Find the real part and the imaginary part of the following complex number. \[ \frac{-\sqrt{11}+\sqrt{11 i}}{10} \]

To find the real part and the imaginary part of the complex number \[ \frac{-\sqrt{11}+\sqrt{11 i}}{10}, \] we first need to simplify the term \(\sqrt{11 i}\). Recall that \(i = e^{i\frac{\pi}{2}}\). Therefore, we can express \(11 i\) as: \[ 11 i = 11 e^{i\frac{\pi}{2}}. \] Taking the square root, we have: \[ \sqrt{11 i} = \sqrt{11} \sqrt{e^{i\frac{\pi}{2}}} = \sqrt{11} e^{i\frac{\pi}{4}}. \] Using Euler's formula, we can express \(e^{i\frac{\pi}{4}}\) as: \[ e^{i\frac{\pi}{4}} = \cos\left(\frac{\pi}{4}\right) + i \sin\left(\frac{\pi}{4}\right) = \frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}. \] Thus, we have: \[ \sqrt{11 i} = \sqrt{11} \left(\frac{\sqrt{2}}{2} + i \frac{\sqrt{2}}{2}\right) = \frac{\sqrt{22}}{2} + i \frac{\sqrt{22}}{2}. \] Now substituting \(\sqrt{11 i}\) back into the original expression, we get: \[ -\sqrt{11} + \sqrt{11 i} = -\sqrt{11} + \left(\frac{\sqrt{22}}{2} + i \frac{\sqrt{22}}{2}\right). \] Combining the real parts, we have: \[ -\sqrt{11} + \frac{\sqrt{22}}{2}. \] Thus, the expression becomes: \[ \frac{-\sqrt{11} + \frac{\sqrt{22}}{2} + i \frac{\sqrt{22}}{2}}{10}. \] Now we can separate the real and imaginary parts: 1. The real part is: \[ \frac{-\sqrt{11} + \frac{\sqrt{22}}{2}}{10}. \] 2. The imaginary part is: \[ \frac{\frac{\sqrt{22}}{2}}{10} = \frac{\sqrt{22}}{20}. \] Therefore, the real part and the imaginary part of the complex number are: \[ \text{Real part: } \frac{-\sqrt{11} + \frac{\sqrt{22}}{2}}{10}, \quad \text{Imaginary part: } \frac{\sqrt{22}}{20}. \]