Find $(1+i)^i$ in simpler terms, without imaginary exponents.

Question

I was asked to find $(1+i)^i$, I don't know what to do when there is an imaginary component in the exponent.

since $1+i=\sqrt{2}e^{-\frac{1}{4}i \pi}$ then $(1+i)^i = \sqrt{2}^i e^{\frac{1}{4} \pi}$ but now we run into the same problem again, what is $\sqrt{2}^i$?

Answer 1

If $x$ is any positive number then $x^i = e^{i\ln x} = \cos(\ln x) + i\sin(\ln x)$.