Explanation Given information: x 3 + 8 = 0 Concept Involved: Every real (or complex) number has exactly n distinct nth roots, a surprising and attractive fact about numbers. The key to finding these roots is trigonometric form for complex numbers . Suppose that z and w are complex numbers such that w is an nth root of z. That is, w = z n If we raise both sides of this last equation to nth power, we have w n = z . Now suppose z = r ( cos θ + i sin θ ) and z = s ( cos α + i sin α ) . Substituting these expressions into the equation w n = z , we have [ s ( cos α + i sin α ) ] n = r ( cos θ + i sin θ ) We can rewrite the left side of this last equation de Moivre’s Theorem. s n ( cos n α + i sin n α ) = r ( cos θ + i sin θ ) The only way these two expressions can be equal is if their moduli are equal and their arguments are coterminal. To summarize, we find the nth roots of a complex number by first finding the real nth root of the modulus. The angle for the first root will be θ / n , and the angles for the remaining roots are found by adding multiples of 360 ° / n The values of k will range from 0 to n - 1, because when k = n we would be adding 360 ° n ⋅ n = 360 ° which would be coterminal with the first angle. Theorem (Roots) The nth roots of the complex number z = r ( cos θ + i sin θ ) = r c i s θ are given by w k = r 1 / n [ cos ( θ n + 360 ° n k ) + i sin ( θ n + 360 ° n k ) ] = r 1 / n c i s ( θ n + 360 ° n k ) where k = 0 , 1 , 2 , ... , n − 1. The root w 0 , corresponding to k = 0 , is called the principal nth root of z. Calculation: x 3 + 8 = 0 x 3 + 8 − 8 = 0 − 8 x 3 = − 8 The solutions to this equation are the cube roots of -2. We already know that one of the cube roots of -8 is -2. There are two other complex cube roots as well. To find them, we write -1 in trigonometric form and then apply the formula from our theorem on roots. TABLE 1 Steps Description − 8 Complex number in Standard Form x + y i ⇒ − 8 x = − 8 y = 0 Identifying x and y by comparison r = x 2 + y 2 = ( − 8 ) 2 + 0 2 = 64 = 8 Identifying x and y cos θ = x r = − 8 8 = − 1 sin θ = y r = 0 8 = 0 Identifying cos θ and sin θ

8th Edition

ISBN: 9781305652224

Author: Charles P. McKeague, Mark D. Turner

Publisher: Cengage Learning

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Solution Summary: The author explains that every real (or complex) number has exactly nth roots, a surprising and attractive fact about numbers.

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