1. **Stating the problem:** Solve the equation $$z^{50} - 1 = 0$$ where $$z$$ is a complex number. 2. **Formula and important rules:** The equation $$z^n = 1$$ has $$n$$ complex roots called the nth roots of unity. They are given by: $$z_k = \cos\left(\frac{2\pi k}{n}\right) + i \sin\left(\frac{2\pi k}{n}\right) = e^{i \frac{2\pi k}{n}}$$ for $$k = 0, 1, 2, \ldots, n-1$$. 3. **Applying the formula:** Here, $$n=50$$, so the roots are: $$z_k = e^{i \frac{2\pi k}{50}}$$ for $$k=0,1,2,\ldots,49$$. 4. **Explanation:** These roots lie on the unit circle in the complex plane, equally spaced by an angle of $$\frac{2\pi}{50}$$ radians. 5. **Final answer:** The 50 solutions to $$z^{50} - 1 = 0$$ are: $$\boxed{z_k = e^{i \frac{2\pi k}{50}} \text{ for } k=0,1,\ldots,49}$$ This means all complex numbers on the unit circle at these angles satisfy the equation.