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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
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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° ) )

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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

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What does the absolute value of a complex number represent?

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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 .

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How do we find the product of two complex numbers?

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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.

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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° )

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

z = 7 cis ( π 6 )

7 3 2 + i 7 2

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z = 4 cis ( 7 π 6 )

2 3 2 i

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z = 3 cis ( 240° )

1.5 i 3 3 2

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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° )

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

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

3 4 cis ( 180° )

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

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

5 3 cis ( 17 π 24 )

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

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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° )

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

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

5 cis ( 80° )

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

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

5 cis ( π 3 )

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

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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° )

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Find z 4 when z = 2 cis ( 70° ) .

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Find z 2 when z = 3 cis ( 120° ) .

9 cis ( 240° )

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Find z 2 when z = 4 cis ( π 4 ) .

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Find z 4 when z = cis ( 3 π 16 ) .

cis ( 3 π 4 )

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Find z 3 when z = 3 cis ( 5 π 3 ) .

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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° )

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

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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 )

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

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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 )

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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.

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

3.61 e 0.59 i

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

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

2 + 3.46 i

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

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

4.33 2.50 i

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Questions & Answers

how to understand calculus?
Jenica Reply
Hey I am new to precalculus, and wanted clarification please on what sine is as I am floored by the terms in this app? I don't mean to sound stupid but I have only completed up to college algebra.
rachel Reply
I don't know if you are looking for a deeper answer or not, but the sine of an angle in a right triangle is the length of the opposite side to the angle in question divided by the length of the hypotenuse of said triangle.
Marco
can you give me sir tips to quickly understand precalculus. Im new too in that topic. Thanks
Jenica
if you remember sine, cosine, and tangent from geometry, all the relationships are the same but they use x y and r instead (x is adjacent, y is opposite, and r is hypotenuse).
Natalie
the standard equation of the ellipse that has vertices (0,-4)&(0,4) and foci (0, -15)&(0,15) it's standard equation is x^2 + y^2/16 =1 tell my why is it only x^2? why is there no a^2?
Reena Reply
what is foci?
Reena Reply
This term is plural for a focus, it is used for conic sections. For more detail or other math questions. I recommend researching on "Khan academy" or watching "The Organic Chemistry Tutor" YouTube channel.
Chris
how to determine the vertex,focus,directrix and axis of symmetry of the parabola by equations
Bryssen Reply
i want to sure my answer of the exercise
meena Reply
what is the diameter of(x-2)²+(y-3)²=25
Den Reply
how to solve the Identity ?
Barcenas Reply
what type of identity
Jeffrey
Confunction Identity
Barcenas
how to solve the sums
meena
hello guys
meena
For each year t, the population of a forest of trees is represented by the function A(t) = 117(1.029)t. In a neighboring forest, the population of the same type of tree is represented by the function B(t) = 86(1.025)t.
Shakeena Reply
by how many trees did forest "A" have a greater number?
Shakeena
32.243
Kenard
how solve standard form of polar
Rhudy Reply
what is a complex number used for?
Drew Reply
It's just like any other number. The important thing to know is that they exist and can be used in computations like any number.
Steve
I would like to add that they are used in AC signal analysis for one thing
Scott
Good call Scott. Also radar signals I believe.
Steve
They are used in any profession where the phase of a waveform has to be accounted for in the calculations. Imagine two electrical signals in a wire that are out of phase by 90°. At some times they will interfere constructively, others destructively. Complex numbers simplify those equations
Tim
Is there any rule we can use to get the nth term ?
Anwar Reply
how do you get the (1.4427)^t in the carp problem?
Gabrielle Reply
A hedge is contrusted to be in the shape of hyperbola near a fountain at the center of yard.the hedge will follow the asymptotes y=x and y=-x and closest distance near the distance to the centre fountain at 5 yards find the eqution of the hyperbola
ayesha Reply
A doctor prescribes 125 milligrams of a therapeutic drug that decays by about 30% each hour. To the nearest hour, what is the half-life of the drug?
Sandra Reply
Practice Key Terms 4

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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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