How to re-write $(1+i)^{2022}$ to the form a+ib?

Question

I need to write $(1+i)^{2022}$ in the form of a + ib where $a,b\in\mathbb{R}$

I had a similar exercise rewriting $i^{2022}$, which i did by $i^{2022}=(i^2)^{1011}=(-1)^{1011}=-1$

But this method doesn't work for $(1+i)^{2022}$

Hope someone can help me out.

Answer 1

You have $$(1+i)^{2022}=2^{\frac{2022}{2}}e^{\frac{i\pi 2022}{4}}=2^{1011}i^{1011}=2^{1011}i^3=-2^{1011} i$$