We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

SAT Mathematics : Solving Problems with Roots

Study concepts, example questions & explanations for SAT Mathematics

Create An Account

All SAT Mathematics Resources

137 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 Next →

Example Question #11 : Solving Problems With Roots

$\frac{5x^{2}}{\sqrt[3]{x^{2}}}=$

Possible Answers:

$5x\sqrt[3]{x}$

$5x\sqrt[3]{x^{2}}$

$125x^{4}$

$5\sqrt[3]{x}$

Correct answer:

$5x\sqrt[3]{x}$

Explanation:

This problem emphasizes an important concept about mixing exponents and roots: when you see exponents and roots mixed, be ready to treat roots as fractional exponents. That way all of your terms are in a common form, and you can directly apply exponent rules to all terms.

Here, that means that you'll treat $\sqrt[3]{x^{2}}$ as $x^{\frac{2}{3}}$ . With that, the question then becomes:

$\frac{5x^{2}}{x^{\frac{2}{3}}}=$

Now you can use the exponent rule that $\frac{a^{b}}{a^{c}}=a^{b-c}$ (when you divide exponents of the same base, you subtract the exponents) to turn this into:

$5x^{(2-\frac{2}{3})}=5^{\frac{4}{3}}$

While you might now look to immediately convert back to the form of the answer choices (which each use radical signs to represent roots), $\sqrt[3]{x^{4}}$ is not an option. Which brings up another important point about mixing roots with exponents: generally when you're in exponent form you should stay there as long as possible, as the exponential form tends to provide you with more algebraic flexibility.

Here with $5x^{\frac{4}{3}}$ , you can pull out the $x^{1}$ (or just ) to break apart the mixed-number exponent. You can rephrase this as $5x(x^{\frac{1}{3}})$ , which then allows you to convert to an expression that looks like the answer choices. With the only fractional exponent left, $x^{\frac{1}{3}}$ , translating to $\sqrt[3]{x}$ , you can re-express the entire expression as $5x\sqrt[3]{x}$ .

Report an Error

Example Question #12 : Solving Problems With Roots

$\sqrt[3]{\frac{2}{3a}}=$

Possible Answers:

$\frac{\sqrt[3]{6a}}{3a}$

$\frac{\sqrt[3]{18a^{2}}}{3a}$

Correct answer:

$\frac{\sqrt[3]{18a^{2}}}{3a}$

Explanation:

This problem creates confusion for people because they are not comfortable rationalizing the cube root in the denominator. To see how to do this properly, first view the given expression like this:

$\frac{\sqrt[3]{2}}{\sqrt[3]{3a}}$

and then ask what you would need to multiply the denominator by (and thus the numerator so that you're effectively multiplying by ) to make the expression under the root sign a perfect cube in the denominator. In this case, you must multiply top and bottom by $\sqrt[3]{9a^{2}}$ as shown below:

$\frac{\sqrt[3]{2}}{\sqrt[3]{3a}}\times\frac{\sqrt[3]{9a^{2}}}{\sqrt[3]{27a^{3}}}=\frac{\sqrt[3]{18a^{2}}}{\sqrt[3]{27a^{3}}}=\frac{\sqrt[3]{18a^{2}}}{3a}$ .

Report an Error

Example Question #13 : Solving Problems With Roots

$\frac{\sqrt{6}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}}=$

Possible Answers:

$\frac{5}{6}$

$6\sqrt{3}-6\sqrt{2}$

$\frac{6}{5}$

$2\sqrt{3}+3\sqrt{2}$

Correct answer:

$6\sqrt{3}-6\sqrt{2}$

Explanation:

To simplify this multi-denominator expression, you should first multiply by one in the form $\frac{\sqrt{6}}{\sqrt{6}}$ to eliminate the two denominators in the bottom of the expression. This multiplication results in the following:

$\frac{\sqrt{6}}{\frac{1}{\sqrt{2}}+\frac{1}{\sqrt{3}}}(\frac{\sqrt{6}}{\sqrt{6}})=\frac{6}{\frac{\sqrt{6}}{\sqrt{2}}+\frac{\sqrt{6}}{\sqrt{3}}}=\frac{6}{\sqrt{3}+\sqrt{2}}$

Now you must multiply by one again to remove the two roots in the denominator. To do that you must recognize the difference of squares and multiply the top and the bottom by the conjugate of $\sqrt{3}+\sqrt{2}$ as shown below:

$\frac{6}{\sqrt{3}+\sqrt{2}}(\frac{\sqrt{3}-\sqrt{2}}{\sqrt{3}-\sqrt{2}})=\frac{6\sqrt{3}-6\sqrt{2}}{3-2}=6\sqrt{3}-6\sqrt{2}$

Therefore, the correct answer is $6\sqrt{3}-6\sqrt{2}$ .

