Call Now to Set Up Tutoring:

(888) 888-0446

ACT Math : Expressions

Study concepts, example questions & explanations for ACT Math

Create An Account

All ACT Math Resources

14 Diagnostic Tests 767 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 3 4 5 6 7 8 9 10 11 Next →

Example Question #1 : How To Divide Rational Expressions

Simplify:

$\frac{(-5)\cdot(5x+30)}{(-x-6)\cdot(5x)}$

Possible Answers:

$\frac{5+x}{5-x}$

$\frac{25}{x-5}$

$\frac{5}{x}$

$-\frac{5}{x}$

$\frac{25}{x+5}$

Correct answer:

$\frac{5}{x}$

Explanation:

To begin, factor out the greatest common factor from each of the binomials to check for compatibility:

$\frac{(-5)\cdot(5x+30)}{(-x-6)\cdot(5x)} = \frac{(-5)\cdot5(x+6)}{-1(x+6)\cdot(5x)}$ Factor.

Next, eliminate the common factors and simplify.

$\frac{(-5)\cdot5(x+6)}{-1(x+6)\cdot(5x)} = \frac{-25}{-5x}$ Eliminate and simplify.

Lastly, clean up the fraction.

$\frac{-25}{-5x} = \frac{5}{x}$

Thus, our answer is $\frac{5}{x}$ .

Report an Error

Example Question #2 : How To Divide Rational Expressions

Simplify the expression:

$\frac{(3ab-9ax)}{(9a^2b-27a^2x)}$

Possible Answers:

$\frac{3a}{9a^2}$

$\frac{9a^2}{3a}$

$\textup{The expression is prime, and cannot be simplified.}$

$\frac{1}{3a}$

Correct answer:

$\frac{1}{3a}$

Explanation:

This problem looks intimidating, but can be solved with our standard tool; factoring the GCF out of each binomial.

$\frac{(3ab-9ax)}{(9a^2b-27a^2x)} = \frac{3a(b-3x)}{9a^2(b-3x)}$

Eliminate the obvious binomial:

$\frac{3a(b-3x)}{9a^2(b-3x)} = \frac{3a}{9a^2}$

But we're not through yet! Notice that we can further simplify this fraction:

$\frac{(3a)}{(9a^2)} = \frac{3a}{3a\cdot 3a}$

Eliminate another pair, and the solution is now obvious:

$\frac{3a}{3a\cdot 3a} = \frac{1}{3a}$

Thus, our answer is $\frac{1}{3a}$ .

Report an Error

Example Question #1 : How To Multiply Rational Expressions

Multiply $\left ( 3x-1\right )\cdot \left ( x+4\right )$

Possible Answers:

$3x^{2}+11x-4$

$3x^{2}+11x+4$

$3x^{2}+13x+4$

$3x^{2}-11x-4$

$3x^{2}+13x-4$

Correct answer:

$3x^{2}+11x-4$

Explanation:

When multiplying two binomials, it's best to remember the mnemonic FOIL, which stands for First, Outer, Inner, and Last. Going in the order of FOIL, we multiply the first term of each binomial, then the outer terms of the expression, then the inner terms of the expression, and finally the last term of each binomial. For this question we get: $F : 3x\cdot x = 3x^{2}, O : 3x\cdot 4 = 12x, I : -1\cdot x = -x, L: -1\cdot 4 = -4$ Now that we have 'FOILed' our expression, we simply add our results to get the final answer. $3x^{2} + 12x - x -4 = 3x^{2}+11x-4$

Report an Error

Example Question #1 : How To Multiply Rational Expressions

Give the product in simplified form: $\frac{9x}{y^{2}} \cdot \frac{x^{3}}{12} \cdot \frac{2}{xy}$

Possible Answers:

$\frac{2x^{3}y^{3}}{3}$

$\frac{2x^{3}}{3y^{3}}$

$\frac{3}{2x^{3}y^{3}}$

$\frac{3y^{3}}{2x^{3}}$

$\frac{3x^{3}}{2y^{3}}$

Correct answer:

$\frac{3x^{3}}{2y^{3}}$

Explanation:

Use cross-cancellation as follows:

