Explanation Given: The provided number is 32 . Formula used: De Moivre’s theorem: r = z k n z k = r n [ cos ( θ + 2 π k n ) + i sin ( θ + 2 π k n ) ] Calculation: Roots of a complex number represented in polar form can be found by the application of De Moivre’s theorem. De Moivre’s theorem for finding complex roots states that every complex number has two distinct complex square roots, three distinct complex cube roots, four distinct complex fourth roots, and so on. Each root has the same modulus. r = z k n z k = r n [ cos ( θ + 2 π k n ) + i sin ( θ + 2 π k n ) ] where k is the number of distinct n th root and θ is in radians. There are exactly five fifth roots of 32 as per De Moivre’s theorem. First write the provided number in rectangular form as follows: 32 = 32 + 0 i Convert the above expression into polar form as follows: 32 = r ( cos θ + i sin θ ) = 32 ( cos 0 + i sin 0 ) Apply De Moivre’s theorem to find all fifth roots of 32 . z k = 32 5 [ cos ( 0 + 2 π k 5 ) + i sin ( 0 + 2 π k 5 ) ] Here, k = 0 , 1 , 2 , 3 , 4 . Insert the above values of k to find five distinct fifth roots of the provided number. For k = 0 , z 0 = 32 5 [ cos ( 0 + 2 π ( 0 ) 5 ) + i sin ( 0 + 2 π ( 0 ) 5 ) ] = 2 [ cos 0 + i sin 0 ] = 2 For k = 1 , z 1 = 32 5 [ cos ( 0 + 2 π ( 1 ) 5 ) + i sin ( 0 + 2 π ( 1 ) 5 ) ] = 2 [ cos ( 2 π 5 ) + i sin ( 2 π 5 ) ] Substitute the values of cos ( 2 π 5 ) and sin ( 2 π 5 ) : z 1 = 2 [ 0.3090 + i 0.9511 ] = 0

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Chapter 7.5, Problem 71E

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To calculate: Complex fifth roots of 32 in rectangular form.

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Chapter 7.5, Problem 70E

Chapter 7.5, Problem 72E

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