1. **Problem statement:** Given the polar form of a complex number $Z = \sqrt{2} \left(\cos \frac{\pi}{4} + i \sin \frac{\pi}{4}\right)$, find $Z^{10}$. 2. **Formula used:** For a complex number in polar form $Z = r(\cos \theta + i \sin \theta)$, the power $Z^n$ is given by De Moivre's theorem: $$Z^n = r^n \left(\cos(n\theta) + i \sin(n\theta)\right)$$ 3. **Apply the formula:** Here, $r = \sqrt{2}$ and $\theta = \frac{\pi}{4}$, and $n=10$. Calculate $r^{10}$: $$r^{10} = (\sqrt{2})^{10} = (2^{1/2})^{10} = 2^{\frac{10}{2}} = 2^5 = 32$$ Calculate $n\theta$: $$10 \times \frac{\pi}{4} = \frac{10\pi}{4} = \frac{5\pi}{2}$$ 4. **Evaluate the trigonometric functions:** $$\cos \frac{5\pi}{2} = \cos \left(2\pi + \frac{\pi}{2}\right) = \cos \frac{\pi}{2} = 0$$ $$\sin \frac{5\pi}{2} = \sin \left(2\pi + \frac{\pi}{2}\right) = \sin \frac{\pi}{2} = 1$$ 5. **Write the final expression:** $$Z^{10} = 32 (0 + i \times 1) = 32i$$ **Final answer:** $$Z^{10} = 32i$$