Call Now to Set Up Tutoring:

(888) 888-0446

ISEE Upper Level Quantitative : Variables and Exponents

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 #1 : How To Find The Exponent Of Variables

Expand: $(x + 2)^{10}$

Which is the greater quantity?

(a) The coefficient of

(b) The coefficient of $x^{2}$

Possible Answers:

It is impossible to tell from the information given.

(b) is greater.

(a) is greater.

The two quantities are equal.

Correct answer:

(b) is greater.

Explanation:

Using the Binomial Theorem, if $(x + 2)^{n}$ is expanded, the $x^{r}$ term is

$_{n}\textrm{C} _{r} \cdot x^{r} \cdot 2 ^{n-r}$

$= \frac{n!}{(n-r)!r!} \cdot 2 ^{n-r} \cdot x^{r}$ .

This makes $\frac{n!}{(n-r)!r!} \cdot 2 ^{n-r}$ the coefficient of $x^{r}$ .

We compare the values of this expression at n = 10 for both r = 1 and r = 2 .

(a) If n = 10 and r = 1 , the coefficient is

$\frac{10!}{(10-1)!1!} \cdot 2 ^{10-1} = \frac{10!}{9!} \cdot 2 ^{9} =10 \cdot 512 =5,120$ .

This is the coefficient of .

(b) If n = 10 and r = 2 , the coefficient is

$\frac{10!}{(10-2)!2!} \cdot 2 ^{10-2} = \frac{10!}{8!2!} \cdot 2 ^{8} = \frac{ 9 \cdot 10}{2} \cdot 256 =11,520$ .

This is the coefficient of $x^{2}$ .

(b) is the greater quantity.

Report an Error

Example Question #3 : How To Find The Exponent Of Variables

Consider the expression $\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}$

Which is the greater quantity?

(a) The expression evaluated at x = 6, y = 10

(b) The expression evaluated at x = y = 8

Possible Answers:

(b) is greater

(a) is greater

It is impossible to tell from the information given

(a) and (b) are equal

Correct answer:

(b) is greater

Explanation:

Use the properties of powers to simplify the expression:

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}$

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

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

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

$= x^{6} \cdot x ^{3}\cdot y ^{3} \cdot y ^{6}$

$= x^{6+3} \cdot y ^{3+ 6}$

$= x^{9} y ^{9} = (xy)^{9}$

(a) If x = 6, y = 10 , then

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}= (xy)^{9} = ( 6 \cdot 10)^{9} = 60^{9}$

(b) If x = y = 8 , then

$\left (x^{2} y \right )^{3} \left (x y ^{2} \right )^{3}= (xy)^{9} = ( 8 \cdot 8)^{9} = 64^{9}$

(b) is greater.

Report an Error

Example Question #4 : How To Find The Exponent Of Variables

Which of the following expressions is equivalent to

$\left (100 + \sqrt{x} \right )^{2}$ ?

Possible Answers:

200+x

200-x

None of the other answers is correct.

10,000+x

10,000-x

Correct answer:

None of the other answers is correct.

Explanation:

Use the square of a binomial pattern as follows:

$\left (100 + \sqrt{x} \right )^{2}$

$= 100^{2}+ 2 \cdot 100 \cdot \sqrt {x} +(\sqrt{x})^{2}$

$= 10,000+ 2 00 \sqrt {x} +x$

This expression is not equivalent to any of the choices.

Report an Error

Example Question #3 : How To Find The Exponent Of Variables

$y = (x+ 3)^{2}$

y = 16z

z = w - 6

Express in terms of .

Possible Answers:

$w = 16 x^{2}+ 96x+ 138$

$w = 16 x^{2}+ 96x+ 105$

$w = 16 x^{2}+ 96x+ 150$

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{25}{16}$

Correct answer:

$w = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

Explanation:

$y = (x+ 3)^{2}$

$= x^{2}+ 2 \cdot 3 \cdot x + 3^{2}$

$= x^{2}+ 6 x + 9$

y = 16z , so

$16z = x^{2}+ 6 x + 9$

$\frac{16z}{16} = \frac{x^{2}+ 6 x + 9}{16}$

$z = \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16}$

z = w - 6 , so

w = z+ 6

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16}+6$

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{9}{16} + \frac{96}{16}$

$= \frac{1}{16}x^{2}+ \frac{3}{8}x+ \frac{105}{16}$

Report an Error

Example Question #11 : Variables And Exponents

$a ^{3} = -17$ . Which is the greater quantity?

(a) $a^{6}$

(b)

Possible Answers:

(b) is the greater quantity

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

(a) and (b) are equal

(a) is the greater quantity

Correct answer:

(a) is the greater quantity

Explanation:

By the Power of a Power Principle,

$(a ^{3} )^{2} = a ^{3 \cdot 2 } = a ^{6}$

Therefore,

$a ^{6} = (a ^{3} )^{2} = (-17)^{2} = 17^{2} = 17 \cdot 17 = 289$

It follows that $a ^{6} > - 34$

Report an Error

Example Question #911 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

is a real number such that $t ^{6} = 121$ . Which is the greater quantity?

(a) $t^{3}$

(b) 11

Possible Answers:

(b) is the greater quantity

(a) and (b) are equal

(a) is the greater quantity

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

Correct answer:

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

Explanation:

By the Power of a Power Principle,

$(t ^{3} ) ^{2} = t^{3 \cdot 2 } = t ^{6} = 121$

Therefore, $t ^{3}$ is a square root of 121, of which there are two - 11 and . Since it is possible for a third (odd-numbered) power of a real number to be positive or negative, we cannot eliminate either possibility, so either

t = 11

t = -11 < 11 .

