We are open Saturday and Sunday!

Call Now to Set Up Tutoring:

(888) 888-0446

GRE Subject Test: Math : Imaginary Roots of Negative Numbers

Study concepts, example questions & explanations for GRE Subject Test: Math

Create An Account

All GRE Subject Test: Math Resources

2 Diagnostic Tests 148 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

← Previous 1 2 Next →

Example Question #1 : Irrational Numbers

Evaluate:

$\left (2 + 3i \right )^{3}$

Possible Answers:

-4 6+ 9i

8 + 9i

8 - 27i

-62 + 63 i

-46 + 63 i

Correct answer:

-4 6+ 9i

Explanation:

We can set A = 2, B = 3i in the cube of a binomial pattern:

$\left ( A + B\right ) ^{3} = A ^{3} + 3 A^{2 }B + 3AB^{2} + B^{3}$

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

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

$= 8 + 36i + 54 i^{2} + 27 i^{3}$

= 8 + 36i + 54 (-1) + 27 (-i)

= 8 + 36i - 54 - 27 i

= -46 + 9i

Report an Error

Example Question #4 : Complex Imaginary Numbers

Simplify the following product:

{} (5+3i)(-2+i)

Possible Answers:

{} -10+3i

{}-7-i

{} -13-i

{} -13+11i

Correct answer:

{} -13-i

Explanation:

Multiply these complex numbers out in the typical way:

${}(5+3i)(-2+i) = -10+5i-6i+3i^2$

and recall that i^2=-1 by definition. Then, grouping like terms we get

{} (-10-3)+(5i-6i) = -13-i

which is our final answer.

Report an Error

Example Question #2 : Irrational Numbers

Simplify:

(5+8i)(6-2i)

Possible Answers:

7+16i

16+32i

46+38i

25+8i

Correct answer:

46+38i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(5+8i)(6-2i)=30-10i+48i-16i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

30-10i+48i-16i^2=30+38i-16(-1)=30+38i+16=46+38i

Report an Error

Example Question #11 : Number Theory

Simplify:

(2+5i)(-8-i)

Possible Answers:

42+11i

-11-42i

-18-5i

12+6i

Correct answer:

-11-42i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+5i)(-8-i)=-16-2i-40i-5i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

-16-2i-40i-5i^2=-16-42i-5(-1)=-16+5-42i=-11-42i

Report an Error

Example Question #12 : Number Theory

Simplify:

(2+6i)(5+3i)

Possible Answers:

36-8i

-36-8i

12-8i

-8+36i

Correct answer:

-8+36i

Explanation:

Start by using FOIL. Which means to multiply the first terms together then the outer terms followed by the inner terms and lastly, the last terms.

(2+6i)(5+3i)=10+6i+30i+18i^2

Remember that $i=\sqrt-1$ , so i^2=-1 .

Substitute in for i^2

10+6i+30i+18i^2=10+36i+18(-1)=10-18+36i=-8+36i

Report an Error

Example Question #1 : Equations With Complex Numbers

Solve for and :

$ai^{93}+bi^{35}+ai^{24}-bi^{86}=40+20i$

Possible Answers:

a=40, b=20

a=30, b=10

a=10, b=30

a=15, b=15

a=20, b=40

Correct answer:

a=30, b=10

Explanation:

Remember that

$i = \sqrt{-1}\\ i^2 = \sqrt{-1}^2 = -1\\ i^3 = i^2\cdot i = -1 \cdot i = -i\\ i^4 = i^2 \cdot i^2 = (-1)\cdot(-1) = 1\\ i^5 = i^4\cdot i = 1\cdot i = i = \sqrt{-1}\\ i^6 = i^4\cdot i^2 = 1\cdot i^2 = i^2 =-1$

So the powers of are cyclic. This means that when we try to figure out the value of an exponent of , we can ignore all the powers that are multiples of because they end up multiplying the end result by , and therefore do nothing.

This means that

$ai^{93} + bi^{35} + ai^{24} - bi^{86}\\ = ai^{92+1} + bi^{32+3} + ai^{24+0} - bi^{84+2}\\ = ai^{92}\cdot i^1 + bi^{32}\cdot i^3 + ai^{24}\cdot i^0 - bi^{84}\cdot i^2\\ =a(i^4)^{23}\cdot i^1 + b(i^4)^8\cdot i^3 + a(i^4)^6\cdot i^0 - b(i^4)^{21}\cdot i^2\\ = a (1^{23})\cdot i + b(1^8)\cdot i^3 + a(1^6) - b(1^{21})\cdot i^2$

Now, remembering the relationships of the exponents of , we can simplify this to:

$ai - bi + a - (-b) = ai-bi+a+b\\ = (a+b) + (a-b)i = 40+20i$

Because the elements on the left and right have to correspond (no mixing and matching!), we get the relationships:

a+b=40

a-b=10

No matter how you solve it, you get the values a=30 , b=10 .

Report an Error

Example Question #61 : Algebra

Simplify: $\sqrt{-243}$

Possible Answers:

$9i\sqrt{3}$

$-9i\sqrt{3}$

None of the Above

Correct answer:

$9i\sqrt{3}$

Explanation:

Step 1: Split the $\sqrt{-243}$ into $\sqrt{243}\cdot\sqrt{-1}$ .

Step 2: Recall that $\sqrt{-1}=i$ , so let's replace it.

We now have: $\sqrt{243}\cdot i$ .

Step 3: Simplify $\sqrt{243}$ . To do this, we look at the number on the inside.

$243=3\cdot3\cdot3\cdot3\cdot3$ .

