We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Math : Variables and Exponents

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

Create An Account

All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Variables And Exponents

Rewrite the polynomial in standard form:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

Possible Answers:

$-2x - 3x ^{2}- 4x^{3}+ 7+ 8x^{5} + 9x^{4}$

$9x^{4} + 8x^{5} + 7-2x - 3x ^{2} - 4x^{3}$

$9x^{4} + 8x^{5} + 7- 4x^{3} - 3x ^{2} -2x$

$7-2x- 3x ^{2}- 4x^{3}+ 9x^{4} + 8x^{5}$

$8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x + 7$

Correct answer:

$8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x + 7$

Explanation:

The degree of a term of a polynomial with one variable is the exponent of that variable. The terms of a polynomial in standard form are written in descending order of degree. Therefore, we rearrange the terms by their exponent, from 5 down to 0, noting that we can rewrite the and constant terms with exponents 1 and 0, respectively:

$9x^{4} - 3x ^{2} + 8x^{5} -2x + 7 - 4x^{3}$

$= 9x^{4} - 3x ^{2} + 8x^{5} -2x ^{1} + 7x ^{0} - 4x^{3}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x ^{1}+ 7x^{0}$

$=8x^{5} + 9x^{4}- 4x^{3} - 3x ^{2} -2x+ 7$

Report an Error

Example Question #252 : Algebraic Concepts

Simplify:

$\frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

Possible Answers:

$\frac{4}{x^{2}y^{2}}$

$\frac{4}{x^{4}y^{4}}$

$\frac{4}{x^{3}y^{3}}$

$\frac{4}{xy}$

Correct answer:

$\frac{4}{x^{2}y^{2}}$

Explanation:

$\frac{(x^{2}+y^{2})^{2} - (x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

$=\frac{(x^{4}+2x^{2}y^{2}+y^{4}) - (x^{4}-2x^{2}y^{2}+y^{4}) }{x^{4}y^{4}}$

$=\frac{(x^{4}-x^{4}+2x^{2}y^{2}+2x^{2}y^{2}+y^{4}-y^{4}) }{x^{4}y^{4}}$

$=\frac{4x^{2}y^{2}}{x^{4}y^{4}}$

$=\frac{4}{x^{2}y^{2}}$

Report an Error

Example Question #253 : Algebraic Concepts

Assume that $x,y \neq 0$ . Simplify:

$\frac{(x^{2}+y^{2})^{2} +(x^{2} - y^{2})^{2}}{x^{4}y^{4}}$

Possible Answers:

$\frac{4}{x^{2}y^{2}}$

$2x^{4}+2y^{4}$

$\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

$2x^{4}-2y^{4}$

$\frac{2x^{4}-2y^{4} }{x^{4}y^{4}}$

Correct answer:

$\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

Explanation:

$\frac{(x^{2}+y^{2})^{2} + (x^{2} + y^{2})^{2}}{x^{4}y^{4}}$

$=\frac{(x^{4}+2x^{2}y^{2}+y^{4}) + (x^{4}-2x^{2}y^{2}+y^{4}) }{x^{4}y^{4}}$

$=\frac{x^{4}+x^{4}+2x^{2}y^{2}-2x^{2}y^{2}+y^{4}+y^{4} }{x^{4}y^{4}}$

$=\frac{2x^{4}+2y^{4} }{x^{4}y^{4}}$

Report an Error

Example Question #2 : Variables And Exponents

If $x+y\neq 0$ , simplify:

$\frac{(x-y)^2+4xy}{x+y}$

Possible Answers:

x+y

x-y

Correct answer:

x+y

Explanation:

$\frac{(x-y)^2+4xy}{x+y}=\frac{(x^2-2xy+y^2)+4xy}{x+y}$

$=\frac{(x^2-2xy+4xy+y^2)}{x+y}$

$=\frac{(x^2+2xy+y^2)}{x+y}$

$=\frac{(x+y)^2}{x+y}=x+y$

Report an Error

Example Question #254 : Algebraic Concepts

If $x\neq 0$ , simplify:

$\frac{(x+\frac{1}{x})^3+(x-\frac{1}{x})^3}{\frac{2}{x}}$

Possible Answers:

x^4+3

x+3

x^2+3

Correct answer:

x^4+3

Explanation:

$\frac{(x+\frac{1}{x})^3+(x-\frac{1}{x})^3}{\frac{2}{x}}$

$=\frac{(x^3+3x^2\times \frac{1}{x}+3x\times \frac{1}{x^2}+\frac{1}{x^3})+(x^3-3x^2\times \frac{1}{x}+3x\times \frac{1}{x^2}-\frac{1}{x^3})}{\frac{2}{x}}$

$=(x^3+x^3)+(3x-3x)+(\frac{3}{x}+\frac{3}{x})+(\frac{1}{x^3}-\frac{1}{x^3})$

$=\frac{2x^3+\frac{6}{x}}{\frac{2}{x}}$

$=\frac{\frac{2x^4+6}{x}}{\frac{2}{x}}$

$=\frac{2x^4+6}{2}$

$=\frac{2(x^4+3)}{2}$

=x^4+3

Report an Error

Example Question #1 : Variables And Exponents

Define g(x) = 2x - 5

What is $g(x ^{2} + 3x)$ ?

