Simplify the complex expression to complex numbers (2i)^i

Question

I am getting stuck with this one.... This is how far I got $$2i = 2 \exp\left(i\frac{\pi}{2} + 2k\pi\right)$$ $$(2i)^I = 2^{\frac{i}{2}}$$

I'm having trouble moving past that point

Answer 1

HINT:

Notice that:

$$i^2=-1$$ $$i^3=-i$$ $$i^4=1$$ $$i^5=i$$

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So:

$$(2i)^i=2^ii^i=2^ie^{-\frac{\pi}{2}}=\exp\left(-\frac{\pi}{2}\right)e^{i\ln(2)}$$