<< Chapter < Page Chapter >> Page >

Finding the n Th root of a complex number

Evaluate the cube roots of z = 8 ( cos ( 2 π 3 ) + i sin ( 2 π 3 ) ) .

We have

z 1 3 = 8 1 3 [ cos ( 2 π 3 3 + 2 k π 3 ) + i sin ( 2 π 3 3 + 2 k π 3 ) ] z 1 3 = 2 [ cos ( 2 π 9 + 2 k π 3 ) + i sin ( 2 π 9 + 2 k π 3 ) ]

There will be three roots: k = 0 , 1 , 2. When k = 0 , we have

z 1 3 = 2 ( cos ( 2 π 9 ) + i sin ( 2 π 9 ) )

When k = 1 , we have

z 1 3 = 2 [ cos ( 2 π 9 + 6 π 9 ) + i sin ( 2 π 9 + 6 π 9 ) ]     Add  2 ( 1 ) π 3  to each angle. z 1 3 = 2 ( cos ( 8 π 9 ) + i sin ( 8 π 9 ) )

When k = 2 , we have

z 1 3 = 2 [ cos ( 2 π 9 + 12 π 9 ) + i sin ( 2 π 9 + 12 π 9 ) ] Add  2 ( 2 ) π 3  to each angle. z 1 3 = 2 ( cos ( 14 π 9 ) + i sin ( 14 π 9 ) )

Remember to find the common denominator to simplify fractions in situations like this one. For k = 1 , the angle simplification is

2 π 3 3 + 2 ( 1 ) π 3 = 2 π 3 ( 1 3 ) + 2 ( 1 ) π 3 ( 3 3 ) = 2 π 9 + 6 π 9 = 8 π 9
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find the four fourth roots of 16 ( cos ( 120° ) + i sin ( 120° ) ) .

z 0 = 2 ( cos ( 30° ) + i sin ( 30° ) )

z 1 = 2 ( cos ( 120° ) + i sin ( 120° ) )

z 2 = 2 ( cos ( 210° ) + i sin ( 210° ) )

z 3 = 2 ( cos ( 300° ) + i sin ( 300° ) )

Got questions? Get instant answers now!

Access these online resources for additional instruction and practice with polar forms of complex numbers.

Key concepts

  • Complex numbers in the form a + b i are plotted in the complex plane similar to the way rectangular coordinates are plotted in the rectangular plane. Label the x- axis as the real axis and the y- axis as the imaginary axis. See [link] .
  • The absolute value of a complex number is the same as its magnitude. It is the distance from the origin to the point: | z | = a 2 + b 2 . See [link] and [link] .
  • To write complex numbers in polar form, we use the formulas x = r cos θ , y = r sin θ , and r = x 2 + y 2 . Then, z = r ( cos θ + i sin θ ) . See [link] and [link] .
  • To convert from polar form to rectangular form, first evaluate the trigonometric functions. Then, multiply through by r . See [link] and [link] .
  • To find the product of two complex numbers, multiply the two moduli and add the two angles. Evaluate the trigonometric functions, and multiply using the distributive property. See [link] .
  • To find the quotient of two complex numbers in polar form, find the quotient of the two moduli and the difference of the two angles. See [link] .
  • To find the power of a complex number z n , raise r to the power n , and multiply θ by n . See [link] .
  • Finding the roots of a complex number is the same as raising a complex number to a power, but using a rational exponent. See [link] .

Section exercises

Verbal

A complex number is a + b i . Explain each part.

a is the real part, b is the imaginary part, and i = 1

Got questions? Get instant answers now!

What does the absolute value of a complex number represent?

Got questions? Get instant answers now!

How is a complex number converted to polar form?

Polar form converts the real and imaginary part of the complex number in polar form using x = r cos θ and y = r sin θ .

Got questions? Get instant answers now!

How do we find the product of two complex numbers?

Got questions? Get instant answers now!

What is De Moivre’s Theorem and what is it used for?

z n = r n ( cos ( n θ ) + i sin ( n θ ) ) It is used to simplify polar form when a number has been raised to a power.

Got questions? Get instant answers now!

Algebraic

For the following exercises, find the absolute value of the given complex number.

For the following exercises, write the complex number in polar form.

8 4 i

4 5 cis ( 333.4° )

Got questions? Get instant answers now!

For the following exercises, convert the complex number from polar to rectangular form.

z = 7 cis ( π 6 )

7 3 2 + i 7 2

Got questions? Get instant answers now!

z = 4 cis ( 7 π 6 )

2 3 2 i

Got questions? Get instant answers now!

z = 3 cis ( 240° )

1.5 i 3 3 2

Got questions? Get instant answers now!

For the following exercises, find z 1 z 2 in polar form.

z 1 = 2 3 cis ( 116° ) ;   z 2 = 2 cis ( 82° )

4 3 cis ( 198° )

Got questions? Get instant answers now!

z 1 = 2 cis ( 205° ) ;   z 2 = 2 2 cis ( 118° )

Got questions? Get instant answers now!

z 1 = 3 cis ( 120° ) ;   z 2 = 1 4 cis ( 60° )

3 4 cis ( 180° )

Got questions? Get instant answers now!

z 1 = 3 cis ( π 4 ) ;   z 2 = 5 cis ( π 6 )

Got questions? Get instant answers now!

z 1 = 5 cis ( 5 π 8 ) ;   z 2 = 15 cis ( π 12 )

5 3 cis ( 17 π 24 )

Got questions? Get instant answers now!

z 1 = 4 cis ( π 2 ) ;   z 2 = 2 cis ( π 4 )

Got questions? Get instant answers now!

For the following exercises, find z 1 z 2 in polar form.

z 1 = 21 cis ( 135° ) ;   z 2 = 3 cis ( 65° )

7 cis ( 70° )

Got questions? Get instant answers now!

z 1 = 2 cis ( 90° ) ;   z 2 = 2 cis ( 60° )

Got questions? Get instant answers now!

z 1 = 15 cis ( 120° ) ;   z 2 = 3 cis ( 40° )

5 cis ( 80° )

Got questions? Get instant answers now!

z 1 = 6 cis ( π 3 ) ;   z 2 = 2 cis ( π 4 )

Got questions? Get instant answers now!

z 1 = 5 2 cis ( π ) ;   z 2 = 2 cis ( 2 π 3 )

5 cis ( π 3 )

Got questions? Get instant answers now!

z 1 = 2 cis ( 3 π 5 ) ;   z 2 = 3 cis ( π 4 )

Got questions? Get instant answers now!

For the following exercises, find the powers of each complex number in polar form.

Find z 3 when z = 5 cis ( 45° ) .

125 cis ( 135° )

Got questions? Get instant answers now!

Find z 4 when z = 2 cis ( 70° ) .

Got questions? Get instant answers now!

Find z 2 when z = 3 cis ( 120° ) .

9 cis ( 240° )

Got questions? Get instant answers now!

Find z 2 when z = 4 cis ( π 4 ) .

Got questions? Get instant answers now!

Find z 4 when z = cis ( 3 π 16 ) .

cis ( 3 π 4 )

Got questions? Get instant answers now!

Find z 3 when z = 3 cis ( 5 π 3 ) .

Got questions? Get instant answers now!

For the following exercises, evaluate each root.

Evaluate the cube root of z when z = 27 cis ( 240° ) .

3 cis ( 80° ) , 3 cis ( 200° ) , 3 cis ( 320° )

Got questions? Get instant answers now!

Evaluate the square root of z when z = 16 cis ( 100° ) .

Got questions? Get instant answers now!

Evaluate the cube root of z when z = 32 cis ( 2 π 3 ) .

2 4 3 cis ( 2 π 9 ) , 2 4 3 cis ( 8 π 9 ) , 2 4 3 cis ( 14 π 9 )

Got questions? Get instant answers now!

Evaluate the square root of z when z = 32 cis ( π ) .

Got questions? Get instant answers now!

Evaluate the cube root of z when z = 8 cis ( 7 π 4 ) .

2 2 cis ( 7 π 8 ) , 2 2 cis ( 15 π 8 )

Got questions? Get instant answers now!

Graphical

For the following exercises, plot the complex number in the complex plane.

Technology

For the following exercises, find all answers rounded to the nearest hundredth.

Use the rectangular to polar feature on the graphing calculator to change 5 + 5 i to polar form.

Got questions? Get instant answers now!

Use the rectangular to polar feature on the graphing calculator to change 3 2 i to polar form.

3.61 e 0.59 i

Got questions? Get instant answers now!

Use the rectangular to polar feature on the graphing calculator to change 3 8 i to polar form.

Got questions? Get instant answers now!

Use the polar to rectangular feature on the graphing calculator to change 4 cis ( 120° ) to rectangular form.

2 + 3.46 i

Got questions? Get instant answers now!

Use the polar to rectangular feature on the graphing calculator to change 2 cis ( 45° ) to rectangular form.

Got questions? Get instant answers now!

Use the polar to rectangular feature on the graphing calculator to change 5 cis ( 210° ) to rectangular form.

4.33 2.50 i

Got questions? Get instant answers now!

