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

8th Edition

ISBN: 9780357019788

Author: Stewart

Publisher: CENGAGE L

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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

CALCULUS: EARLY TRANS. (LL)W/WEBASSIGN

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 - Evaluate the expression and write your answer in...Ch. H - Prob. 9E Ch. H - Prob. 10E

Ch. H - Prob. 11E Ch. H - Prob. 12E Ch. H - Prob. 13E Ch. H - Evaluate the expression and write your answer in...Ch. H - Prob. 15E Ch. H - Prob. 16E Ch. H - Prob. 17E Ch. H - Prove the following properties of complex numbers....Ch. H - Prob. 19E Ch. H - Prob. 20E Ch. H - Prob. 21E Ch. H - Find all solutions of the equation. 22. 2x2 2x +...Ch. H - Prob. 23E Ch. H - Find all solutions of the equation. 24....Ch. H - Write the number in polar form with argument...Ch. H - Prob. 26E Ch. H - Prob. 27E Ch. H - Prob. 28E Ch. H - Write the number in polar form with argument...Ch. H - Prob. 30E Ch. H - Prob. 31E Ch. H - Prob. 32E Ch. H - Find the indicated power using De Moivres Theorem....Ch. H - Prob. 34E Ch. H - Prob. 35E Ch. H - Prob. 36E Ch. H - Prob. 37E Ch. H - Prob. 38E Ch. H - Prob. 39E Ch. H - Prob. 40E Ch. H - Prob. 41E Ch. H - Prob. 42E Ch. H - Prob. 43E Ch. H - Prob. 44E Ch. H - Prob. 45E Ch. H - Prob. 46E Ch. H - Prob. 47E Ch. H - Use Eulers formula to prove the following formulas...Ch. H - If u(x) = f(x) + ig(x) is a complex-valued...Ch. H - (a) If u is a complex-valued function of a real...

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The imaginary number i has the property that i2 = ___________________ .

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