(a) Explanation Calculation: Consider the complex number, − 1024 . Calculate the modulus of z with the use of formula, z = x 2 + y 2 , as shown below, z = − 1024 2 + 0 2 = 1024 2 = 1024 Since r = z , r = 1024 . Here, y is zero and x is negative, that means, − 1024 lies in the second quadrant Now, calculate for θ with the use of the formula, θ = π + arctan y x , as shown below, θ = π + arctan 0 − 1024 = π + arctan tan 0 = π + 0 = π So, the trigonometric form of z is 1024 cos π + i sin π . Use the formula of n th root of a complex number to calculate the sixth root of the considered complex number. If n is a positive integer and complex number z = r cos θ + i sin θ has exactly n roots, then n th root is given by, z k = r n cos θ + 2 π k n + i sin θ + 2 π k n where, k = 0 , 1 , 2 , … , n − 1 . Substitute 1024 for r , π for θ and 5 for n in the above formula, z k = 1024 5 cos π + 2 π k 5 + i sin π + 2 π k 5 = 4 cos π + 2 π k 5 + i sin π + 2 π k 5 Thus, z k = 4 cos π + 2 π k 5 + i sin π + 2 π k 5 (b) To determine To calculate: The standard form of fifth roots of complex number, − 1024 . (c) To determine To graph: The fifth roots of complex number − 1024 .

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ISBN: 8220103611893

Author: Larson

Publisher: CENGAGE L

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Chapter 4, Problem 100RE

(a)

To determine

To calculate: The fifth roots of complex number, −1024.

(b)

To determine

To calculate: The standard form of fifth roots of complex number, −1024.

(c)

To determine

To graph: The fifth roots of complex number −1024.

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Chapter 4, Problem 99RE

Chapter 4, Problem 101RE

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The fifth roots of complex number , − 1024 .

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To calculate: The fifth roots of complex number, −1024.

To calculate: The standard form of fifth roots of complex number, −1024.

To graph: The fifth roots of complex number −1024.

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