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 = − 4 as follows. r = | − 4 | = ( − 4 ) 2 + ( 0 ) 2 = 16 = 4 θ = tan − 1 ( 0 − 4 ) = tan − 1 ( 0 ) = 0 The complex number lies on the negative x -axis, thus from Table 1 the angle becomes 0 + π = π . Thus, the polar form of z = − 4 becomes, z = ( 4 ) e i ( π ) = 4 e i π Obtain the value of expression ( − 4 ) 1 2 as follows

Calculus: Early Transcendentals (3rd Edition)

3rd Edition

ISBN: 9780134763644

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

Publisher: PEARSON

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Chapter C, Problem 47E

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To compute: The roots of the quantity (−4)12.

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Chapter C, Problem 46E

Chapter C, Problem 48E

Chapter C Solutions

Calculus: Early Transcendentals (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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