$\frac{9x}{y^{2}} \cdot \frac{x^{3}}{12} \cdot \frac{2}{xy}$

$= \frac{3x}{y^{2}} \cdot \frac{x^{2}}{2} \cdot \frac{1}{ y}$

$= \frac{3x^{3}}{2y^{3}}$

Report an Error

Example Question #2 : How To Multiply Rational Expressions

Give the product in simplified form:

$\frac{4x}{7y} \cdot \frac{y^{2}- 7y}{x^{2}+ 4x}$

Possible Answers:

$\frac{ y - 1 }{ x+ 1 }$

$\frac{ 4y - 28 }{ 7x+ 28 }$

$\frac{ 4y - 1 }{ 7x+ 1 }$

$\frac{ y - 4 }{ x+ 7 }$

$\frac{ 4y }{ 7x }$

Correct answer:

$\frac{ 4y - 28 }{ 7x+ 28 }$

Explanation:

Factor the numerator and denominator in the second fraction, cross-cancel common factors, then multiply out.

$\frac{4x}{7y} \cdot \frac{y^{2}- 7y}{x^{2}+ 4x}$

$= \frac{4x}{7y} \cdot \frac{y \left (y - 7 \right ) }{x\left ( x+ 4 \right )}$

$= \frac{4 }{7 } \cdot \frac{ y - 7 }{ x+ 4 }$

$= \frac{ 4y - 28 }{ 7x+ 28 }$

Report an Error

Example Question #3 : How To Multiply Rational Expressions

Give the product in simplified form:

$\left (x^{2} - 9 \right ) \cdot \frac{x^{2}+6x+9}{x^{2}-6x+9}$

Possible Answers:

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

$\frac{ x^{3} +27}{x-3}$

$\frac{ x^{3} + 9 x^{2} +9x +27}{x-3}$

$\frac{ x^{3} + 9 x^{2} +27 x +27}{x-3}$

$x^{2} + 3x+9$

Correct answer:

$\frac{ x^{3} + 9 x^{2} +27 x +27}{x-3}$

Explanation:

Factor the first polynomial and the numerator and denominator in the second fraction, cross-cancel common factors, then multiply out.

$\left (x^{2} - 9 \right ) \cdot \frac{x^{2}+6x+9}{x^{2}-6x+9}$

$=\frac{x^{2} - 9 }{1} \cdot \frac{x^{2}+6x+9}{x^{2}-6x+9}$

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

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

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

x+3 can be cubed using the cube of a binomial pattern:

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

$= x^{3} + 9 x^{2} +27 x +27$

so the above rational expression can be rewritten as

$\frac{ x^{3} + 9 x^{2} +27 x +27}{x-3}$

Report an Error

Example Question #4 : How To Multiply Rational Expressions

Give the product in simplified form:

$\frac{x^{3}- 8}{x^{3}+8} \cdot \frac{x+2}{x-2}$

Possible Answers:

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

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

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

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

Correct answer:

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

Explanation:

Factor the numerator and denominator in the first fraction, cross-cancel common factors, then multiply out.

$\frac{x^{3}- 8}{x^{3}+8} \cdot \frac{x+2}{x-2}$

$=\frac{\left (x-2 \right )\left (x ^{2}+2x+4 \right )}{ \left (x+2 \right )\left (x ^{2}-2x+4 \right )} \cdot \frac{x+2}{x-2}$

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

Report an Error

Example Question #5 : How To Multiply Rational Expressions

Give the product in simplified form:

$\frac{x^{2}- 4x+4}{x^{2}-16} \cdot \frac{ x^{2}+ 8x + 16}{ x^{2}- 4 }$

Possible Answers:

$\frac{x^{2} +2x-8}{ x^{2} -2x-8 }$

$\frac{x^{2} -6x+8}{x^{2}+6x+8 }$

$\frac{x^{2}+6x+8 }{x^{2} -6x+8}$

$\frac{x^{2} -2x-8}{ x^{2} +2x-8 }$

Correct answer:

$\frac{x^{2} +2x-8}{ x^{2} -2x-8 }$

Explanation:

Factor the numerators and denominators, cross-cancel common factors, then multiply out. Both numerators are perfect square trinomials; both denominators are differences of squares.

$\frac{x^{2}- 4x+4}{x^{2}-16} \cdot \frac{ x^{2}+ 8x + 16}{ x^{2}- 4 }$

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

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

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

$=\frac{x^{2} +2x-8}{ x^{2} -2x-8 }$

Report an Error

Example Question #6 : How To Multiply Rational Expressions

Give the product in simplified form:

$\frac{x^{2}-x}{x+1} \cdot \frac{x^{2}+1}{ x^{2}-2x+1}$

Possible Answers:

$\frac{x^{3} }{ x^{2}-1 }$

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

$\frac{x^{2}+x}{ x -1 }$

$\frac{x^{2} }{ x -1 }$

Correct answer:

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

Explanation:

Factor the numerators and denominators, cross-cancel common factors, then multiply out. Note that the second numerator is prime.

$\frac{x^{2}-x}{x+1} \cdot \frac{x^{2}+1}{ x^{2}-2x+1}$

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

$=\frac{x }{x+1} \cdot \frac{x^{2}+1}{ x-1 }$

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

Report an Error

Example Question #21 : Expressions

Compute the following: $\frac{x+2}{2-x}+\frac{x-2}{2-x}$

Possible Answers:

$\frac{2x}{1-2x}$

$\frac{x}{2-x}$

$\frac{2x}{2-x}$

Correct answer:

$\frac{2x}{2-x}$

Explanation:

Notice that the denominator are the same for both terms. Since they are both the same, the fractions can be added. The denominator will not change in this problem.

$\frac{x+2}{2-x}+\frac{x-2}{2-x} = \frac{x+2+x-2}{2-x} = \frac{2x}{2-x}$

Report an Error

← Previous 1 2 3 4 5 6 7 8 9 10 11 Next →

View ACT Math Tutors

Janet
Certified Tutor

Union College, Bachelor of Science, Chemistry. Clarkson University, Master in Public Health, Health Services Administration.

View ACT Math Tutors

Christopher
Certified Tutor

Boston University, Bachelor in Arts, Mathematics Teacher Education. University of Massachusetts-Boston, Masters in Education,...

View ACT Math Tutors

Anna
Certified Tutor

Seattle University, Bachelor of Science, Mathematics.

All ACT Math Resources

14 Diagnostic Tests 767 Practice Tests Question of the Day Flashcards Learn by Concept

ACT Math Tutors in Top Cities:

Atlanta ACT Math Tutors, Austin ACT Math Tutors, Boston ACT Math Tutors, Chicago ACT Math Tutors, Dallas Fort Worth ACT Math Tutors, Denver ACT Math Tutors, Houston ACT Math Tutors, Kansas City ACT Math Tutors, Los Angeles ACT Math Tutors, Miami ACT Math Tutors, New York City ACT Math Tutors, Philadelphia ACT Math Tutors, Phoenix ACT Math Tutors, San Diego ACT Math Tutors, San Francisco-Bay Area ACT Math Tutors, Seattle ACT Math Tutors, St. Louis ACT Math Tutors, Tucson ACT Math Tutors, Washington DC ACT Math Tutors

Popular Courses & Classes

GMAT Courses & Classes in Washington DC, Spanish Courses & Classes in Houston, GMAT Courses & Classes in Dallas Fort Worth, Spanish Courses & Classes in Seattle, Spanish Courses & Classes in Chicago, Spanish Courses & Classes in Boston, SAT Courses & Classes in San Diego, LSAT Courses & Classes in Boston, MCAT Courses & Classes in San Francisco-Bay Area, Spanish Courses & Classes in San Francisco-Bay Area

Popular Test Prep

ISEE Test Prep in San Francisco-Bay Area, ACT Test Prep in Atlanta, GRE Test Prep in Seattle, ISEE Test Prep in Denver, ACT Test Prep in New York City, GMAT Test Prep in Dallas Fort Worth, ISEE Test Prep in Boston, ISEE Test Prep in Chicago, LSAT Test Prep in Houston, GRE 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

ACT Math : Expressions

Study concepts, example questions & explanations for ACT Math

All ACT Math Resources

Example Questions

Example Question #1 : How To Divide Rational Expressions

Example Question #2 : How To Divide Rational Expressions

Example Question #1 : How To Multiply Rational Expressions

Example Question #1 : How To Multiply Rational Expressions

Example Question #2 : How To Multiply Rational Expressions

Example Question #3 : How To Multiply Rational Expressions

Example Question #4 : How To Multiply Rational Expressions

Example Question #5 : How To Multiply Rational Expressions

Example Question #6 : How To Multiply Rational Expressions

Example Question #21 : Expressions

All ACT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: