S. Hanley Rd, Suite 300
St. Louis, MO 63105

Or fill out the form below:

Do not fill in this field

Contact Information

* Full legal name

Company name

* Phone number

* Email address

Address

* Address1

Address2

* City

* State

* Zipcode

* Country

Complaint Details

* URL of copyrighted work (where your original material is located)

* Please describe the copyrighted work so that it may be easily identified

I am the owner, or an agent authorized to act on behalf of the owner of an exclusive right that is allegedly infringed.

I have a good faith belief that the use of the material in the manner complained of is not authorized by the copyright owner, its agent, or the law.

This notification is accurate.

I acknowledge that there may be adverse legal consequences for making false or bad faith allegations of copyright infringement by using this process.

* Signature (your digital signature is legally binding)

HIGH SCHOOL

GRADUATE SCHOOL

K-8

Search 50+ Tests

math tutoring

science tutoring

foreign languages

elementary tutoring

other

Search 350+ Subjects

SAT II Math I : Number Theory

Study concepts, example questions & explanations for SAT II Math I

All SAT II Math I Resources

Example Questions

Example Question #11 : Real And Complex Numbers

Example Question #41 : Number Theory

Example Question #41 : Number Theory

All SAT II Math I Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: