Consider the following. fifth roots of \( 4(1-i) \) (a) Find the roots of the complex number. (Enter your answers in trigonometric form. Let \( 0 \leq \theta<2 \pi \).)

To find the fifth roots of the complex number \( 4(1-i) \), we first need to express it in polar (or trigonometric) form. 1. **Magnitude**: \[ |4(1-i)| = 4|1-i| = 4\sqrt{1^2 + (-1)^2} = 4\sqrt{2} \] 2. **Argument**: The argument \( \theta \) of \( 1-i \) can be calculated as: \[ \theta = \tan^{-1}\left(\frac{-1}{1}\right) = \tan^{-1}(-1) = -\frac{\pi}{4} \] However, to express it within \( 0 \leq \theta < 2\pi \): \[ \text{Adjusted } \theta = 2\pi - \frac{\pi}{4} = \frac{7\pi}{4} \] Putting it together, we have the polar form: \[ 4(1-i) = 4\sqrt{2} \left( \cos\left(\frac{7\pi}{4}\right) + i \sin\left(\frac{7\pi}{4}\right) \right) \] Now to find the fifth roots, we apply the formula for roots of complex numbers: \[ z_k = r^{1/n} \left( \cos\left(\frac{\theta + 2k\pi}{n}\right) + i \sin\left(\frac{\theta + 2k\pi}{n}\right) \right) \] where \( r = 4\sqrt{2} \), \( \theta = \frac{7\pi}{4} \), \( n = 5 \), and \( k = 0, 1, 2, 3, 4 \). 1. Calculate \( r^{1/5} \): \[ r^{1/5} = (4\sqrt{2})^{1/5} = 2^{3/5} \cdot 2^{1/10} = 2^{\frac{6}{10}} = 2^{\frac{3}{5}} \] 2. Now calculate the roots \( z_k \), for \( k = 0, 1, 2, 3, 4 \): \[ z_k = 2^{\frac{3}{5}} \left( \cos\left(\frac{\frac{7\pi}{4} + 2k\pi}{5}\right) + i \sin\left(\frac{\frac{7\pi}{4} + 2k\pi}{5}\right) \right) \] Calculating \( z_k \) for each \( k \): - For \( k=0 \): \[ z_0 = 2^{\frac{3}{5}} \left( \cos\left(\frac{7\pi/4}{5}\right) + i \sin\left(\frac{7\pi/4}{5}\right) \right) \] - For \( k=1 \): \[ z_1 = 2^{\frac{3}{5}} \left( \cos\left(\frac{7\pi/4 + 2\pi}{5}\right) + i \sin\left(\frac{7\pi/4 + 2\pi}{5}\right) \right) \] - For \( k=2 \): \[ z_2 = 2^{\frac{3}{5}} \left( \cos\left(\frac{7\pi/4 + 4\pi}{5}\right) + i \sin\left(\frac{7\pi/4 + 4\pi}{5}\right) \right) \] - For \( k=3 \): \[ z_3 = 2^{\frac{3}{5}} \left( \cos\left(\frac{7\pi/4 + 6\pi}{5}\right) + i \sin\left(\frac{7\pi/4 + 6\pi}{5}\right) \right) \] - For \( k=4 \): \[ z_4 = 2^{\frac{3}{5}} \left( \cos\left(\frac{7\pi/4 + 8\pi}{5}\right) + i \sin\left(\frac{7\pi/4 + 8\pi}{5}\right) \right) \