Questions & Answers

what is f(x)=
Karim Reply
I don't understand
Joe
Typically a function 'f' will take 'x' as input, and produce 'y' as output. As 'f(x)=y'. According to Google, "The range of a function is the complete set of all possible resulting values of the dependent variable (y, usually), after we have substituted the domain."
Thomas
Sorry, I don't know where the "Â"s came from. They shouldn't be there. Just ignore them. :-)
Thomas
GREAT ANSWER THOUGH!!!
Darius
Thanks.
Thomas
Â
Thomas
It is the  that should not be there. It doesn't seem to show if encloses in quotation marks. "Â" or 'Â' ... Â
Thomas
Now it shows, go figure?
Thomas
what is this?
unknown Reply
i do not understand anything
unknown
lol...it gets better
Darius
I've been struggling so much through all of this. my final is in four weeks 😭
Tiffany
this book is an excellent resource! have you guys ever looked at the online tutoring? there's one that is called "That Tutor Guy" and he goes over a lot of the concepts
Darius
thank you I have heard of him. I should check him out.
Tiffany
is there any question in particular?
Joe
I have always struggled with math. I get lost really easy, if you have any advice for that, it would help tremendously.
Tiffany
Sure, are you in high school or college?
Darius
Hi, apologies for the delayed response. I'm in college.
Tiffany
how to solve polynomial using a calculator
Ef Reply
So a horizontal compression by factor of 1/2 is the same as a horizontal stretch by a factor of 2, right?
KARMEL Reply
The center is at (3,4) a focus is at (3,-1), and the lenght of the major axis is 26
Rima Reply
The center is at (3,4) a focus is at (3,-1) and the lenght of the major axis is 26 what will be the answer?
Rima
I done know
Joe
What kind of answer is that😑?
Rima
I had just woken up when i got this message
Joe
Can you please help me. Tomorrow is the deadline of my assignment then I don't know how to solve that
Rima
i have a question.
Abdul
how do you find the real and complex roots of a polynomial?
Abdul
@abdul with delta maybe which is b(square)-4ac=result then the 1st root -b-radical delta over 2a and the 2nd root -b+radical delta over 2a. I am not sure if this was your question but check it up
Nare
This is the actual question: Find all roots(real and complex) of the polynomial f(x)=6x^3 + x^2 - 4x + 1
Abdul
@Nare please let me know if you can solve it.
Abdul
I have a question
juweeriya
hello guys I'm new here? will you happy with me
mustapha
The average annual population increase of a pack of wolves is 25.
Brittany Reply
how do you find the period of a sine graph
Imani Reply
Period =2π if there is a coefficient (b), just divide the coefficient by 2π to get the new period
Am
if not then how would I find it from a graph
Imani
by looking at the graph, find the distance between two consecutive maximum points (the highest points of the wave). so if the top of one wave is at point A (1,2) and the next top of the wave is at point B (6,2), then the period is 5, the difference of the x-coordinates.
Am
you could also do it with two consecutive minimum points or x-intercepts
Am
I will try that thank u
Imani
Case of Equilateral Hyperbola
Jhon Reply
ok
Zander
ok
Shella
f(x)=4x+2, find f(3)
Benetta
f(3)=4(3)+2 f(3)=14
lamoussa
14
Vedant
pre calc teacher: "Plug in Plug in...smell's good" f(x)=14
Devante
8x=40
Chris
Explain why log a x is not defined for a < 0
Baptiste Reply
the sum of any two linear polynomial is what
Esther Reply
divide simplify each answer 3/2÷5/4
Momo Reply
divide simplify each answer 25/3÷5/12
Momo
how can are find the domain and range of a relations
austin Reply
the range is twice of the natural number which is the domain
Morolake
A cell phone company offers two plans for minutes. Plan A: $15 per month and $2 for every 300 texts. Plan B: $25 per month and $0.50 for every 100 texts. How many texts would you need to send per month for plan B to save you money?
Diddy Reply
6000
Robert
more than 6000
Robert
For Plan A to reach $27/month to surpass Plan B's $26.50 monthly payment, you'll need 3,000 texts which will cost an additional $10.00. So, for the amount of texts you need to send would need to range between 1-100 texts for the 100th increment, times that by 3 for the additional amount of texts...
Gilbert
...for one text payment for 300 for Plan A. So, that means Plan A; in my opinion is for people with text messaging abilities that their fingers burn the monitor for the cell phone. While Plan B would be for loners that doesn't need their fingers to due the talking; but those texts mean more then...
Gilbert
can I see the picture
Zairen Reply
How would you find if a radical function is one to one?
Peighton Reply
Practice Key Terms 4

Get the best Precalculus course in your pocket!





Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Precalculus' conversation and receive update notifications?

Ask