By writing Z in polar form, we will see that:
![z^3 = C](https://tex.z-dn.net/?f=z%5E3%20%3D%20C)
<h3>
How to get the value of Z³?</h3>
First, we can see that:
z = (-1 - i)
If we write it in polar form, we get:
![z = \sqrt{2} *e^{i(\pi + \pi/4)](https://tex.z-dn.net/?f=z%20%3D%20%5Csqrt%7B2%7D%20%2Ae%5E%7Bi%28%5Cpi%20%2B%20%5Cpi%2F4%29)
If we apply the power 3, we get:
![z^3 = (\sqrt{2} *e^{i(\pi + \pi/4)})^3\\\\z^3 = 2^{3/2}*e^{i*(3\pi + 3\pi/4)}\\\\z^3 = 2^{3/2}*e^{i*15\pi/4}](https://tex.z-dn.net/?f=z%5E3%20%3D%20%28%5Csqrt%7B2%7D%20%2Ae%5E%7Bi%28%5Cpi%20%2B%20%5Cpi%2F4%29%7D%29%5E3%5C%5C%5C%5Cz%5E3%20%3D%202%5E%7B3%2F2%7D%2Ae%5E%7Bi%2A%283%5Cpi%20%2B%203%5Cpi%2F4%29%7D%5C%5C%5C%5Cz%5E3%20%3D%202%5E%7B3%2F2%7D%2Ae%5E%7Bi%2A15%5Cpi%2F4%7D)
Notice that:
![\frac{15\pi}{4}](https://tex.z-dn.net/?f=%5Cfrac%7B15%5Cpi%7D%7B4%7D)
Is an angle equivalent to:
![\frac{15\pi}{4} - 2\pi = \frac{15\pi}{4} - \frac{8\pi}{4} = \frac{7\pi}{4} = 1.75\pi](https://tex.z-dn.net/?f=%5Cfrac%7B15%5Cpi%7D%7B4%7D%20-%202%5Cpi%20%3D%20%5Cfrac%7B15%5Cpi%7D%7B4%7D%20-%20%5Cfrac%7B8%5Cpi%7D%7B4%7D%20%3D%20%5Cfrac%7B7%5Cpi%7D%7B4%7D%20%3D%201.75%5Cpi)
Because the angle is measured from the positive x-axis, this means that we will have:
![z^3 = C](https://tex.z-dn.net/?f=z%5E3%20%3D%20C)
If you want to learn more about complex numbers:
brainly.com/question/10662770
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