Report an Error

Example Question #14 : Solving Problems With Roots

$(\sqrt{11}+\sqrt{11}+\sqrt{11})^{2}=$

Possible Answers:

121

Correct answer:

Explanation:

All of the exponent rules deal with multiplying, rather than adding, bases. In order to turn this into a multiplication question, we count apples (or chickens, or $y=\sqrt{(11)}$ s, or whatever). How many $\sqrt{(11)}$ ’s are there here? Three. This expression can be rewritten as $(3*\sqrt{(11)})^{2}$ . Now the exponent rules will apply; $(3*\sqrt{(11)})^{2}=3^{2}*(\sqrt{(11)})^{2}=9*11=99$ . The answer is .

Report an Error

Example Question #15 : Solving Problems With Roots

Simplify:

$\frac{x-1}{4(\sqrt{x}+1)}$

Possible Answers:

$\frac{\sqrt{x}-1}{4}$

$\frac{(x-1)\sqrt{x}-1}{4}$

$\frac{\sqrt{x}+1}{4}$

$\frac{(x-1)\sqrt{x}+1}{4}$

Correct answer:

$\frac{\sqrt{x}-1}{4}$

Explanation:

To solve this problem we must recognize that (x-1) can be broken down into $(\sqrt{x}-1)(\sqrt{x}+1)$

After breaking (x-1) into $(\sqrt{x}-1)(\sqrt{x}+1)$ we see that the $(\sqrt{x}+1)$ terms in numerator and denominator cancel out.

This leaves us with $\frac{\sqrt{x}-1}{4}$

Report an Error

← Previous 1 2 Next →

View Tutors

Sekou
Certified Tutor

University of New Haven, Bachelor in Arts, Mathematics.

View Tutors

Nicole
Certified Tutor

University of Wisconsin-Madison, Bachelor in Arts, Spanish.

View Tutors

Rodney
Certified Tutor

University of Washington, Master's/Graduate, Economics.

All SAT Mathematics Resources

137 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Los Angeles, Reading Tutors in Miami, Algebra Tutors in San Diego, Computer Science Tutors in Denver, Reading Tutors in Denver, Computer Science Tutors in Miami, Physics Tutors in Seattle, English Tutors in San Diego, SSAT Tutors in Atlanta, Reading Tutors in Atlanta

Popular Courses & Classes

ISEE Courses & Classes in Phoenix, GMAT Courses & Classes in San Francisco-Bay Area, LSAT Courses & Classes in Seattle, ISEE Courses & Classes in Washington DC, GRE Courses & Classes in Houston, GRE Courses & Classes in Chicago, LSAT Courses & Classes in Denver, GMAT Courses & Classes in Denver, Spanish Courses & Classes in Washington DC, ISEE Courses & Classes in San Francisco-Bay Area

Popular Test Prep

SAT Test Prep in New York City, SSAT Test Prep in Denver, ISEE Test Prep in San Francisco-Bay Area, ISEE Test Prep in Los Angeles, SAT Test Prep in San Diego, GRE Test Prep in Seattle, ISEE Test Prep in Washington DC, GRE Test Prep in Chicago, MCAT Test Prep in Washington DC, ISEE Test Prep in Atlanta

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

SAT Mathematics : Solving Problems with Roots

Study concepts, example questions & explanations for SAT Mathematics

All SAT Mathematics Resources

Example Questions

Example Question #11 : Solving Problems With Roots

Example Question #12 : Solving Problems With Roots

Example Question #13 : Solving Problems With Roots

Example Question #14 : Solving Problems With Roots

Example Question #15 : Solving Problems With Roots

All SAT Mathematics Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: