Call Now to Set Up Tutoring:

(888) 888-0446

Precalculus : Polar Coordinates and 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 3 4 5 6 7 8 9 … 14 15 Next →

Example Question #1 : Find The Product Of Complex Numbers

Find the value of ${}(1+i)^{100}$ ,where the complex number is given by ${}i^2=-1$ .

Possible Answers:

${}2^{101}$

${}i2^{100}$

${}-2^{100}$

${}-i2^{100}$

${}-2^{50}$

Correct answer:

${}-2^{50}$

Explanation:

We note that ${}(1+i)^{2}=1+2i+i^2=1+2i-1=2i$ by FOILing.

We also know that:

$(z)^{nm}=(z^{n})^{m}$

We have by using the above rule: n=2 , m=50

$(1+i)^{100}=((1+i)^{2})^{50}$

Since we know that,

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

We have then:

$(1+i)^{100}=((1+i)^{2})^{50}$ $=(2i)^{50}$

Since we know that:

$(ab)^{n}=a^{n}b^{n}$ , we use a=2 ,b=i

We have then:

$2^{50}({i}^2)^{25}=2^{50}(-1)^{25} =-2^{50}}$

Report an Error

Example Question #1 : Polar Coordinates And Complex Numbers

Compute the following sum:

${}1+i+i^2+i^3....+i^{2015}$ . Remember {}i is the complex number satisfying ${}i^{2}=-1$ .

Possible Answers:

{}-i

{}i

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

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

Correct answer:

Explanation:

Note that this is a geometric series.

Therefore we have:

${}1+i+i^2....+i^{2015}=\frac{1-i^{2016}}{1-i}$

Note that,

${}i^{2016}$ = ${}({i}^{2})^{1008}$ and since ${}i^2=-1$ we have ${}{-1}^{1008}=1$ .

$\frac{1-1}{1-i}=\frac{0}{1-i}=0$

this shows that the sum is 0.

Report an Error

Example Question #3 : Find The Product Of Complex Numbers

Find the following product.

${}(1-i)^{n}(1+i)^n$

Possible Answers:

${}2i^{n}$

${}2^{n-1}$

${}2^{n}$

${}-2i^{n}$

{}2

Correct answer:

${}2^{n}$

Explanation:

Note that by FOILing the two binomials we get the following:

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

Therefore,

${}(1-i)^{n}(1-i)^{n}=2^{n}$

Report an Error

Example Question #4 : Find The Product Of Complex Numbers

Compute the magnitude of ${}(1+i)^{n}$ .

Possible Answers:

${}(\sqrt{2})^{n+1}$

${}(\sqrt{2})^n-1$

${}(\sqrt{2})^n$

${}(1-\sqrt{2})^n$

${}(\sqrt{2})^n+1$

Correct answer:

${}(\sqrt{2})^n$

Explanation:

We have

${}|(1+i)|=\sqrt{2}$ .

We know that ${}|z^n|=|z|^n$

Thus this gives us,

${}|(1+i)^n|=|1+i|^{n}=(\sqrt{2})^{n}$ .

Report an Error

Example Question #5 : Find The Product Of Complex Numbers

Evaluate:

$(2+i)\cdot(1+4i)$

Possible Answers:

9i-2

i^2

5i-7

i+4

Correct answer:

9i-2

Explanation:

To evaluate this problem we need to FOIL the binomials.

(2+i)(1+4i)

=2(1)+2(4i)+1(i)+4i^2

=2+9i+4i^2

Now recall that i^2=-1

Thus,

=2+9i-4

=9i-2

Report an Error

Example Question #1 : Polar Coordinates And Complex Numbers

Find the product , if

$a=3+7i\:\:\:b=1-i$ .

Possible Answers:

3+4i

10+4i

3-3i

3-7i

Correct answer:

10+4i

Explanation:

To find the product , FOIL the complex numbers. FOIL stands for the multiplication of the Firsts, Outers, Inners, and Lasts.

Using this method we get the following,

ab=(3+7i)(1-i)=3-3i+7i-7i^2=3+4i-7i^2

and because i^2=-1

=3+4i-7(-1)=3+4i+7=10+4i .

Report an Error

Example Question #1 : Find The Product Of Complex Numbers

Simplify: $i^4 \times i^3$

Possible Answers:

$i+\sqrt{-1}$

Correct answer:

Explanation:

The expression $i^4 \times i^3$ can be rewritten as:

$i^4 \times i^3 = i^2\times i^2\times i^2\times i$

Since $i=\sqrt{-1}$ , the value of i^2 = -1 .

$i^2\times i^2\times i^2\times i = (-1) (-1) (-1)(i) = -i$

The correct answer is:

Report an Error

Example Question #8 : Find The Product Of Complex Numbers

Find the product of the two complex numbers

$\small 3+7i$ and $\small 7+10i$

Possible Answers:

$\small 21+70i$

$\small -49+79i$

$\small 21+79i+70i^2$

$\small 10+17i$

Correct answer:

$\small -49+79i$

Explanation:

The product is

$\small (3+7i)(7+10i)=21+30i+49i+70i^2=21-70+79i=-49+79i$

Report an Error

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

← Previous 1 2 3 4 5 6 7 8 9 … 14 15 Next →

View Pre-Calculus Tutors

Melody
Certified Tutor

George Fox University, Bachelor of Science, Biology, General.

View Pre-Calculus Tutors

Michael
Certified Tutor

University of Ghana Legon, Bachelor in Arts, Statistics. University of Louisiana-Monroe, Non Degree Bachelors, Education.

View Pre-Calculus Tutors

Andrew
Certified Tutor

University of Wisconsin-Stevens Point, Bachelor of Science, Mathematics. University of Wisconsin-Oshkosh, Master of Science, ...

All Precalculus Resources

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

Popular Subjects

SAT Tutors in Chicago, Statistics Tutors in Dallas Fort Worth, Biology Tutors in Atlanta, LSAT Tutors in San Francisco-Bay Area, MCAT Tutors in New York City, Biology Tutors in Washington DC, ACT Tutors in Dallas Fort Worth, GMAT Tutors in Philadelphia, Computer Science Tutors in Denver, LSAT Tutors in Seattle

Popular Courses & Classes

ACT Courses & Classes in San Diego, MCAT Courses & Classes in Boston, MCAT Courses & Classes in Dallas Fort Worth, ACT Courses & Classes in Chicago, LSAT Courses & Classes in Phoenix, MCAT Courses & Classes in Seattle, Spanish Courses & Classes in Houston, Spanish Courses & Classes in Miami, MCAT Courses & Classes in Houston, GMAT Courses & Classes in Washington DC

Popular Test Prep

SAT Test Prep in Houston, GRE Test Prep in Los Angeles, ISEE Test Prep in Los Angeles, GMAT Test Prep in Denver, ISEE Test Prep in San Francisco-Bay Area, MCAT Test Prep in Philadelphia, GRE Test Prep in Dallas Fort Worth, SAT Test Prep in Atlanta, SSAT Test Prep in Chicago, LSAT 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

Precalculus : Polar Coordinates and Complex Numbers

Study concepts, example questions & explanations for Precalculus

All Precalculus Resources

Example Questions

Example Question #1 : Find The Product Of Complex Numbers

Example Question #1 : Polar Coordinates And Complex Numbers

Example Question #3 : Find The Product Of Complex Numbers

Example Question #4 : Find The Product Of Complex Numbers

Example Question #5 : Find The Product Of Complex Numbers

Example Question #1 : Polar Coordinates And Complex Numbers

Example Question #1 : Find The Product Of Complex Numbers

Example Question #8 : Find The Product Of Complex Numbers

Example Question #1 : Find The Quotient Of Complex Numbers

Example Question #1 : 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: