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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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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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