1. **Problem Statement:** Convert the complex number $-5 + 5i$ into polar and exponential form. 2. **Formula and Rules:** - Polar form: $r(\cos \theta + i \sin \theta)$ where $r = \sqrt{x^2 + y^2}$ and $\theta = \tan^{-1}(\frac{y}{x})$. - Exponential form: $re^{i\theta}$. - Note: Adjust $\theta$ based on the quadrant of the complex number. 3. **Calculate $r$:** $$r = \sqrt{(-5)^2 + 5^2} = \sqrt{25 + 25} = \sqrt{50} = 5\sqrt{2}$$ 4. **Calculate $\theta$:** $$\theta = \tan^{-1}\left(\frac{5}{-5}\right) = \tan^{-1}(-1)$$ Since the point $(-5,5)$ is in the second quadrant, $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$. 5. **Polar form:** $$5\sqrt{2} \left(\cos \frac{3\pi}{4} + i \sin \frac{3\pi}{4}\right)$$ 6. **Exponential form:** $$5\sqrt{2} e^{i \frac{3\pi}{4}}$$