Explanation Theorem used: Roots of a complex number : Let z = r ( cos θ + i sin θ ) and n be a positive integer. Then z has the n distinct n th roots w k = r 1 n [ cos ( θ + 2 k π n ) + i sin ( θ + 2 k π n ) ] where k = 0 , 1 , 2 , … , n − 1 . Calculation: Rewrite the complex number i in polar form. The polar form of the complex number z = a + b i is z = r ( cos θ + i sin θ ) where r = | z | = a 2 + b 2 and tan θ = b a . Consider the complex number i . Here, i = 0 + i . So, a = 0 , b = 1 and n = 3 . Obtain the argument of the complex number i . tan θ = 1 0 θ = tan − 1 ( 1 0 ) θ = π 2 Thus, the argument of the complex number i is θ = π 2

Single Variable Calculus: Early Transcendentals, Volume I

8th Edition

ISBN: 9781305270343

Author: James Stewart

Publisher: Cengage Learning

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Question

Chapter H, Problem 39E

To determine

To find: The roots of the cube root of i and sketch the roots in the complex plane.

Expert Solution & Answer

Chapter H Solutions

Single Variable Calculus: Early Transcendentals, Volume I

Ch. H - Prob. 1E Ch. H - Prob. 2E Ch. H - Prob. 3E Ch. H - Prob. 4E Ch. H - Prob. 5E Ch. H - Prob. 6E Ch. H - Prob. 7E Ch. H - Prob. 8E Ch. H - Prob. 9E Ch. H - Prob. 10E

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