Possible Answers:

$2 x ^{2} + 6x - 5$

$x ^{2} + 6x - 5$

$2 x ^{2} + 6x - 10$

$2 x ^{2} + 3x - 5$

Correct answer:

$2 x ^{2} + 6x - 5$

Explanation:

Substitute $x ^{2} + 3x$ for in the definition:

g(x) = 2x - 5

$g(x ^{2} + 3x ) = 2 \left ( x ^{2} + 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 \left ( x ^{2}\right ) + 2 \left ( 3x \right ) - 5$

$g(x ^{2} + 3x ) = 2 x ^{2} + 6x - 5$

Report an Error

Example Question #1 : How To Add Exponential Variables

Add:

x^3+4x^2-2x^3+3x+2x^2+4

Possible Answers:

x^3+6x^2+3x+4

-x^3+6x^2+3x+4

3x^3+6x^2+3x+4

-3x^3+6x^2+3x+4

Correct answer:

-x^3+6x^2+3x+4

Explanation:

x^3+4x^2-2x^3+3x+2x^2+4

=(x^3-2x^3)+(4x^2+2x^2)+3x+4

=-x^3+6x^2+3x+4

Report an Error

Example Question #255 : Algebraic Concepts

Simplify:

(x+y)^2+2x^2-2y^2

Possible Answers:

x^2+2xy-y^2

3x^2-2xy-y^2

3x^2+2xy+y^2

3x^2+2xy-y^2

-3x^2+2xy-y^2

Correct answer:

3x^2+2xy-y^2

Explanation:

We can expand the first term using FOIL:

(x+y)^2=x^2+2xy+y^2

(x+y)^2+2x^2-2y^2=x^2+2xy+y^2+2x^2-2y^2

Reorder the expression to group like-terms together.

(x^2+2x^2)+2xy+(y^2-2y^2)

Simplify by combining like-terms.

3x^2+2xy-y^2

Report an Error

Example Question #6 : Variables And Exponents

Simplify:

(2x-y)^2-(x-2y)^2

Possible Answers:

3x^2+3y^2-4xy

3x^2+3y^2

x^2+y^2

3x^2-3y^2

3x^2+3y^2+4xy

Correct answer:

3x^2-3y^2

Explanation:

Expand each term by using FOIL:

(2x-y)^2-(x-2y)^2 = [(2x)^2-(2) (2x)(y)+(y)^2]-[(x)^2+(2) (x)(2y)-(2y)^2]

=4x^2-4xy+y^2-x^2+4xy-4y^2

Rearrange to group like-terms together.

(4x^2-x^2)+(-4xy+4xy)+(y^2-4y^2)

Simplify by combining like-terms.

3x^2-3y^2

Report an Error

Example Question #7 : Variables And Exponents

Simplify:

4x^3y-x^2y^2-3x^3y+2x^2y^2

Possible Answers:

-x^3y+x^2y^2

-x^3y-x^2y^2

x^3y-x^2y^2

x^3y+x^2y^2

The expression can not be simplified further

Correct answer:

x^3y+x^2y^2

Explanation:

Start by reordering the expression to group like-terms together.

4x^3y-x^2y^2-3x^3y+2x^2y^2

(4x^3y-3x^3y)+(-x^2y^2+2x^2y^2)

Combine like-terms to simplify.

(4x^3y-3x^3y)+(-x^2y^2+2x^2y^2)

x^3y+x^2y^2

Report an Error

View Tutors

Pavan
Certified Tutor

University of Tennessee, Knoxville, Master's/Graduate, Mechanical Engineering. Pennsylvania State University, Master's/Gradua...

View Tutors

Ana
Certified Tutor

Grand Canyon University, Bachelors, Education. Grand Canyon University, Masters, Education.

View Tutors

Raynette
Certified Tutor

Boise State University, Bachelor in Arts, Elementary School Teaching. Colorado Christian University, Master of Arts, Counselo...

All ISEE Upper Level Math Resources

6 Diagnostic Tests 244 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Physics Tutors in San Diego, English Tutors in Philadelphia, LSAT Tutors in Phoenix, Math Tutors in Washington DC, Physics Tutors in San Francisco-Bay Area, Spanish Tutors in Denver, GMAT Tutors in Seattle, Reading Tutors in Miami, Statistics Tutors in Phoenix, ISEE Tutors in Denver

Popular Courses & Classes

Spanish Courses & Classes in Denver, ACT Courses & Classes in Los Angeles, Spanish Courses & Classes in Philadelphia, LSAT Courses & Classes in Dallas Fort Worth, SAT Courses & Classes in New York City, SAT Courses & Classes in Houston, GMAT Courses & Classes in Los Angeles, Spanish Courses & Classes in Washington DC, ISEE Courses & Classes in Denver, ACT Courses & Classes in Houston

Popular Test Prep

GMAT Test Prep in Chicago, MCAT Test Prep in Houston, SSAT Test Prep in Seattle, MCAT Test Prep in Phoenix, ACT Test Prep in Houston, SAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Chicago, ACT Test Prep in Los Angeles, SAT Test Prep in Washington DC, MCAT Test Prep in Los Angeles

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 Math : Variables and Exponents

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

All ISEE Upper Level Math Resources

Example Questions

Example Question #1 : Variables And Exponents

Example Question #252 : Algebraic Concepts

Example Question #253 : Algebraic Concepts

Example Question #2 : Variables And Exponents

Example Question #254 : Algebraic Concepts

Example Question #1 : Variables And Exponents

Example Question #1 : How To Add Exponential Variables

Example Question #255 : Algebraic Concepts

Example Question #6 : Variables And Exponents

Example Question #7 : Variables And Exponents

All ISEE Upper Level Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: