Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Find the Quotient of Complex Numbers

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

← Previous 1 2 Next →

Example Question #1 : Find The Quotient Of Complex Numbers

Let $a=(1+i)^{n}$ , b=(1-i)^n . Find a simple form of $\frac{a}{b}$ .

Possible Answers:

$2i^{n+1}$

$4i^{n+1}$

$-i^{n}$

$-i^{n+1}$

$i^{n}$

Correct answer:

$i^{n}$

Explanation:

We remark first that:

$\frac{1+i}{1-i}=\frac{1+i}{1-i} \frac{1+i}{1+i}$ $=\frac{(1+i)^2}{2}$

and we know that :

(1+i)^2=1+2i-i^2=2i .

This means that:

$\frac{1+i}{1-i}=\frac{2i}{2}=i$

$\left(\frac{1+i}{1-i}\right)^n=i^{n}$

Report an Error

Example Question #1 : Find The Quotient Of Complex Numbers

What is $\frac{(i^2)}{1-i}$ ?

Possible Answers:

$1+\frac{i}{i^2}$

i^2

$-1+\frac{1}{1-i}$

-2i

Correct answer:

$-1+\frac{1}{1-i}$

Explanation:

Since i^=-1 ,

the problem becomes,

$\frac{-1}{1-i}$

$=-1+\frac{1}{1-i}$

Report an Error

Example Question #1 : Find The Quotient Of Complex Numbers

Write

$\small \frac{1+2i}{1+i}$

in the form $\small a+bi$ for some real numbers $\small a$ and $\small b$ .

Possible Answers:

$\small 1+2i$

$\small \small \frac{3}{\sqrt{2}}+\frac{1}{\sqrt{2}}i$

$\small \small \frac{3}{2}-\frac{1}{2}i$

$\small \frac{3}{2}+\frac{1}{2}i$

Correct answer:

$\small \frac{3}{2}+\frac{1}{2}i$

Explanation:

The correct answer is

$\small \frac{3}{2}+\frac{1}{2}i$

Using simple algebra and multiplying the expression by the complex conjugate of the denominator, we get:

$\small \small \frac{1+2i}{1+i}=\frac{1+2i}{1+i}\cdot\frac{1-i}{1-i}= (1-i)\frac{1+2i}{2}$

$\small =\frac{1-i+2i-2i^2}{2}=\frac{1+i-(-2)}{2}=\frac{3+i}{2}=\frac{3}{2}+\frac{1}{2}i$

Report an Error

Example Question #4 : Find The Quotient Of Complex Numbers

Divide:

$\frac{4+5i}{5-i}$

Possible Answers:

$\frac{29}{26}+\frac{15i}{26}$

$\frac{15}{26}+\frac{29i}{26}$

$\frac{15}{26}-\frac{29i}{26}$

$\frac{5}{8}+\frac{29i}{24}$

Correct answer:

$\frac{15}{26}+\frac{29i}{26}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 5+i .

$\left(\frac{4+5i}{5-i}\right)\left(\frac{5+i}{5+i}\right)$

Now, distribute and simplify.

$\left(\frac{4+5i}{5-i}\right)\left(\frac{5+i}{5+i}\right)=\frac{20+29i+5i^2}{25-i^2}$

Recall that i^2=-1

$\frac{20+29i+5(-1)}{25-(-1)}=\frac{15+29i}{26}=\frac{15}{26}+\frac{29i}{26}$

Report an Error

Example Question #5 : Find The Quotient Of Complex Numbers

Divide:

$\frac{7+i}{2-i}$

Possible Answers:

$\frac{13}{5}+\frac{9i}{5}$

$\frac{13}{5}-\frac{9i}{5}$

$\frac{15}{2}+3i$

$\frac{5}{13}+\frac{5}{9i}$

Correct answer:

$\frac{13}{5}+\frac{9i}{5}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 2+i .

$\left(\frac{7+i}{2-i}\right)\left(\frac{2+i}{2+i}\right)$

Now, distribute and simplify.

$\left(\frac{7+i}{2-i}\right)\left(\frac{2+i}{2+i}\right)=\frac{14+9i+i^2}{4-i^2}$

Recall that i^2=-1

$\frac{14+9i+i^2}{4-i^2}=\frac{14+9i+(-1)}{4-(-1)}=\frac{13+9i}{5}=\frac{13}{5}+\frac{9i}{5}$

Report an Error

Example Question #1 : Find The Quotient Of Complex Numbers

Divide.

$\frac{2-9i}{3+i}$

Possible Answers:

$\frac{3}{4}+\frac{29i}{4}$

$-\frac{3}{10}-\frac{29i}{10}$

$-\frac{10}{3}-\frac{10}{29i}$

$\frac{3}{10}+\frac{29i}{10}$

Correct answer:

$-\frac{3}{10}-\frac{29i}{10}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3-i .

$\left(\frac{2-9i}{3+i}\right)\left(\frac{3-i}{3-i}\right)$

Now, distribute and simplify.

$\left(\frac{2-9i}{3+i}\right)\left(\frac{3-i}{3-i}\right)=\frac{6-29i+9i^2}{9-i^2}$

Recall that i^2=-1

$\frac{6-29i+9i^2}{9-i^2}=\frac{6-29i+9(-1)}{9-(-1)}=\frac{-3-29i}{10}=-\frac{3}{10}-\frac{29i}{10}$

Report an Error

Example Question #7 : Find The Quotient Of Complex Numbers

Divide.

$\frac{4+3i}{6-i}$

Possible Answers:

$\frac{9}{7}+\frac{27}{22i}$

$\frac{21}{37}+\frac{22i}{37}$

$\frac{7}{9}-\frac{22i}{27}$

$\frac{2}{3}+\frac{i}{9}$

Correct answer:

$\frac{21}{37}+\frac{22i}{37}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 6+i .