Therefore, we cannot determine whether is less than 11 or equal to 11.

Report an Error

Example Question #912 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

$x ^{4} = 12$

$y^{2} = 8$

$(xy) ^{4} = ?$

Possible Answers:

192

768

Correct answer:

768

Explanation:

By the Power of a Product Principle,

$(xy) ^{4} = x^{4} \cdot y ^{4}$

Also, by the Power of a Power Principle

$y^{4} = y^{2 \cdot 2} = (y^{2})^{2}$

Therefore,

$(xy) ^{4} = x^{4} \cdot (y^{2})^{2} = 12 \cdot 8 ^{2} = 12 \cdot 64 = 768$

Report an Error

Example Question #12 : Variables And Exponents

is a negative number. Which is the greater quantity?

(a) $-16 a ^{4}$

(b) $(-2 a )^{4}$

Possible Answers:

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

(a) is the greater quantity

(a) and (b) are equal

(b) is the greater quantity

Correct answer:

(b) is the greater quantity

Explanation:

Any nonzero number raised to an even power, such as 4, is a positive number. Therefore,

$-16 a ^{4} = -16 \cdot a ^{4}$ is the product of a negative number and a positive number, and is therefore negative.

By the same reasoning, $(-2 a )^{4}$ is a positive number.

It follows that $(-2 a )^{4} > -16 a ^{4}$ .

Report an Error

Example Question #914 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

$t ^{3} = -24$

Evaluate $\left ( \frac{2}{t} \right ) ^{6}$ .

Possible Answers:

$-\frac{1}{72}$

$\frac{1}{72}$

$-\frac{1}{9}$

$\frac{1}{9}$

Correct answer:

$\frac{1}{9}$

Explanation:

By the Power of a Power Principle,

$(t ^{3} )^{2} = t ^{3 \cdot 2 } = t ^{6}$

By way of the Power of a Quotient Principle,

$\left ( \frac{2}{t} \right ) ^{6} = \frac{2^{6}}{t^{6} } = \frac{2^{6}}{(t^{3}) ^{2}} = \frac{64}{576} = \frac{64 \div 64 }{576 \div 64} = \frac{1}{9}$ .

Report an Error

Example Question #915 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

and are both real numbers.

$t^{4} = 121$

$u ^{4} = 81$

Evaluate (2t - 3u) (2t+3u ) .

Possible Answers:

Correct answer:

Explanation:

(2t - 3u) (2t+3u ) , as the product of a sum and a difference, can be rewritten using the difference of squares pattern:

(2t - 3u) (2t+3u )

$=(2t )^{2}-( 3u)^{2}$

$=2^{2} \cdot t ^{2}- 3^{2}\cdot u^{2}$

$=2^{2}t ^{2}- 3^{2}u^{2}$

$=4t ^{2}- 9u^{2}$

By the Power of a Power Principle,

$(t^{2} ) ^{2} = t^{2 \cdot 2 } = t^{4}$

Therefore, $t^{2}$ is a square root of $t^{4}$ - that is, a square root of 121. 121 has two square roots, and 121, but since is real, $t ^{2}$ must be the positive choice, 11. Similarly, $u^{2}$ is the positive square root of 81, which is 9.

The above expression can be evaluated as

$4t ^{2}- 9u^{2} = 4 (11) - 9 (9) = 44 - 81 = -37$ .

Report an Error

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.

View Tutors

Gregory
Certified Tutor

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

All ISEE Upper Level Quantitative Resources

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

Popular Subjects

Computer Science Tutors in Phoenix, MCAT Tutors in Dallas Fort Worth, SSAT Tutors in New York City, Math Tutors in Philadelphia, Spanish Tutors in Miami, MCAT Tutors in Seattle, SSAT Tutors in Washington DC, GRE Tutors in San Diego, Algebra Tutors in Washington DC, Computer Science Tutors in San Francisco-Bay Area

Popular Courses & Classes

Spanish Courses & Classes in Denver, GMAT Courses & Classes in Boston, SAT Courses & Classes in Denver, ISEE Courses & Classes in New York City, ACT Courses & Classes in Atlanta, MCAT Courses & Classes in Boston, ISEE Courses & Classes in Boston, GMAT Courses & Classes in San Francisco-Bay Area, GRE Courses & Classes in Miami, GMAT Courses & Classes in Miami

Popular Test Prep

ISEE Test Prep in New York City, SSAT Test Prep in Miami, SAT Test Prep in Denver, MCAT Test Prep in Dallas Fort Worth, GRE Test Prep in Denver, GMAT Test Prep in Boston, ACT Test Prep in Atlanta, ISEE Test Prep in Dallas Fort Worth, ACT Test Prep in Houston, SAT Test Prep in Washington DC

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 : Variables and Exponents

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

All ISEE Upper Level Quantitative Resources

Example Questions

Example Question #1 : How To Find The Exponent Of Variables

Example Question #3 : How To Find The Exponent Of Variables

Example Question #4 : How To Find The Exponent Of Variables

Example Question #3 : How To Find The Exponent Of Variables

Example Question #11 : Variables And Exponents

Example Question #911 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #912 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #12 : Variables And Exponents

Example Question #914 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

Example Question #915 : Isee Upper Level (Grades 9 12) Quantitative Reasoning

All ISEE Upper Level Quantitative Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: