Explanation Given: Equation is z 4 + 8 2 = − 8 2 i Formula Used: Let the complex numberin polar form be given as z = r ( cos θ + i sin θ ) Then, for the integer n ≥ 2 , z has n distinct roots of the form w k = r n [ cos θ + 2 π k n + i sin θ + 2 π k n ] , k = 0 , 1 , 2..... n − 1 Calculation: Equation is given as z 4 + 8 2 = − 8 2 i z 4 = − 8 2 − 8 2 i z = ( − 8 2 − 8 2 i ) 1 4 This means we need to calculate the fifth roots of complex number ( − 8 2 − 8 2 i ) Converting the complex number into polar form r = ( − 8 2 ) 2 + ( − 8 2 ) 2 r = 128 + 128 r = 256 r = 16 And tan θ = − 8 2 − 8 2 tan θ = 1 θ = tan − 1 ( 1 ) θ lies in 3 r d quadrant Thus, θ = π + π 4 = 4 π + π 4 = 5 π 4 So, the polar form of complex number is z = 16 ( cos 5 π 4 + i sin 5 π 4 ) Fourth roots of z = 16 ( cos 5 π 4 + i sin 5 π 4 ) are calculated as w k = 16 4 [ cos 5 π 4 + 2 π k 4 + i sin 5 π 4 + 2 π k 4 ] , k = 0 , 1 , 2 , 3 w k = ( 2 ) 4 4 [ cos 5 π + 8 π k

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ISBN: 9781265934590

Author: Miller

Publisher: MCG CUSTOM

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Chapter 8.3, Problem 95PE

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To find all the complex solutions to the below equation and write the solutions in polar form

z4+82=−82i

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Chapter 8.3, Problem 94PE

Chapter 8.3, Problem 96PE

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To find all the complex solutions to the below equation and write the solutions in polar form z 4 + 8 2 = − 8 2 i

Solution Summary: The author explains the formula used to find all the complex solutions to the below equation and write the solutions in polar form z4+8sqrt2.

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To find all the complex solutions to the below equation and write the solutions in polar form

z4+82=−82i

Want to see the full answer?

Chapter 8 Solutions

To find all the complex solutions to the below equation and write the solutions in polar form z 4 + 8 2 = − 8 2 i

Solution Summary: The author explains the formula used to find all the complex solutions to the below equation and write the solutions in polar form z4+8sqrt2.

Videos

To find all the complex solutions to the below equation and write the solutions in polar form z4+82=−82i

Want to see the full answer?

Chapter 8 Solutions

To find all the complex solutions to the below equation and write the solutions in polar form

z4+82=−82i