Explanation Theorem used: DeMoivre Theorem: It states that if z = r e i θ , then the n th roots of z are z 1 n = r 1 n e i ( θ + 2 π k ) n or r 1 n ( cos ( θ + 2 k π n ) + i sin ( θ + 2 k π n ) ) , for k = 0 , 12 , ... n . Formula used: The polar form of z = x + i y is z = r e i θ , where r = x 2 + y 2 and θ = tan − 1 ( y x ) . Suppose α = tan − 1 | y x | argument of a complex number z = x + i y is given as, z lies in first quadrant θ = α z lies in second quadrant θ = π − α z lies in third quadrant θ = − π + α z lies in fourth quadrant θ = − α Table 1 Calculation: Compute r and θ for the complex number z = 2 i as follows. r = | 2 i | = ( 0 ) 2 + ( 2 ) 2 = 4 = 2 θ = tan − 1 ( 2 0 ) = tan − 1 ( ∞ ) = π 2 The complex number lies on the positive y -axis, thus from Table 1 the angle remains as π 2 . Thus, the polar form of z = 2 i becomes, z = ( 2 ) e i ( π 2 ) = 2 e i π 2 Obtain the value of expression ( 2 i ) 1 3 as follows

Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

3rd Edition

ISBN: 9780136207764

Author: Briggs, William, Cochran, Lyle, Gillett, Bernard, SCHULZ, Eric

Publisher: PEARSON

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Chapter C, Problem 48E

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Chapter C, Problem 47E

Chapter C, Problem 49E

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Calculus, Early Transcendentals, Single Variable Loose-Leaf Edition Plus MyLab Math with Pearson eText - 18-Week Access Card Package

Ch. C - Prob. 11E Ch. C - Prob. 12E Ch. C - Prob. 13E Ch. C - Prob. 14E Ch. C - Prob. 15E Ch. C - Prob. 16E Ch. C - Prob. 17E Ch. C - Prob. 18E Ch. C - Prob. 19E Ch. C - Prob. 20E Ch. C - Eulers formula Evaluate the following expressions....Ch. C - Eulers formula Evaluate the following expressions....Ch. C - Prob. 23E Ch. C - Eulers formula Evaluate the following expressions....Ch. C - Eulers formula Evaluate the following expressions....Ch. C - Prob. 26E Ch. C - Prob. 27E Ch. C - Prob. 28E Ch. C - Prob. 29E Ch. C - Prob. 30E Ch. C - Prob. 31E Ch. C - Prob. 32E Ch. C - Prob. 33E Ch. C - Prob. 34E Ch. C - Prob. 35E Ch. C - Prob. 36E Ch. C - Prob. 37E Ch. C - Prob. 38E Ch. C - Prob. 39E Ch. C - Prob. 40E Ch. C - Prob. 41E Ch. C - Prob. 42E Ch. C - Prob. 43E Ch. C - Prob. 44E Ch. C - Prob. 45E Ch. C - Prob. 46E Ch. C - Prob. 47E Ch. C - Prob. 48E Ch. C - Explorations and Challenges Evaluating roots...Ch. C - Prob. 50E Ch. C - Prob. 51E Ch. C - Prob. 52E Ch. C - Prob. 53E Ch. C - Prob. 54E Ch. C - Prob. 55E Ch. C - Prob. 56E Ch. C - Solving polynomial equations Find all roots of the...Ch. C - Solving polynomial equations Find all roots of the...Ch. C - Prob. 59E Ch. C - Prob. 60E Ch. C - Prob. 61E

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The roots of the quantity ( 2 i ) 1 3 .

Solution Summary: The author explains the DeMoivre Theorem, which states that if z=reitheta , then the n th roots of are