Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Rational Exponents

Study concepts, example questions & explanations for Precalculus

Create An Account

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Simplify Expressions With Rational Exponents

Simplify

$\frac{9^{\frac{2}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}$

Possible Answers:

$\frac{9}{4}$

$\frac{3}{4}$

$\frac{9}{2}$

$\frac{27}{2}$

Correct answer:

$\frac{3}{4}$

Explanation:

$\frac{9^{\frac{2}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}=\frac{(3^{2})^{\frac{2}{3}}\cdot3^{\frac{5}{3}}}{6^{2}}=\frac{3^{\frac{4}{3}}\cdot 3^{\frac{5}{3}}}{6^{2}}$

$=\frac{3^\frac{9}{3}}{6^2}=\frac{3^3}{6^2}=\frac{27}{36}=\frac{3}{4}$ .

Report an Error

Example Question #2 : Simplify Expressions With Rational Exponents

Simplify the expression.

$\frac{x^{3/2}y^{7/3}}{x^3 y^{2/9}}$

Possible Answers:

$x^{9/2}y^{23/9}$

$\frac{x^{3/2}}{y^{19/9}}$

$x^{3/2}y^{19/9}$

$\frac{y^{19/9}}{x^{3/2}}$

$\frac{1}{x^{9/2}y^{14/27}}$

Correct answer:

$\frac{y^{19/9}}{x^{3/2}}$

Explanation:

Using the properties of exponents, we can either choose to subtract the exponents of the corresponding bases or rewrite the expression using negative exponents as such:

$x^{3/2}x^{-3}y^{7/3}y^{-2/9}$

Here, we combine the terms with corresponding bases by adding the exponents together to get

$x^{-3/2}y^{19/9}$

Placing the x term (since it has a negative exponent) in the denominator will result in the correct answer. It can be shown that simply subtracting the exponents of corresponding bases will result in the same answer.

$\frac{y^{19/9}}{x^{3/2}}$

Report an Error

Example Question #3 : Rational Exponents

Simplify the expression $4^{2.5}$ .

Possible Answers:

None of the other answers.

Correct answer:

Explanation:

We proceed as follows

$4^{2.5}$

Write 2.5 as a fraction

$4^{5/2}$

The denominator of the fraction is a , so it becomes a square root.

$(^2\sqrt{4})^5$

Take the square root.

2^5

Raise to the 5th power.

Report an Error

Example Question #41 : Exponential And Logarithmic Functions

What is the value of $32^{\frac{3}{5}}$ ?

Possible Answers:

19.2

Correct answer:

Explanation:

Recall that when considering rational exponents, the denominator of the fraction tells us the "root" of the expression.

Thus in this case we are taking the fifth root of .

The fifth root of is , because $2\cdot2\cdot2\cdot2\cdot2 = 32$ .

Thus, we have reduced our expression to $2^3 = 2\cdot2\cdot2 = 8$ .

Report an Error

Example Question #5 : Simplify Expressions With Rational Exponents

Simplify the expression: $\frac{2x^3y^4}{6x^4y}$

Possible Answers:

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

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

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

$\frac{3}{xy^3}$

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

Correct answer:

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

Explanation:

$\frac{2x^3y^4}{6x^4y}$

Simplify the constants:

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

Subtract the "x" exponents:

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

$x^3-x^4=x^{3-4}=x^{-1}=\frac{1}{x}$ This is how the x moves to the denominator.

Finally subtract the "y" exponents:

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

Report an Error

Example Question #6 : Simplify Expressions With Rational Exponents

Solve:

$3x^{\frac{3}{4}} = x^{\frac{1}{2}}$

Possible Answers:

$x = \frac{3}{4}$

x = 3

x = 9

$x = \frac{1}{81}$

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

Correct answer:

$x = \frac{1}{81}$

Explanation:

To remove the fractional exponents, raise both sides to the second power and simplify:

$(3x^{\frac{3}{4}})^{2} = (x^{\frac{1}{2}})^{2}$

$9x^{\frac{3}{2}} = x$

$\frac{9x^{\frac{3}{2}}}{x} = \frac{x}{x}$

$9x^{\frac{1}{2}} = 1$

Now solve for :

$\frac{9x^{\frac{3}{2}}}{9} = \frac{1}{9}$

$x^{\frac{1}{2}} = \frac{1}{9}$

$(x^{\frac{1}{2}})^{2} = (\frac{1}{9})^{2}$

$x = \frac{1}{81}$

Report an Error

Example Question #1 : Simplify Expressions With Rational Exponents

Solve:

