A complex number is defined by \( z=1+i \). (a) Express \( z \) in polar form. (b) Use de Moivre's theorem to evaluate \( z^{8} \).

To solve the problem, we will follow the steps outlined for each part. ### Part (a): Express \( z \) in polar form 1. **Identify the complex number**: \[ z = 1 + i \] 2. **Calculate the modulus \( r \)**: The modulus of a complex number \( z = a + bi \) is given by: \[ r = \sqrt{a^2 + b^2} \] Here, \( a = 1 \) and \( b = 1 \): \[ r = \sqrt{1^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2} \] 3. **Calculate the argument \( \theta \)**: The argument \( \theta \) is given by: \[ \theta = \tan^{-1}\left(\frac{b}{a}\right) \] Thus, \[ \theta = \tan^{-1}\left(\frac{1}{1}\right) = \tan^{-1}(1) = \frac{\pi}{4} \] 4. **Express in polar form**: The polar form of a complex number is given by: \[ z = r(\cos \theta + i \sin \theta) \] Therefore, substituting the values we found: \[ z = \sqrt{2}\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right) \] ### Part (b): Use de Moivre's theorem to evaluate \( z^{8} \) 1. **Apply de Moivre's theorem**: De Moivre's theorem states that: \[ z^n = r^n \left(\cos(n\theta) + i \sin(n\theta)\right) \] For \( n = 8 \): \[ z^8 = \left(\sqrt{2}\right)^8 \left(\cos\left(8 \cdot \frac{\pi}{4}\right) + i \sin\left(8 \cdot \frac{\pi}{4}\right)\right) \] 2. **Calculate \( \left(\sqrt{2}\right)^8 \)**: \[ \left(\sqrt{2}\right)^8 = (2^{1/2})^8 = 2^{4} = 16 \] 3. **Calculate \( 8 \cdot \frac{\pi}{4} \)**: \[ 8 \cdot \frac{\pi}{4} = 2\pi \] 4. **Evaluate \( \cos(2\pi) \) and \( \sin(2\pi) \)**: \[ \cos(2\pi) = 1, \quad \sin(2\pi) = 0 \] 5. **Combine the results**: \[ z^8 = 16 \left(1 + 0i\right) = 16 \] ### Final Answers: (a) The polar form of \( z \) is: \[ z = \sqrt{2}\left(\cos\frac{\pi}{4} + i \sin\frac{\pi}{4}\right) \] (b) The value of \( z^{8} \) is: \[ z^{8} = 16 \]