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 , 1 , 2 , ... 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 z ) . Calculation: Compute r and θ for the complex number z = 2 + 2 i as follows. r = | 2 + 2 i | = ( 2 ) 2 + ( 2 ) 2 = 4 + 4 = 8 That is, r = 2 2 . θ = tan − 1 ( 2 2 ) = tan − 1 ( 1 ) = π 4 The complex number lies in the first quadrant as both the x axis and y axis are positive, thus the angle is π 4 . Thus, the polar form of z = 2 + 2 i becomes, z = ( 2 2 ) e i ( π 4 ) = 2 2 e i π 4 Obtain the value of expression ( 2 + 2 i ) 1 3 as follows

Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)

3rd Edition

ISBN: 9780134770512

Author: William L. Briggs, Lyle Cochran, Bernard Gillett, Eric Schulz

Publisher: PEARSON

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Chapter C, Problem 52E

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To compute: The roots of the quantity (2+2i)13.

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Chapter C, Problem 51E

Chapter C, Problem 53E

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Calculus: Early Transcendentals, Books A La Carte Edition (3rd Edition)

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 + 2 i ) 1 3 .

Solution Summary: The author explains the DeMoivre Theorem, which states that if z=x+iy, then the n th roots of are z1n