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

Possible Answers:

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

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

x = 6

x = 8

x = 1

Correct answer:

x = 8

Explanation:

To remove the rational exponent, cube both sides of the equation:

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

Now simplify both sides of the equation:

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

$x^{1} = 2^{3}$

x = 8

Report an Error

Example Question #1 : Simplify Expressions With Rational Exponents

Simplify and rewrite with positive exponents:

$\frac{2a^3b^{28}c^2d^2e^4f}{3a^2b^{27}cd^3e^5f^2}$

Possible Answers:

$\frac{2def}{3abc}$

$\frac{2def}{3a^6bc^3}$

$\frac{2a^6bc^3}{3def}$

$\frac{2abcd^{-1}e^{-1}f^{-1}}{3}$

$\frac{2abc}{3def}$

Correct answer:

$\frac{2abc}{3def}$

Explanation:

$\frac{2a^3b^{28}c^2d^2e^4f}{3a^2b^{27}cd^3e^5f^2}$

When dividing two exponents with the same base we subtract the exponents:

$\frac{2abcd^{-1}e^{-1}f^{-1}}{3}$

Negative exponents are dealt with based on the rule

$a^{-m}=\frac{1}{a^m}$ :

$\mathbf{\frac{2abc}{3def}}$

Report an Error

Example Question #3 : Simplify Expressions With Rational Exponents

Simplify the function:

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

Possible Answers:

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

$y = (x+2)^{1/4}$

y = (x+2)

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

Correct answer:

y = (x+2)

Explanation:

When an exponent is raised to the power of another exponent, just multiply the exponents together.

$y = (x^a)^b = x^{ab}$

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

Report an Error

Example Question #2 : Simplify Expressions With Rational Exponents

Simplify:

$\frac{8xy^2}{2x^4y}*x^2y^3$

Possible Answers:

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

4xy^4

None of the other answers.

$\frac{4y^6}{x}$

4xy^6

Correct answer:

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

Explanation:

Subtract the "x" exponents and the "y" exponents vertically. Then add the exponents horizontally if they have the same base (subtract the "x" and subtract the "y" ones). Finally move the negative exponent to the denominator.

$\frac{8xy^2}{2x^4y}*x^2y^3\rightarrow 4x^{-3}y*x^2y^3\rightarrow 4x^{-1}y^4=\frac{4y^4}{x}$

Report an Error

View Pre-Calculus Tutors

Chase
Certified Tutor

Western Governors University, Bachelor of Science, Mathematics.

View Pre-Calculus Tutors

Jody
Certified Tutor

The University of Texas at Austin, Bachelor of Science, Mathematics.

View Pre-Calculus Tutors

Margaret
Certified Tutor

All Precalculus Resources

12 Diagnostic Tests 380 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

LSAT Tutors in Phoenix, GMAT Tutors in Phoenix, French Tutors in New York City, SAT Tutors in Miami, Biology Tutors in Atlanta, Computer Science Tutors in Dallas Fort Worth, Biology Tutors in New York City, GMAT Tutors in San Diego, SAT Tutors in Seattle, Computer Science Tutors in Atlanta

Popular Courses & Classes

ISEE Courses & Classes in New York City, ACT Courses & Classes in Miami, SAT Courses & Classes in Houston, SSAT Courses & Classes in Chicago, Spanish Courses & Classes in New York City, GMAT Courses & Classes in Boston, GMAT Courses & Classes in Denver, GMAT Courses & Classes in Miami, GMAT Courses & Classes in San Francisco-Bay Area, SSAT Courses & Classes in San Diego

Popular Test Prep

LSAT Test Prep in Atlanta, GRE Test Prep in Houston, ACT Test Prep in San Francisco-Bay Area, MCAT Test Prep in San Diego, MCAT Test Prep in New York City, GRE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Philadelphia, LSAT Test Prep in Phoenix, GRE Test Prep in Washington DC, 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

Precalculus : Rational Exponents

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Simplify Expressions With Rational Exponents

Example Question #2 : Simplify Expressions With Rational Exponents

Example Question #3 : Rational Exponents

Example Question #41 : Exponential And Logarithmic Functions

Example Question #5 : Simplify Expressions With Rational Exponents

Example Question #6 : Simplify Expressions With Rational Exponents

Example Question #1 : Simplify Expressions With Rational Exponents

Example Question #1 : Simplify Expressions With Rational Exponents

Example Question #3 : Simplify Expressions With Rational Exponents

Example Question #2 : Simplify Expressions With Rational Exponents

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: