Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : How to multiply exponential variables

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

Create An Account

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #31 : Variables And Exponents

and are positive integers. Which is the greater quantity?

(A) $A^{2}+B^{2}$

(B) $(A+B)^{2}$

Possible Answers:

It is impossible to tell which is greater from the information given

(A) and (B) are equal

(A) is greater

(B) is greater

Correct answer:

(B) is greater

Explanation:

$(A+B)^{2} = A^{2} + 2AB + B^{2}$

Since and are positive,

2AB > 0

$2AB +A^{2}+B^{2}> 0 +A^{2}+B^{2}$

$A^{2}+2AB +B^{2}> A^{2}+B^{2}$

$(A+B)^{2} > A^{2}+B^{2}$ for all positive and , making (B) greater.

Report an Error

Example Question #32 : Variables And Exponents

and are negative integers. Which is the greater quantity?

(A) $A^{2}+B^{2}$

(B) $(A+B)^{2}$

Possible Answers:

It is impossible to tell which is greater from the information given

(B) is greater

(A) is greater

(A) and (B) are equal

Correct answer:

(B) is greater

Explanation:

$(A+B)^{2} = A^{2} + 2AB + B^{2}$

Since and are both negative,

2AB > 0 .

$2AB +A^{2}+B^{2} > 0 +A^{2}+B^{2}$

$A^{2}+2AB +B^{2} > A^{2}+B^{2}$

$(A+B)^{2} > A^{2}+B^{2}$ for all negative and , making (B) greater.

Report an Error

Example Question #33 : Variables And Exponents

and are positive integers. Which is the greater quantity?

(A) $A^{2}-B^{2}$

(B) $(A-B)^{2}$

Possible Answers:

It is impossible to tell which is greater from the information given

(A) and (B) are equal

(B) is greater

(A) is greater

Correct answer:

It is impossible to tell which is greater from the information given

Explanation:

It is impossible to tell which is greater.

Case 1: A = 4, B= 4

Then

$A^{2}-B^{2} =4^{2}-4^{2} = 16-16 = 0$

and

$(A-B)^{2} = (4-4)^{2} = 0^{2} = 0$ .

This makes (A) and (B) equal.

Case 2: A = 5, B= 4

Then

$A^{2}-B^{2} =5^{2}-4^{2} = 25-16 = 9$

and

$(A-B)^{2} = (5-4)^{2} = 1^{2} = 1$ .

This makes (A) the greater quantity.

Report an Error

Example Question #41 : Variables And Exponents

and are positive integers greater than 1.

Which is the greater quantity?

(A) $(x+y -1)^{2}$

(B) $(x-y+1)^{2}$

Possible Answers:

It is impossible to determine which is greater from the information given

(A) and (B) are equal

(B) is greater

(A) is greater

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute u = y - 1 . Since y > 1 , must be positive, and this problem is to compare $(x+u)^{2}$ and $(x-u)^{2}$ .

$(x+u)^{2} = x^{2} + 2ux + u^{2}$

and

$(x-u)^{2} = x^{2} - 2ux + u^{2}$

Since 2, , and are positive, by closure, 2ux > 0 > -2ux , and by the addition property of inequality,

$x^{2} + 2ux + u^{2} > x^{2} - 2ux + u^{2}$

$(x+u)^{2} > (x-u)^{2}$

Substituting back:

$(x+y -1)^{2} > (x-y+1)^{2}$

(A) is the greater quantity.

Report an Error

Example Question #42 : Variables And Exponents

and are positive integers greater than 1.

Which is the greater quantity?

(A) $\left (x + y + 4\right ) ^{2}$

(B) $\left (x + 2y + 2\right ) ^{2}$

Possible Answers:

It is impossible to determine which is greater from the information given

(B) is greater

(A) and (B) are equal

(A) is greater

Correct answer:

It is impossible to determine which is greater from the information given

Explanation:

Case 1: x = 2, y = 2

Then

$\left (x + y + 4\right ) ^{2} = \left (2 + 2 + 4\right ) ^{2} = 8 ^{2} = 64$

and

$\left (x + 2y + 2\right ) ^{2} = \left (2 + 2 \cdot 2 + 2\right ) ^{2} = 8 ^{2} = 64$

This makes the quantities equal.

Case 2:

x = 2, y = 3

Then

$\left (x + y + 4\right ) ^{2} = \left (2 + 3 + 4\right ) ^{2} = 9 ^{2} = 81$

and

$\left (x + 2y + 2\right ) ^{2} = \left (2 + 2 \cdot 3 + 2\right ) ^{2} = 10 ^{2} = 100$

This makes (B) greater.

Therefore, it is not clear which quantity, if either, is greater.

Report an Error

Example Question #43 : Variables And Exponents

and are positive integers greater than 1.

Which is the greater quantity?

(A) $\left (x + y + 1\right ) ^{2}$

(B) (x+ 1 ) (x + 2y + 1)

Possible Answers:

(A) and (B) are equal

(B) is greater

(A) is greater

It is impossible to determine which is greater from the information given

Correct answer:

(A) is greater

Explanation:

One way to look at this problem is to substitute u = x +y+1 . The expressions to be compared are

$\left (x + y + 1\right ) ^{2} = u ^{2}$

and

$(x+ 1 ) (x + 2y + 1) =(u - y ) (u+y)= u^{2}- y^{2}$

Since is positive, so is $y^{2}$ , and

$u^{2}> u^{2}- y^{2}$

Substituting back,

$\left (x + y + 1\right ) ^{2} > (x+ 1 ) (x + 2y + 1)$ ,

making (A) greater.

Report an Error

Example Question #44 : Variables And Exponents

Factor:

$4x ^{2} - 9 y^{2} + 14x - 21y$

Possible Answers:

(2x-3y-7)(2x+3y)

(2x+3y+7)(2x-3y)

(2x+3y-7)(2x+3y)

(2x-3y-7)(2x-3y)

The expression is a prime polynomial.

Correct answer:

(2x+3y+7)(2x-3y)

Explanation:

We can rewrite as follows:

$4x ^{2} - 9 y^{2} + 14x - 21y$

$= (4x ^{2} - 9 y^{2}) +( 14x - 21y)$

Each group can be factored - the first as the difference of squares, the second as a pair with a greatest common factor. This becomes

(2x+3y)(2x-3y)+7 ( 2x - 3y) ,

which, by distribution, becomes

(2x+3y+7)(2x-3y)

Report an Error

Example Question #45 : Variables And Exponents

is a positive number; is the additive inverse of .

Which is the greater quantity?

(a) $a ^{2} - b^{2}$

(b) $a - b\;$

Possible Answers:

(b) is the greater quantity

(a) is the greater quantity

(a) and (b) are equal

It is impossible to determine which is greater from the information given

Correct answer:

(b) is the greater quantity

Explanation:

If is the additive inverse of , then, by definition,

a+ b = 0 .

$a^{2} - b^{2}$ , as the difference of the squares of two expressions, can be factored as follows:

$a^{2} - b^{2} = (a+b) (a-b )$

Since a+ b = 0 , it follows that

$a^{2} - b^{2} = 0 \cdot (a-b ) = 0$

Another consequence of being the additive inverse of is that

b= -a , so

a - b = a - (-a) = a+ a = 2a

is positive, so is as well.

It follows that $a - b > a ^{2} - b^{2}$ .

Report an Error

View Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Tutors

Gabby
Certified Tutor

University of Oregon, Bachelor in Arts, Journalism. University of Oregon, Current Grad Student, Communication, General.

View Tutors

Eva
Certified Tutor

Technicka Univerzita Liberec, Bachelor of Education, Education.

All ISEE Upper Level Quantitative Resources

8 Diagnostic Tests 234 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Biology Tutors in Houston, SAT Tutors in Houston, Algebra Tutors in Denver, Statistics Tutors in Seattle, Reading Tutors in Dallas Fort Worth, Statistics Tutors in Houston, Algebra Tutors in San Diego, Physics Tutors in Atlanta, Computer Science Tutors in San Francisco-Bay Area, GRE Tutors in Houston

Popular Courses & Classes

GMAT Courses & Classes in San Diego, ISEE Courses & Classes in Washington DC, ISEE Courses & Classes in Boston, GMAT Courses & Classes in Chicago, Spanish Courses & Classes in Atlanta, SSAT Courses & Classes in Atlanta, ISEE Courses & Classes in Denver, ACT Courses & Classes in San Francisco-Bay Area, SAT Courses & Classes in Atlanta, Spanish Courses & Classes in Denver

Popular Test Prep

SAT Test Prep in Washington DC, SSAT Test Prep in New York City, LSAT Test Prep in Houston, GMAT Test Prep in San Diego, SSAT Test Prep in Philadelphia, GMAT Test Prep in Chicago, GRE Test Prep in New York City, LSAT Test Prep in Denver, ISEE Test Prep in San Diego, LSAT Test Prep in Miami

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

ISEE Upper Level Quantitative : How to multiply exponential variables

Study concepts, example questions & explanations for ISEE Upper Level Quantitative

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #31 : Variables And Exponents

Example Question #32 : Variables And Exponents

Example Question #33 : Variables And Exponents

Example Question #41 : Variables And Exponents

Example Question #42 : Variables And Exponents

Example Question #43 : Variables And Exponents

Example Question #44 : Variables And Exponents

Example Question #45 : Variables And Exponents

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: