GMAT Math : Simplifying Algebraic Expressions

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

← Previous 1 2 3 4 5 6 Next →

Example Question #1331 : Gmat Quantitative Reasoning

Which of the following is equal to the expresion $\sqrt{16x} - \sqrt{25x} + \sqrt{98x}$ ?

Possible Answers:

$12 \sqrt{x}$

$7x\sqrt{2}-\sqrt{x}$

$13 \sqrt{x}$

$7\sqrt{2x}-\sqrt{x}$

$6 \sqrt{2x}$

Correct answer:

$7\sqrt{2x}-\sqrt{x}$

Explanation:

$\sqrt{16x} - \sqrt{25x} + \sqrt{98x}$

$=\sqrt{16} \cdot \sqrt{x} - \sqrt{25} \cdot \sqrt{x} + \sqrt{49} \cdot \sqrt{2x}$

$=4\sqrt{x} - 5 \sqrt{x} +7 \sqrt{2x}$

$=\left (4 - 5 \right ) \sqrt{x} +7 \sqrt{2x}$

$=- \sqrt{x} +7 \sqrt{2x}$

$= 7 \sqrt{2x} - \sqrt{x}$

Report an Error

Example Question #21 : Simplifying Algebraic Expressions

Simplify: $7x^{5} + 7x^{4}- 4 x - 4$

Possible Answers:

$14x^{9} - 4 x - 4$

$7x^{9} - 4 x - 4$

$14x^{9} - 8 x$

The polynomial cannot be simplified further.

$7x^{9} - 8 x$

Correct answer:

The polynomial cannot be simplified further.

Explanation:

None of the terms of the polynomial have the same degree - the exponents of in the terms are, in order, 5, 4, 1, and 0. Therefore, the four terms are all unlike terms, and none can be combined. The polynomial is already in simplified form.

Report an Error

Example Question #21 : Simplifying Algebraic Expressions

Simplify:

$\left ( x - 1\right ) + \left ( x-1\right )^{2} + \left ( x-1\right )^{3}$

Possible Answers:

$x^{3}+4x^{2} -4x- 1$

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

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

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

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

Correct answer:

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

Explanation:

$\left (x-1 \right )^{2}$

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

$= x^{2}- 2 x + 1$

$\left (x-1 \right )^{3}$

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

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

$\left ( x - 1\right ) + \left ( x-1\right )^{2} + \left ( x-1\right )^{3}$

$= \left ( x - 1\right ) + \left (x^{2}- 2 x + 1\right ) + \left (x^{3} -3 x^{2} + 3 x - 1\right )$

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

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

Report an Error

Example Question #1335 : Problem Solving Questions

Simplify: $(x-7)^{2} - (x+14)^{2}$

Possible Answers:

14x-147

-42x +245

-21x -147

-42x -147

14x+245

Correct answer:

-42x -147

Explanation:

Use the perfect square trinomial pattern to expand, then collect like terms:

$(x-7)^{2} - (x+14)^{2}$

$=(x^{2} - 2\cdot 7 \cdot x +7^{2} ) - (x^{2} + 2\cdot 14 \cdot x +14^{2} )$

$=(x^{2} -14 x +49) - (x^{2} + 28x +196 )$

$=x^{2} -14 x +49 - x^{2} - 28x -196$

$=x^{2}- x^{2} -14 x- 28x +49 -196$

= -42x -147

Report an Error

Example Question #1336 : Problem Solving Questions

Simplify: $(x+3)+(x+3)^{2}+ (x+3)^{3}$

Possible Answers:

$x^{3}+ 9 x^{2} + 27 x + 39$

$x^{3}+ 10x^{2} +34 x + 39$

$x^{3}+ 9 x^{2} + 27 x + 38$

$x^{3}+ 9x^{2} +34 x + 39$

$x^{3}+10 x^{2} + 27 x + 38$

Correct answer:

$x^{3}+ 10x^{2} +34 x + 39$

Explanation:

Expand $(x+3)^{2}$ and $(x+3)^{3}$ :

$(x+3)^{2}$

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

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

$(x+3)^{3}$

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

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

Add:

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

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

$\underline{ \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; \; x\; \; \; +3}$

$x^{3}+ 10x^{2} +34 x + 39$

Report an Error

Example Question #1337 : Problem Solving Questions

Which of the following is equal to $\sqrt{6x-6} \cdot \sqrt{15x-15} \cdot \sqrt{35x-35}$ ?

Possible Answers:

$15x\sqrt{14x-14}-15\sqrt{14x-14}$

$14x\sqrt{15x-15}-14\sqrt{15x-15}$

$210 x\sqrt{x-1}-210 \sqrt{x-1}$

210x-210

$210x^{2}-420x+210$

Correct answer:

$15x\sqrt{14x-14}-15\sqrt{14x-14}$

Explanation:

Factor all radicands, multiply, then simplify:

$\sqrt{6x-6} \cdot \sqrt{15x-15} \cdot \sqrt{35x-35}$

$= \sqrt{2 \cdot 3\cdot (x-1)} \cdot \sqrt{3 \cdot 5 \cdot (x-1)} \cdot \sqrt{5\cdot 7\cdot (x-1)}$

$= \sqrt{2 \cdot 3\cdot3 \cdot 5 \cdot 5\cdot 7\cdot(x-1)\cdot (x-1)\cdot (x-1)}$

$=\sqrt{3\cdot3} \cdot \sqrt{5 \cdot 5} \cdot \sqrt{(x-1)\cdot (x-1)} \cdot \sqrt{2 \cdot 7 \cdot (x-1)}$

$=3 \cdot5 \cdot (x-1) \cdot \sqrt{14x-14}$

$= (15x-15) \cdot \sqrt{14x-14}$

$= 15x\sqrt{14x-14}-15\sqrt{14x-14}$

Report an Error

Example Question #21 : Simplifying Algebraic Expressions

Simplify the expression:

(x-1)(x+1)+(x-2)(x+2)+(x-3)(x+3)

Possible Answers:

$3x^{2}-11$

$3x^{2}-6x -1 4$

$3x^{2}-12x -1 4$

$3x^{2}$

$3x^{2}-1 4$

Correct answer:

$3x^{2}-1 4$

Explanation:

Each of the three products added is a product of the sum and the difference of the same two expressions, so each can be simplified using that pattern:

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

$(x-2)(x+2)= x^{2}- 2^{2} = x^{2}-4$

$(x-3)(x+3)= x^{2}- 3^{2} = x^{2}-9$

Add:

(x-1)(x+1)+(x-2)(x+2)+(x-3)(x+3)

$=(x^{2}-1) +(x^{2}-4) +(x^{2}-9)$

$=x^{2}+x^{2}+x^{2}-1 -4 -9$

$=3x^{2}-1 4$

Report an Error

Example Question #21 : Simplifying Algebraic Expressions

Simplify: $\sqrt[3]{ \sqrt[4]{x ^{20}} }$

You may assume is positive.

Possible Answers:

$x^{8}$

$x^{2} \sqrt[7]{x^{6}}$

$x^{13}$

None of the other choices gives the correct answer.

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

Correct answer:

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

Explanation:

$\sqrt[m]{x^{n}}$ is equal to $x^{\frac{m}{n}}$ , so $\sqrt[4]{x ^{20}}$ can be rewritten as $x^{\frac{20}{4}}$ or $x^{5}$ .

The original expression can be rewritten as $\sqrt[3]{ x ^{5}}$ , which itself can be rewritten by noting that 5 divided by 3 yields quotient 1 and remainder 2. Therefore, the simplified form is $x^{1} \sqrt[3]{x^{2}}$ , or $x \sqrt[3]{x^{2}}$ .

