Real Numbers Raised to Imaginary Powers?

The basic formula is $e^{i\theta}=\cos\theta+i\sin\theta$, so your example would be

$$3^i=e^{i\ln3}=\cos(\ln3)+i\sin(\ln3)$$

Using euler's formula :

$$c^{a+bi}=c^ac^{bi}=c^ae^{bi \ln (c)}$$

$$=c^a \left((\cos (b \ln (c))+ i \sin(b \ln (c) \right))$$