Step 4: Take the factorization of 243 and take out any pairs of numbers. For any pair of numbers that we find, we only take of the numbers out.

We have a pair of 3's , so a is outside the radical.
We have another pair of 3's , so one more three is put outside the radical.

We need to multiply everything that we bring outside: 3*3=9

Step 5: The goes with the 9...

Step 6: The last after taking out pairs gets put back into a square root and is written right after the

It will look something like this: $9i\sqrt{3}$

Report an Error

Example Question #1 : Imaginary Numbers & Complex Functions

$Simplify: 5\sqrt{12i^{2}}$

Possible Answers:

$5i\sqrt{12}$

$10i\sqrt{}3$

$2i\sqrt{3}$

$5\sqrt{-12}$

Correct answer:

$10i\sqrt{}3$

Explanation:

There are two ways to simplify this problem:

Method 1:

$We\ know\ that\ i^2=-1,\ when\ we\ replace\ that\ we\ get:$

$5\sqrt{-12}$

$=5i\sqrt{12}(take\ the\ i\ out)$

$=5i\sqrt{4}\sqrt{3} (the\ perfect\ square\ 4\ will\ go\ into\ 12)$

$=10i{\sqrt{3}}\ reduce$

Method 2:

$We\ know\ that\ the\ square\ root\ of\ anything\ squared\ is\ just\ itself$

$\sqrt{i^2}=i$

$5\sqrt{12i^{2}}=5i\sqrt{12}$

$=5i\sqrt{4}\sqrt{3} (the\ perfect\ square\ 4\ will\ go\ into\ 12)$

$=10i{\sqrt{3}}\ reduce$

Report an Error

Example Question #1 : Imaginary Numbers & Complex Functions

$Simplify: \sqrt{-49}-\sqrt{-16}$

Possible Answers:

-11i

4-7i

7-4i

Correct answer:

Explanation:

$First\ simplify\ each\ radical:$

$\sqrt{-49}-\sqrt{-16}=7i-4i$

$Next\ combine\ like\ terms:$

7i-4i=3i

Report an Error

Example Question #1 : Imaginary Roots Of Negative Numbers

$Simplify: \sqrt{-72}$

Possible Answers:

$-6{\sqrt{2}}$

$-6i\sqrt{2}$

$6i{\sqrt{2}}$

$6{\sqrt{-2}}$

Correct answer:

$6i{\sqrt{2}}$

Explanation:

$When\ reducing\ the\ square\ root\ of\ a\ negative\ number\ -\ we\ must\ first\$

$i=\sqrt{-1}$

$\sqrt{-72}=\sqrt{-1}\sqrt{72}=i\sqrt{72}$

$Once\ we\ have\ taken\ out\ the\ i,\ then\ we\ can\ simply\ break\ down$ $the\ number\ under\ the\ radical\ by\ finding\ the\ largest\ perfect\ square\ that\ will go\ into\ it.$

$36\ will\ go\ into\ 72\ so\ we\ break\ this\ down\ to\ become:$

$i\sqrt{36}\sqrt{2}=6i\sqrt{2}$

Report an Error

← Previous 1 2 Next →

View Tutors

Gregory
Certified Tutor

Pennsylvania State University-Main Campus, Bachelor of Science, Chemical Engineering. Carnegie Mellon University, Doctor of P...

View Tutors

Alexia
Certified Tutor

University of Missouri-Columbia, Bachelor of Science, Biochemistry. Washington University in St Louis, Master of Science, Bio...

View Tutors

Mike
Certified Tutor

Concordia University-Ann Arbor, Master of Arts Teaching, Mathematics Teacher Education.

All GRE Subject Test: Math Resources

2 Diagnostic Tests 148 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Computer Science Tutors in New York City, MCAT Tutors in San Francisco-Bay Area, Reading Tutors in Los Angeles, GMAT Tutors in Los Angeles, Math Tutors in Boston, GRE Tutors in Phoenix, Biology Tutors in New York City, French Tutors in Los Angeles, Reading Tutors in Houston, Reading Tutors in Atlanta

Popular Courses & Classes

MCAT Courses & Classes in Phoenix, SSAT Courses & Classes in New York City, SSAT Courses & Classes in Miami, GRE Courses & Classes in Philadelphia, SAT Courses & Classes in Atlanta, GMAT Courses & Classes in Boston, MCAT Courses & Classes in Houston, SAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Dallas Fort Worth, MCAT Courses & Classes in Miami

Popular Test Prep

SAT Test Prep in Miami, GRE Test Prep in Phoenix, SAT Test Prep in Boston, SAT Test Prep in Chicago, GMAT Test Prep in Miami, ACT Test Prep in Washington DC, LSAT Test Prep in Houston, MCAT Test Prep in Phoenix, SSAT Test Prep in Washington DC, ISEE Test Prep in Philadelphia

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

GRE Subject Test: Math : Imaginary Roots of Negative Numbers

Study concepts, example questions & explanations for GRE Subject Test: Math

All GRE Subject Test: Math Resources

Example Questions

Example Question #1 : Irrational Numbers

Example Question #4 : Complex Imaginary Numbers

Example Question #2 : Irrational Numbers

Example Question #11 : Number Theory

Example Question #12 : Number Theory

Example Question #1 : Equations With Complex Numbers

Example Question #61 : Algebra

Example Question #1 : Imaginary Numbers & Complex Functions

Example Question #1 : Imaginary Numbers & Complex Functions

Example Question #1 : Imaginary Roots Of Negative Numbers

All GRE Subject Test: Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: