Look at this equation: \[ g^{3}=1 \] What is \( g \), the cube root of 1 ? \( g=\square \)

The equation \( g^3 = 1 \) has three distinct cube roots of unity in the complex number system. The primary root is \( g = 1 \), but the other two are found using Euler's formula: \( g = e^{2\pi i / 3} \) and \( g = e^{4\pi i / 3} \), which can be simplified to \( g = -\frac{1}{2} + \frac{\sqrt{3}}{2}i \) and \( g = -\frac{1}{2} - \frac{\sqrt{3}}{2}i \), respectively. So, \( g = 1, \; -\frac{1}{2} + \frac{\sqrt{3}}{2}i, \; -\frac{1}{2} - \frac{\sqrt{3}}{2}i \). In practical terms, the cube roots of 1 play a crucial role in various fields, including engineering and physics. For example, they are used in analyzing alternating current circuits or signals, where the phase shifts can be represented using complex numbers. Understanding these roots helps in comprehending how alternating currents interact with resistors, capacitors, and inductors in real-world applications!