Report an Error

Example Question #1340 : Problem Solving Questions

Simplify the expression:

$\sqrt[5]{x^{-8}y^{7}}$

Possible Answers:

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

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

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

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

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

Correct answer:

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

Explanation:

First, rewrite the radicand with positive exponents, then take the fifth root of the numerator and the denominator:

$\sqrt[5]{x^{-8}y^{7}} = \sqrt[5]{\frac{y^{7}}{x^{8}}} = \frac{ \sqrt[5]{ y^{7}} }{ \sqrt[5]{ x^{8}}}$

To simplify each expression, divide the exponent by the index 5, and note its quotient and remainder. 7 divided by 5 is 1 with remainder 2; 8 divided by 5 is 1 with remainder 3. Subsequently,

$\frac{ \sqrt[5]{ y^{7}} }{ \sqrt[5]{ x^{8}}} =\frac{y \sqrt[5]{ y^{2}} }{x \sqrt[5]{ x^{3}}}$

The denominator can be rationalized by multiplying both halves by $\sqrt[5]{ x^{2}}$ :

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

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

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

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

Report an Error

Example Question #21 : Simplifying Algebraic Expressions

Simplify:

$(x+5)(x^{2}-5x+25) - (x-5)(x^{2}+5x+25)$

Possible Answers:

$25x^{2}+250$

The correct answer is not among the other choices.

$2x^{3} +10x$

$5x^{2}+250$

$2x^{3} +50x$

Correct answer:

The correct answer is not among the other choices.

Explanation:

The problem is easier if you recognize the two products $(x+5)(x^{2}-5x+25)$ and $(x-5)(x^{2}+5x+25)$ as the factorizations of the sum and difference of two cubes, respectively:

$(x+5)(x^{2}-5x+25) = (x+5)(x^{2}-5 \cdot x +5^{2}) = x^{3}+5^{3}= x^{3}+125$

$(x-5)(x^{2}+5x+25) = (x-5)(x^{2}+5 \cdot x +5^{2}) = x^{3}-5^{3}= x^{3}-125$

Therefore,

$(x+5)(x^{2}-5x+25) - (x-5)(x^{2}+5x+25)$

$= (x^{3} +125) - (x^{3} -125)$

$= x^{3} +125 - x^{3} +125$

= 250

This is not one of the responses.

Report an Error

← Previous 1 2 3 4 5 6 Next →

All GMAT Math Resources

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

Popular Subjects

Statistics Tutors in Houston, SSAT Tutors in Philadelphia, English Tutors in Phoenix, GMAT Tutors in Denver, Biology Tutors in San Diego, Physics Tutors in Houston, Algebra Tutors in San Francisco-Bay Area, SSAT Tutors in Seattle, SSAT Tutors in Atlanta, Physics Tutors in Los Angeles

Popular Courses & Classes

MCAT Courses & Classes in Los Angeles, GRE Courses & Classes in Atlanta, GMAT Courses & Classes in Phoenix, SAT Courses & Classes in Washington DC, SSAT Courses & Classes in San Diego, LSAT Courses & Classes in Dallas Fort Worth, GMAT Courses & Classes in Philadelphia, ACT Courses & Classes in Phoenix, MCAT Courses & Classes in Dallas Fort Worth, ISEE Courses & Classes in Los Angeles

Popular Test Prep

LSAT Test Prep in Los Angeles, SSAT Test Prep in Boston, ISEE Test Prep in Los Angeles, LSAT Test Prep in Atlanta, GMAT Test Prep in Phoenix, LSAT Test Prep in Houston, ISEE Test Prep in New York City, GRE Test Prep in Phoenix, SSAT Test Prep in New York City, GRE Test Prep in Houston

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 : Simplifying Algebraic Expressions

Study concepts, example questions & explanations for GMAT Math

All GMAT Math Resources

Example Questions

Example Question #1331 : Gmat Quantitative Reasoning

Example Question #21 : Simplifying Algebraic Expressions

Example Question #21 : Simplifying Algebraic Expressions

Example Question #1335 : Problem Solving Questions

Example Question #1336 : Problem Solving Questions

Example Question #1337 : Problem Solving Questions

Example Question #21 : Simplifying Algebraic Expressions

Example Question #21 : Simplifying Algebraic Expressions

Example Question #1340 : Problem Solving Questions

Example Question #21 : Simplifying Algebraic Expressions

All GMAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details