GMAT Math : Powers & Roots of Numbers

Study concepts, example questions & explanations for GMAT Math

Create An Account

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1581 : Gmat Quantitative Reasoning

Solve: $\frac{(0.5)^{6}}{(0.5)^{9}}$ $\displaystyle \frac{(0.5)^{6}}{(0.5)^{9}}$

Possible Answers:

$\dpi{100} \small 8$ $\displaystyle \dpi{100} \small 8$

$\dpi{100} \small 0.125$ $\displaystyle \dpi{100} \small 0.125$

$\dpi{100} \small 125$ $\displaystyle \dpi{100} \small 125$

$\dpi{100} \small 4$ $\displaystyle \dpi{100} \small 4$

$\dpi{100} \small 0.25$ $\displaystyle \dpi{100} \small 0.25$

Correct answer:

$\dpi{100} \small 8$ $\displaystyle \dpi{100} \small 8$

Explanation:

Solve $\dpi{100} \small : \frac{0.5^{6}}{0.5^{9}} = 0.5^{6-9}= 0.5^{-3}=\frac{1}{0.5^{3}}=\frac{1}{0.125} = 8$ $\displaystyle \dpi{100} \small : \frac{0.5^{6}}{0.5^{9}} = 0.5^{6-9}= 0.5^{-3}=\frac{1}{0.5^{3}}=\frac{1}{0.125} = 8$

Report an Error

Example Question #1581 : Problem Solving Questions

$\frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} =$ $\displaystyle \frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} =$

Possible Answers:

$\frac{yz}{x}$ $\displaystyle \frac{yz}{x}$

$\frac{y^{2}z}{x}$ $\displaystyle \frac{y^{2}z}{x}$

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

$\frac{yz}{x^{2}}$ $\displaystyle \frac{yz}{x^{2}}$

$\dpi{100} \small cannot\ be\ simplified$ $\displaystyle \dpi{100} \small cannot\ be\ simplified$

Correct answer:

$\frac{y^{2}z}{x}$ $\displaystyle \frac{y^{2}z}{x}$

Explanation:

$\frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} = \frac{y^{3-1}z^{4-3}}{x^{3-2}} = \frac{y^{2}z}{x}$ $\displaystyle \frac{x^{2}y^{3}z^{4}}{x^{3}yz^{3}} = \frac{y^{3-1}z^{4-3}}{x^{3-2}} = \frac{y^{2}z}{x}$

Report an Error

Example Question #3 : Understanding Powers And Roots

Solve: $\small (\sqrt{5}+\sqrt{4})^2$ $\displaystyle \small (\sqrt{5}+\sqrt{4})^2$

Possible Answers:

$\small 9$ $\displaystyle \small 9$

$\small 20$ $\displaystyle \small 20$

$\small 9+4\sqrt5$ $\displaystyle \small 9+4\sqrt5$

$\small 20+4\sqrt5$ $\displaystyle \small 20+4\sqrt5$

Correct answer:

$\small 9+4\sqrt5$ $\displaystyle \small 9+4\sqrt5$

Explanation:

First, FOIL:

$\small (\sqrt5+\sqrt4)^2=(\sqrt5+\sqrt4)(\sqrt5+\sqrt4)=5+\sqrt20+\sqrt20+4$ $\displaystyle \small (\sqrt5+\sqrt4)^2=(\sqrt5+\sqrt4)(\sqrt5+\sqrt4)=5+\sqrt20+\sqrt20+4$

$\small =9+2\sqrt20$ $\displaystyle \small =9+2\sqrt20$

Factor out $\small \sqrt4$ $\displaystyle \small \sqrt4$

$\small =9+4\sqrt5$ $\displaystyle \small =9+4\sqrt5$

Report an Error

Example Question #4 : Understanding Powers And Roots

Solve: $\small \frac{10^9-10^7}{99}$ $\displaystyle \small \frac{10^9-10^7}{99}$

Possible Answers:

$\small 10^7$ $\displaystyle \small 10^7$

$\small \frac{10^7}{99}$ $\displaystyle \small \frac{10^7}{99}$

$\small \frac{10}{99}$ $\displaystyle \small \frac{10}{99}$

$\small \frac{100}{99}$ $\displaystyle \small \frac{100}{99}$

Correct answer:

$\small 10^7$ $\displaystyle \small 10^7$

Explanation:

First factor. $\small \frac{10^9-10^7}{99}\ =\ \frac{(10^7)(10^2-1)}{99}$ $\displaystyle \small \frac{10^9-10^7}{99}\ =\ \frac{(10^7)(10^2-1)}{99}$

Simplify. $\small \frac{(10^7)(10^2-1)}{99}\ =\ \frac{(10^7)(100-1)}{99}\ =\ \frac{(10^7)(99)}{99}\ =\ 10^7$ $\displaystyle \small \frac{(10^7)(10^2-1)}{99}\ =\ \frac{(10^7)(100-1)}{99}\ =\ \frac{(10^7)(99)}{99}\ =\ 10^7$

Report an Error

Example Question #1 : Powers & Roots Of Numbers

If $\small x^2=8$ $\displaystyle \small x^2=8$, what is $\small x^4?$ $\displaystyle \small x^4?$

Possible Answers:

$\displaystyle 16$

$\displaystyle 32$

$\displaystyle 64$

124 $\displaystyle 124$

Correct answer:

$\displaystyle 64$

Explanation:

$\small x^4=(x^2)^2=(8)^2=64$ $\displaystyle \small x^4=(x^2)^2=(8)^2=64$

Report an Error

Example Question #6 : Understanding Powers And Roots

Evaluate: $11^{100} - 11^{99}$ $\displaystyle 11^{100} - 11^{99}$

Possible Answers:

$10 \cdot 11^{99}$ $\displaystyle 10 \cdot 11^{99}$

$11^{98}$ $\displaystyle 11^{98}$

$10 \cdot 11^{98}$ $\displaystyle 10 \cdot 11^{98}$

$\displaystyle 11$

$11^{99}$ $\displaystyle 11^{99}$

Correct answer:

$10 \cdot 11^{99}$ $\displaystyle 10 \cdot 11^{99}$

Explanation:

$11^{100} - 11^{99} = 11 \cdot 11^{99} - 1\cdot 11^{99} = (11-1) \cdot 11^{99} = 10 \cdot 11^{99}$ $\displaystyle 11^{100} - 11^{99} = 11 \cdot 11^{99} - 1\cdot 11^{99} = (11-1) \cdot 11^{99} = 10 \cdot 11^{99}$

Report an Error

Example Question #1 : Powers & Roots Of Numbers

Simplify this expression as much as possible:

$\sqrt{9x}+\sqrt{12x}+\sqrt{27x}$ $\displaystyle \sqrt{9x}+\sqrt{12x}+\sqrt{27x}$

Possible Answers:

$3\sqrt{x}+5\sqrt{3x}$ $\displaystyle 3\sqrt{x}+5\sqrt{3x}$

$\sqrt{3x}+\sqrt{15x}$ $\displaystyle \sqrt{3x}+\sqrt{15x}$

$8\sqrt{3x}$ $\displaystyle 8\sqrt{3x}$

$8\sqrt{x}$ $\displaystyle 8\sqrt{x}$

The expression cannot be simplified further

Correct answer:

$3\sqrt{x}+5\sqrt{3x}$ $\displaystyle 3\sqrt{x}+5\sqrt{3x}$

Explanation:

$\sqrt{9x}+\sqrt{12x}+\sqrt{27x}$ $\displaystyle \sqrt{9x}+\sqrt{12x}+\sqrt{27x}$

$=\sqrt{9}\cdot \sqrt{x}+\sqrt{4}\cdot \sqrt{3x}+\sqrt{9}\cdot \sqrt{3x}$ $\displaystyle =\sqrt{9}\cdot \sqrt{x}+\sqrt{4}\cdot \sqrt{3x}+\sqrt{9}\cdot \sqrt{3x}$

$=3 \sqrt{x}+2 \sqrt{3x}+3 \sqrt{3x}$ $\displaystyle =3 \sqrt{x}+2 \sqrt{3x}+3 \sqrt{3x}$

$=3 \sqrt{x}+\left (2 +3 \right )\sqrt{3x}$ $\displaystyle =3 \sqrt{x}+\left (2 +3 \right )\sqrt{3x}$

$=3 \sqrt{x}+5\sqrt{3x}$ $\displaystyle =3 \sqrt{x}+5\sqrt{3x}$

Report an Error

Example Question #7 : Understanding Powers And Roots

If the side length of a cube is tripled, how does the volume of the cube change?

Possible Answers:

Volume becomes 27 times larger.

The volume doesn't change.

Volume becomes 3 times larger.

Not enough informatin is given.

Volume becomes 9 times larger.

Correct answer:

Volume becomes 27 times larger.

Explanation:

The equation for the volume of a cube is $L^{3}$ $\displaystyle L^{3}$. If the length is tripled, it becomes $(3L)^{3}$ $\displaystyle (3L)^{3}$, and $3^{3}=27$ $\displaystyle 3^{3}=27$, so the volume increases by 27 times the original size.

Report an Error

Example Question #5 : Understanding Powers And Roots

Simplify $\sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$ $\displaystyle \sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$

Possible Answers:

$\displaystyle 2$

$\displaystyle 4$

$\displaystyle 64$

$\displaystyle 32$

$\displaystyle 8$

Correct answer:

$\displaystyle 2$

Explanation:

This can either be done by brute force (slow) or by recognizing the properties of roots and exponents (fast). Roots are simply fractional exponents: $\sqrt{x}=x^{\frac{1}{2}}$ $\displaystyle \sqrt{x}=x^{\frac{1}{2}}$, $\sqrt[3]{x}=x^{\frac{1}{3}}$ $\displaystyle \sqrt[3]{x}=x^{\frac{1}{3}}$, etc. so they can be done in any order.

So we see a cube root, we can immediately cancel that with the exponent of 3. taking us from here: $\sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$ $\displaystyle \sqrt{\sqrt[4]{\sqrt[3]{4^{2}}^{2}}^{3}}$ to $\sqrt{\sqrt[4]{(4^2)^2}}$ $\displaystyle \sqrt{\sqrt[4]{(4^2)^2}}$. We now simplify $\displaystyle (4^2)^2 = 4^4$ to get $\sqrt{\sqrt[4]{(4^4)}} = \sqrt4 = 2$ $\displaystyle \sqrt{\sqrt[4]{(4^4)}} = \sqrt4 = 2$

Report an Error

Example Question #10 : Understanding Powers And Roots

In the sequence 1, 3, 9, 27, 81, … , each term after the first is three times the previous term. What is the sum of the 9th and 10th terms in the sequence?

Possible Answers:

$(3)^{10}$ $\displaystyle (3)^{10}$

$4(3)^{8}$ $\displaystyle 4(3)^{8}$

$4(3)^{9}$ $\displaystyle 4(3)^{9}$

$6(3)^{9}$ $\displaystyle 6(3)^{9}$

$6(3)^{8}$ $\displaystyle 6(3)^{8}$

Correct answer:

$4(3)^{8}$ $\displaystyle 4(3)^{8}$

Explanation:

We can rewrite the sequence as $(3)^{0}$ $\displaystyle (3)^{0}$, $(3)^{1}$ $\displaystyle (3)^{1}$, $(3)^{2}$ $\displaystyle (3)^{2}$, $(3)^{3}$ $\displaystyle (3)^{3}$, $(3)^{4}$ $\displaystyle (3)^{4}$, … ,

and we can see that the 9th term in the sequence is $(3)^{8}$ $\displaystyle (3)^{8}$ and the 10th term in the sequence is $(3)^{9}$ $\displaystyle (3)^{9}$. Therefore, the sum of the 9th and 10th terms would be

$3^{8}+3^{9}=3^{8}\cdot \left ( 1+3 \right )=4\cdot 3^{8}$ $\displaystyle 3^{8}+3^{9}=3^{8}\cdot \left ( 1+3 \right )=4\cdot 3^{8}$

Report an Error

All GMAT Math Resources

22 Diagnostic Tests 693 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

SSAT Tutors in Dallas Fort Worth, SSAT Tutors in Los Angeles, ISEE Tutors in Phoenix, Statistics Tutors in Seattle, English Tutors in San Francisco-Bay Area, Calculus Tutors in Seattle, Calculus Tutors in Philadelphia, Physics Tutors in Houston, French Tutors in Atlanta, SAT Tutors in Chicago

Popular Courses & Classes

LSAT Courses & Classes in Houston, MCAT Courses & Classes in Washington DC, LSAT Courses & Classes in San Francisco-Bay Area, MCAT Courses & Classes in San Diego, Spanish Courses & Classes in New York City, SAT Courses & Classes in Miami, GMAT Courses & Classes in Washington DC, Spanish Courses & Classes in Philadelphia, ISEE Courses & Classes in Atlanta, Spanish Courses & Classes in San Francisco-Bay Area

Popular Test Prep

MCAT Test Prep in Miami, GRE Test Prep in Philadelphia, GMAT Test Prep in Washington DC, GMAT Test Prep in Philadelphia, GMAT Test Prep in Los Angeles, MCAT Test Prep in New York City, SAT Test Prep in Houston, LSAT Test Prep in San Diego, GRE Test Prep in Miami, MCAT Test Prep in Chicago

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)

Email address:
Your name:
Feedback:

GMAT Math : Powers & Roots of Numbers

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1581 : Gmat Quantitative Reasoning

Example Question #1581 : Problem Solving Questions

Example Question #3 : Understanding Powers And Roots

Example Question #4 : Understanding Powers And Roots

Example Question #1 : Powers & Roots Of Numbers

Example Question #6 : Understanding Powers And Roots

Example Question #1 : Powers & Roots Of Numbers

Example Question #7 : Understanding Powers And Roots

Example Question #5 : Understanding Powers And Roots

Example Question #10 : Understanding Powers And Roots

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details