1. **State the problem:** Calculate the value of $(-1+i)^{2i}$ where $i$ is the imaginary unit. 2. **Recall the formula:** For a complex number $z$ and a complex exponent $w$, we use the formula: $$z^w = e^{w \log z}$$ where $\log z$ is the complex logarithm of $z$. 3. **Find the polar form of $z = -1 + i$:** - Calculate the modulus: $$r = |z| = \sqrt{(-1)^2 + 1^2} = \sqrt{1 + 1} = \sqrt{2}$$ - Calculate the argument (angle): $$\theta = \arctan\left(\frac{1}{-1}\right) = \arctan(-1)$$ Since the point $(-1,1)$ is in the second quadrant, the angle is: $$\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$$ 4. **Express $z$ in polar form:** $$z = r e^{i\theta} = \sqrt{2} e^{i \frac{3\pi}{4}}$$ 5. **Calculate $z^{2i}$:** $$(-1+i)^{2i} = \left(\sqrt{2} e^{i \frac{3\pi}{4}}\right)^{2i} = e^{2i \log \left(\sqrt{2} e^{i \frac{3\pi}{4}}\right)}$$ 6. **Calculate the complex logarithm:** $$\log \left(\sqrt{2} e^{i \frac{3\pi}{4}}\right) = \log \sqrt{2} + i \frac{3\pi}{4} = \frac{1}{2} \log 2 + i \frac{3\pi}{4}$$ 7. **Substitute back:** $$e^{2i \left( \frac{1}{2} \log 2 + i \frac{3\pi}{4} \right)} = e^{2i \cdot \frac{1}{2} \log 2 + 2i \cdot i \frac{3\pi}{4}} = e^{i \log 2 - 2 \cdot \frac{3\pi}{4}} = e^{i \log 2 - \frac{3\pi}{2}}$$ 8. **Separate real and imaginary parts in the exponent:** $$e^{i \log 2 - \frac{3\pi}{2}} = e^{- \frac{3\pi}{2}} \cdot e^{i \log 2}$$ 9. **Express $e^{i \log 2}$ using Euler's formula:** $$e^{i \log 2} = \cos(\log 2) + i \sin(\log 2)$$ 10. **Final answer:** $$(-1+i)^{2i} = e^{- \frac{3\pi}{2}} \left( \cos(\log 2) + i \sin(\log 2) \right)$$ This is the exact value in rectangular form.