1. **State the problem:** Convert the complex number $z = \left(\frac{\sqrt{2}}{2} + \sqrt{2}i\right)^8$ into its Cartesian form (a + bi). 2. **Rewrite the complex number:** Note that $\frac{\sqrt{2}}{2} = \cos\frac{\pi}{4}$ and $\sqrt{2} = \sqrt{2} \sin\frac{\pi}{4} \times 2$, but since the imaginary part is $\sqrt{2}i$, let's check the magnitude and argument. 3. **Find magnitude $r$:** $$r = \sqrt{\left(\frac{\sqrt{2}}{2}\right)^2 + (\sqrt{2})^2} = \sqrt{\frac{2}{4} + 2} = \sqrt{\frac{1}{2} + 2} = \sqrt{\frac{5}{2}} = \frac{\sqrt{10}}{2}$$ 4. **Find argument $\theta$:** $$\theta = \tan^{-1}\left(\frac{\sqrt{2}}{\frac{\sqrt{2}}{2}}\right) = \tan^{-1}(2)$$ 5. **Express in polar form:** $$z = r(\cos\theta + i\sin\theta) = \frac{\sqrt{10}}{2}(\cos(\tan^{-1}(2)) + i\sin(\tan^{-1}(2)))$$ 6. **Raise to the 8th power using De Moivre's theorem:** $$z^8 = r^8 (\cos(8\theta) + i\sin(8\theta)) = \left(\frac{\sqrt{10}}{2}\right)^8 (\cos(8\tan^{-1}(2)) + i\sin(8\tan^{-1}(2)))$$ 7. **Calculate $r^8$:** $$r^8 = \left(\frac{\sqrt{10}}{2}\right)^8 = \left(\frac{10^{1/2}}{2}\right)^8 = \frac{10^4}{2^8} = \frac{10000}{256} = \frac{625}{16}$$ 8. **Calculate $8\theta$:** $$8\theta = 8 \times \tan^{-1}(2)$$ 9. **Evaluate $\cos(8\theta)$ and $\sin(8\theta)$:** Since $\tan^{-1}(2) \approx 1.1071487$ radians, $$8\theta \approx 8.85719$$ Using periodicity: $$8\theta - 2\pi \approx 8.85719 - 6.28319 = 2.574$$ Calculate: $$\cos(2.574) \approx -0.84147$$ $$\sin(2.574) \approx 0.54030$$ 10. **Write Cartesian form:** $$z^8 = \frac{625}{16}(-0.84147 + 0.54030i) = -32.85 + 21.14i$$ **Final answer:** $$z^8 \approx -32.85 + 21.14i$$