Call Now to Set Up Tutoring:

(888) 888-0446

PSAT Math : Complex Numbers

Study concepts, example questions & explanations for PSAT Math

Create An Account

All PSAT Math Resources

10 Diagnostic Tests 421 Practice Tests Question of the Day Flashcards Learn by Concept

Example Questions

Example Question #1 : Complex Numbers

Simplify by rationalizing the denominator:

$\frac{20 + 6i}{8i}$

Possible Answers:

$\frac{ 3 }{4} + \frac{ 5}{2}i$

$-\frac{ 3 }{4} + \frac{ 5}{2}i$

$- \frac{ 5}{2} + \frac{ 3 }{4}i$

$\frac{ 3 }{4} - \frac{ 5}{2}i$

$\frac{ 5}{2} - \frac{ 3 }{4}i$

Correct answer:

$\frac{ 3 }{4} - \frac{ 5}{2}i$

Explanation:

Multiply both numerator and denominator by :

$\frac{20 + 6i}{8i}$

$=\frac{\left (20 + 6i \right ) \cdot i}{8i \cdot i}$

$=\frac{ 20i + 6i^{2} }{8i ^{2}}$

$=\frac{ 20i + 6(-1) }{8 (-1)}$

$=\frac{ 20i -6 }{-8}$

$=\frac{ -6+ 20i }{-8}$

$=\frac{ -6 }{-8} + \frac{ 20i }{-8}$

$=\frac{ 3 }{4} - \frac{ 5}{2}i$

Report an Error

Example Question #2 : How To Divide Complex Numbers

Divide:

$-100 \div 5i$

Possible Answers:

-20i

20i

20 - 20i

20 + 20 i

Correct answer:

20i

Explanation:

Rewrite in fraction form, and multiply both numerator and denominator by :

$-100 \div 5i$

$= \frac{-100}{5i}$

$= \frac{-100 \cdot i }{5i \cdot i }$

$= \frac{-100 i }{5i ^{2}}$

$= \frac{-100 i }{5 (-1)}$

$= \frac{-100 i }{-5}$

$= \frac{-100 }{-5}\cdot i$

= 20i

Report an Error

Example Question #2 : Complex Numbers

Simplify by rationalizing the denominator:

$\frac{ 25- 12i}{ - 10i}$

Possible Answers:

$\frac{ 5 }{ 2} + \frac{6}{ 5} i$

$\frac{6}{ 5} - \frac{ 5 }{ 2}i$

$-\frac{ 5 }{ 2} + \frac{6}{ 5} i$

$-\frac{6}{ 5} + \frac{ 5 }{ 2}i$

$\frac{6}{ 5} + \frac{ 5 }{ 2}i$

Correct answer:

$\frac{6}{ 5} + \frac{ 5 }{ 2}i$

Explanation:

Multiply both numerator and denominator by :

$\frac{ 25- 12i}{ - 10i}$

$=\frac{\left ( 25- 12i \right )\cdot i}{ - 10i \cdot i }$

$=\frac{ 25i- 12i ^{2} }{ - 10i ^{2} }$

$=\frac{ 25i- 12 (-1)}{ - 10 (-1) }$

$=\frac{ 25i+ 12}{ 10}$

$=\frac{12+ 25i }{ 10}$

$=\frac{12 }{ 10} + \frac{ 25i }{ 10}$

$=\frac{6}{ 5} + \frac{ 5 }{ 2}i$

Report an Error

Example Question #3 : How To Divide Complex Numbers

Simplify by rationalizing the denominator:

$\frac{ 21i}{3+ i\sqrt{5}}$

Possible Answers:

$-\frac{ 21 \sqrt{5} }{ 4}+\frac{63 }{ 4} i$

$\frac{ 3\sqrt{5} }{ 2} - \frac{ 9 }{ 2} i$

$\frac{ 3\sqrt{5} }{ 2} + \frac{ 9 }{ 2} i$

$\frac{ 21 \sqrt{5} }{ 4}+ \frac{63 }{ 4} i$

$-\frac{ 3\sqrt{5} }{ 2} + \frac{ 9 }{ 2} i$

Correct answer:

$\frac{ 3\sqrt{5} }{ 2} + \frac{ 9 }{ 2} i$

Explanation:

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is $3- i\sqrt{5}$ :

$\frac{ 21i}{3+ i\sqrt{5}}$

$= \frac{ 21i\left (3- i\sqrt{5} \right )}{\left (3+ i\sqrt{5} \right )\left (3- i\sqrt{5} \right )}$

$= \frac{21i \cdot 3- 21i \cdot i\sqrt{5} }{ 3^{2}+ \left ( \sqrt{5} \right )^{2}}$

$= \frac{ 63i - 21i^{2} \sqrt{5}}{ 9+5}$

$= \frac{ 63i - 21 (-1)\sqrt{5}}{ 14}$

$= \frac{ 21 \sqrt{5}+ 63i }{ 14}$

$= \frac{ 21 \sqrt{5} }{ 14}+ \frac{63 i}{ 14}$

$= \frac{ 3\sqrt{5} }{ 2} + \frac{ 9 }{ 2} i$

Report an Error

Example Question #2 : How To Divide Complex Numbers

Simplify by rationalizing the denominator:

$\frac{2+3i}{4+ 3i}$

Possible Answers:

$-\frac{1 }{25} + \frac{ 18 }{25} i$

$\frac{17 }{25} + \frac{ 6 }{25} i$

$\frac{17}{7} + \frac{ 6 }{7} i$

-1 + 18i

$-\frac{1 }{7} + \frac{ 18 }{7} i$

Correct answer:

$\frac{17 }{25} + \frac{ 6 }{25} i$

Explanation:

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is 4 - 3i :

$\frac{2+3i}{4+ 3i}$

$=\frac{\left (2+3i \right )\left (4-3i \right )}{\left (4+ 3i \right )\left (4-3i \right )}$

$=\frac{2 \cdot 4 - 2 \cdot 3i +3i \cdot 4 - 3i \cdot 3i }{4^{2}+3^{2}}$

$=\frac{8 - 6i +12i - 9 i ^{2}}{16+ 9}$

$=\frac{8 - 6i +12i - 9(-1)}{25}$

$=\frac{8 - 6i +12i +9}{25}$

$=\frac{17 + 6i }{25}$

$=\frac{17 }{25} + \frac{ 6 }{25} i$

Report an Error

Example Question #72 : Exponents

Simplify by rationalizing the denominator:

$\frac{15i}{4- 3i}$

Possible Answers:

$\frac{45 }{7} + \frac{ 60}{7}i$

$-\frac{9}{5} + \frac{ 12}{5}i$

$-\frac{45 }{7} + \frac{ 60}{7}i$

$\frac{9}{5} + \frac{ 12}{5}i$

$\frac{9}{5} -\frac{ 12}{5}i$

Correct answer:

$-\frac{9}{5} + \frac{ 12}{5}i$

Explanation:

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is 4 +3i :

$\frac{15i}{4- 3i}$

$= \frac{15i\left (4+ 3i \right )}{\left (4- 3i \right )\left (4+ 3i \right )}$

$= \frac{15i\cdot 4+ 15i\cdot 3i }{4^{2}+3^{2}}$

$= \frac{60 i + 45i^{2} }{16+9}$

$= \frac{60 i + 45 (-1)}{25}$

$= \frac{-45 + 60 i}{25}$

$= \frac{-45 }{25} + \frac{ 60 i}{25}$

$= - \frac{9}{5} + \frac{ 12}{5}i$

Report an Error

Example Question #2 : Complex Numbers

Define an operation $\ast$ as follows:

For all complex numbers a, b ,

$a \ast b = \frac{ai}{b}$ .

Evaluate $(2+ i) \ast (2-i)$ .

Possible Answers:

$\frac{4}{5} -\frac {3 }{5}i$

$- \frac{4}{5} +\frac {3 }{5}i$

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

$\frac {3 }{5}i$

Correct answer:

$- \frac{4}{5} +\frac {3 }{5}i$

Explanation:

$a \ast b = \frac{ai}{b}$

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

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

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

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

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is 2 + i . The above becomes:

$\frac{\left (-1+2i \right )\left (2+i \right )}{\left (2-i \right )\left (2+i \right )}$

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

$= \frac{-2 +\left (-i \right )+ 4i + 2i^{2} }{4+1}$

$= \frac{-2 -i+ 4i + 2(-1) }{5}$

$= \frac{-2 -i+ 4i -2 }{5}$

$= \frac{-4 +3i }{5}$

$=- \frac{4}{5} +\frac {3 }{5}i$

Report an Error

Example Question #2 : Complex Numbers

Define an operation $\ast$ as follows:

For all complex numbers a, b ,

$a \ast b = \frac{ab}{i}$ .

Evaluate $(3+ 4i) \ast (4-3i)$ .

Possible Answers:

7 +24i

25i

7 - 24i

-25i

24-7i

Correct answer:

7 - 24i

Explanation:

$a \ast b = \frac{ab}{i}$

$(3+ 4i) \ast (4-3i)= \frac{(3+ 4i) (4-3i)}{i}$

$= \frac{ 3 \cdot 4 - 3 \cdot 3i + 4i \cdot 4 -4i \cdot 3i}{i}$

$= \frac{ 12 - 9i + 16i -12i ^{2}}{i}$

$= \frac{ 12 - 9i + 16i -12 (-1)}{i}$

$= \frac{ 12 - 9i + 16i +12}{i}$

$= \frac{ 24+7i }{i}$

$= \frac{ \left (24+7i \right ) (-i)}{i (-i)}$

$= \frac{ -24i-7i ^{2} }{ -i^{2}}$

$= \frac{ -24i-7(-1) }{ -(-1)}}$

$= \frac{ -24i+7 }{ 1}$

= 7 - 24i

Report an Error

Example Question #4 : How To Divide Complex Numbers

Define an operation $\ast$ as follows:

For all complex numbers a, b ,

$a \ast b = \frac{ a^{3}}{b^{3} + 1}$

Evaluate $i \ast i$ .

Possible Answers:

1-i

$\frac{ 1}{2 }i$

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

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

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

Correct answer:

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

Explanation:

$a \ast b = \frac{ a^{3}}{b^{3} + 1}$

$i \ast i = \frac{ i^{3}}{i^{3} + 1}$

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

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

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

Multiply the numerator and the denominator by the complex conjugate of the denominator, which is -1 - i . The above becomes:

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

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

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

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

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

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

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

Report an Error

Example Question #1 : How To Multiply Complex Numbers

Which of the following is equal to $\left (3i \right )^{5}$ ?

Possible Answers:

243i

None of the other responses is correct.

15i

-15i

-243i

Correct answer:

243i

Explanation:

By the power of a product property,

$\left (3i \right )^{5} = 3^{5} \cdot i^{5} = 3^{5} \cdot i^{4} \cdot i = 243 \cdot 1 \cdot i = 243 i$

Report an Error

View PSAT Mathematics Tutors

Alice
Certified Tutor

The University of Texas at Dallas, Bachelor of Science, Accounting.

View PSAT Mathematics Tutors

Joseph
Certified Tutor

Bank Street College of Education , Master's/Graduate, Early Childhood and Childhood General Education.

View PSAT Mathematics Tutors

Gregory
Certified Tutor

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

All PSAT Math Resources

10 Diagnostic Tests 421 Practice Tests Question of the Day Flashcards Learn by Concept

Popular Subjects

Statistics Tutors in Atlanta, Biology Tutors in Phoenix, French Tutors in Boston, SSAT Tutors in Boston, English Tutors in Los Angeles, ACT Tutors in Atlanta, Calculus Tutors in Houston, LSAT Tutors in Los Angeles, Algebra Tutors in Atlanta, Chemistry Tutors in Houston

Popular Courses & Classes

ACT Courses & Classes in Seattle, Spanish Courses & Classes in Denver, SAT Courses & Classes in New York City, SSAT Courses & Classes in Phoenix, MCAT Courses & Classes in Los Angeles, SSAT Courses & Classes in Denver, MCAT Courses & Classes in Phoenix, GMAT Courses & Classes in Phoenix, GMAT Courses & Classes in Washington DC, LSAT Courses & Classes in Chicago

Popular Test Prep

SSAT Test Prep in New York City, GMAT Test Prep in New York City, ISEE Test Prep in Los Angeles, SAT Test Prep in Denver, GMAT Test Prep in Denver, MCAT Test Prep in Los Angeles, ACT Test Prep in Seattle, GMAT Test Prep in Miami, ACT Test Prep in San Francisco-Bay Area, GRE 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

PSAT Math : Complex Numbers

Study concepts, example questions & explanations for PSAT Math

All PSAT Math Resources

Example Questions

Example Question #1 : Complex Numbers

Example Question #2 : How To Divide Complex Numbers

Example Question #2 : Complex Numbers

Example Question #3 : How To Divide Complex Numbers

Example Question #2 : How To Divide Complex Numbers

Example Question #72 : Exponents

Example Question #2 : Complex Numbers

Example Question #2 : Complex Numbers

Example Question #4 : How To Divide Complex Numbers

Example Question #1 : How To Multiply Complex Numbers

All PSAT Math Resources

Report an issue with this question

DMCA Complaint

Contact Information

Address

Complaint Details

Email address:
Your name:
Feedback: