Computing complex number

Question

"Compute $(1 + i)^{1000}$.

So far I have: $(1+i)^{4 (2^2 5^3)} $ but I am not sure how to proceed. Ideas?

Answer 1

HINT:

${(i+1)}^{1000} = {({(i+1)}^{2})}^{500} = {(2i)}^{500} = {2}^{500}{i}^{500} = {2}^{500}{(i^2)}^{250} = {2}^{500}{(-1)}^{250}$

Answer 2

Powers of complex numbers are often easier in polar coordinates. The absolute value of $(1+i)$ is $\sqrt{2}$, and its angle is $\pi/4$. So the question becomes $$(\sqrt{2}e^{i\pi/4})^{1000}$$

Answer 3

Here, $\exp(z)$ stands for $e^z$

$$\begin{align}(1+i)^{1000}&=(\sqrt{2}(\cos\frac{\pi}{4}+i\sin\frac{\pi}{4}))^{1000} \\ &= \sqrt2^{1000}\exp(1000\cdot\frac{\pi}{4}i)\\ &=\sqrt{2}^{1000}\exp(250\pi i) \\ &=2^{500}(\cos(250\pi)+ i\sin(250\pi)) \\ &\text{For even multiples of $\pi$, $\cos \theta =1 , \sin \theta = 0$} \\ &=2^{500}(1+0)=2^{500} \end{align}$$