1. **State the problem:** We want to express $ (1+i)^{30} $ in the form $ x + iy $ using De Moivre's theorem. 2. **Recall De Moivre's theorem:** For a complex number in polar form $ r(\cos \theta + i \sin \theta) $, its $n$th power is given by: $$ (r(\cos \theta + i \sin \theta))^n = r^n (\cos n\theta + i \sin n\theta) $$ 3. **Convert $1+i$ to polar form:** - Calculate modulus: $$ r = \sqrt{1^2 + 1^2} = \sqrt{2} $$ - Calculate argument (angle): $$ \theta = \tan^{-1}\left(\frac{1}{1}\right) = \frac{\pi}{4} $$ 4. **Apply De Moivre's theorem:** $$ (1+i)^{30} = (\sqrt{2})^{30} \left( \cos \left(30 \times \frac{\pi}{4}\right) + i \sin \left(30 \times \frac{\pi}{4}\right) \right) $$ 5. **Simplify powers and angles:** $$ (\sqrt{2})^{30} = (2^{1/2})^{30} = 2^{15} = 32768 $$ $$ 30 \times \frac{\pi}{4} = \frac{30\pi}{4} = \frac{15\pi}{2} $$ 6. **Simplify the trigonometric expressions:** Note that $ \cos \left(\frac{15\pi}{2}\right) $ and $ \sin \left(\frac{15\pi}{2}\right) $ can be simplified by subtracting multiples of $2\pi$: $$ \frac{15\pi}{2} - 4\pi = \frac{15\pi}{2} - \frac{8\pi}{2} = \frac{7\pi}{2} $$ $$ \frac{7\pi}{2} - 4\pi = \frac{7\pi}{2} - \frac{8\pi}{2} = -\frac{\pi}{2} $$ So, $$ \cos \left(\frac{15\pi}{2}\right) = \cos \left(-\frac{\pi}{2}\right) = 0 $$ $$ \sin \left(\frac{15\pi}{2}\right) = \sin \left(-\frac{\pi}{2}\right) = -1 $$ 7. **Write the final expression:** $$ (1+i)^{30} = 32768 (0 + i(-1)) = -32768 i $$ **Answer:** $$ x + iy = 0 - 32768 i $$