$\left(\frac{4+3i}{6-i}\right)\left(\frac{6+i}{6+i}\right)$

Now, distribute and simplify.

$\left(\frac{4+3i}{6-i}\right)\left(\frac{6+i}{6+i}\right)=\frac{24+22i+3i^2}{36-i^2}$

Recall that i^2=-1

$\frac{24+22i+3i^2}{36-i^2}=\frac{24+22i+3(-1)}{36-(-1)}=\frac{21+22i}{37}$

Report an Error

Example Question #8 : Find The Quotient Of Complex Numbers

Divide.

$\frac{5-2i}{3-8i}$

Possible Answers:

$\frac{73}{31}-\frac{73}{64i}$

$\frac{21}{29}+\frac{6i}{29}$

$\frac{31}{73}-\frac{34i}{73}$

$\frac{31}{73}+\frac{34i}{73}$

Correct answer:

$\frac{31}{73}+\frac{34i}{73}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3+8i .

$\left(\frac{5-2i}{3-8i}\right)\left(\frac{3+8i}{3+8i}\right)$

Now, distribute and simplify.

$\left(\frac{5-2i}{3-8i}\right)\left(\frac{3+8i}{3+8i}\right)=\frac{15+34i-16i^2}{9-64i^2}$

Recall that i^2=-1

$\frac{15+34i-16i^2}{9-64i^2}=\frac{15+34i-16(-1)}{9-64(-1)}=\frac{31+34i}{73}=\frac{31}{73}+\frac{34i}{73}$

Report an Error

Example Question #9 : Find The Quotient Of Complex Numbers

Divide.

$\frac{5+i}{3+i}$

Possible Answers:

$\frac{8}{5}+\frac{i}{5}$

$\frac{3}{8}-\frac{i}{4}$

$\frac{8}{5}-\frac{i}{5}$

$\frac{5}{8}+\frac{i}{5}$

Correct answer:

$\frac{8}{5}-\frac{i}{5}$

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 3-i .

$\left(\frac{5+i}{3+i}\right)\left(\frac{3-i}{3-i}\right)$

Now, distribute and simplify.

$\left(\frac{5+i}{3+i}\right)\left(\frac{3-i}{3-i}\right)=\frac{15-2i-i^2}{9-i^2}$

Recall that i^2=-1

$\frac{15-2i-i^2}{9-i^2}=\frac{15-2i-(-1)}{9-(-1)}=\frac{16-2i}{10}=\frac{8}{5}-\frac{i}{5}$

Report an Error

Example Question #10 : Find The Quotient Of Complex Numbers

Divide.

$\frac{-9+i}{1-i}$

Possible Answers:

-5-4i

-2+3i

-4-5i

5+4i

Correct answer:

-5-4i

Explanation:

To divide complex numbers, multiply both the numerator and denominator by the conjugate of the denominator.

To find the conjugate, just change the sign in the denominator. The conjugate used will be 1-i .

$\left(\frac{-9+i}{1-i}\right)\left(\frac{1+i}{1+i}\right)$

Now, distribute and simplify.

$\left(\frac{-9+i}{1-i}\right)\left(\frac{1+i}{1+i}\right)=\frac{-9-8i+i^2}{1-i^2}$

Recall that i^2=-1

$\frac{-9-8i+i^2}{1-i^2}=\frac{-9-8i+(-1)}{1-(-1)}=\frac{-10-8i}{2}=-5-4i$

Report an Error

← Previous 1 2 Next →

View Pre-Calculus Tutors

Ravi
Certified Tutor

Indian Institute of Technology - Mumbai, Bachelor of Technology, Mechanical Engineering. University of Cincinnati-Main Campus...

View Pre-Calculus Tutors

Manpreet
Certified Tutor

Panjab University, Bachelor of Science, Physics. Panjab University, Master of Science, Physics.

View Pre-Calculus Tutors

Shelly
Certified Tutor

Delhi University India, Bachelor of Science, Mathematics. Delhi University India, Master of Science, Mathematics.

All Precalculus Resources

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

Popular Subjects

MCAT Tutors in Houston, ISEE Tutors in Denver, Statistics Tutors in New York City, SSAT Tutors in Los Angeles, LSAT Tutors in Seattle, SAT Tutors in Los Angeles, Spanish Tutors in Dallas Fort Worth, Math Tutors in Miami, Reading Tutors in Dallas Fort Worth, Computer Science Tutors in San Diego

Popular Courses & Classes

GMAT Courses & Classes in Houston, ISEE Courses & Classes in Chicago, Spanish Courses & Classes in Seattle, GMAT Courses & Classes in Denver, GMAT Courses & Classes in Washington DC, SSAT Courses & Classes in Dallas Fort Worth, LSAT Courses & Classes in Chicago, SSAT Courses & Classes in Miami, LSAT Courses & Classes in Miami, SSAT Courses & Classes in Atlanta

Popular Test Prep

SAT Test Prep in Houston, SSAT Test Prep in Phoenix, SAT Test Prep in New York City, GRE Test Prep in Houston, SAT Test Prep in Atlanta, GMAT Test Prep in Houston, ACT Test Prep in San Francisco-Bay Area, GMAT Test Prep in Washington DC, GRE Test Prep in Chicago, GRE Test Prep in San Francisco-Bay Area

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 : Find the Quotient of Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #4 : Find The Quotient Of Complex Numbers

Example Question #5 : Find The Quotient Of Complex Numbers

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #7 : Find The Quotient Of Complex Numbers

Example Question #8 : Find The Quotient Of Complex Numbers

Example Question #9 : Find The Quotient Of Complex Numbers

Example Question #10 : Find The Quotient Of Complex Numbers

All Precalculus Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: