Quaternion ^ Quaternion I was looking at Quaternions at Wikipedia - I was trying to find the value of $i^j$ etc...
Wikipedia lists $q^\alpha$ where $\alpha$ is real, but I can't find the value of $i^j$. Any clues?
The answer given here doesn't give an explicit solution.
 A: Expressing $i$ as $e^{i\frac\pi 2+2ni\pi}$ and defining $(e^a)^b=e^{ab}$, we can then write
$$i^j=e^{(i\frac\pi 2+2ni\pi)j}=e^{ij\frac\pi 2+2nij\pi}\\
=e^{k\frac\pi 2+2nk\pi}=k$$
This secondary translation follows directly from the linked definition of exponentiation as a sum.
Following further commented discussion, we can then do some manipulations:
$$i^j=k\implies i^j-ij=0\\
i(i^{j-1}-j)=0\implies i^{j-1}=j\\
i^{2j-1}=i^{j-1}i^j=jk=i$$
This clearly falls flat, as $i^{2j-1}=i^ji^{j-1}=kj=-i$ as well.  Therefore, the simple derivation used above together with assuming that we can define $(e^a)^b=e^{ab}$ fails to produce a consistent model, and cannot be relied upon.
A: Wecan define the exponential of a quaternion $e^q$ , $q \in \mathbb{H}$, see: Exponential Function of Quaternion - Derivation. Than we can use such definition for define $p^q$ , $q,p \in \mathbb{H}$ as $p^q=e^{(\log p)q}$, with some attention to the definition of $\log p$ that is a multivalued function. See the logarithm of quaternion.

Added after the comments.
For a quaternion $q$ theexponential $e^q$ is a well defined quaternion, so also $(e^q)^p$ is well defined, but, in general $(e^q)^p \ne (e^p)^q$ . this is not a strange result since $\mathbb{H}$ is a non commutative ring and also other properies of the exponential function for a field are not valid in a ring.
We know that every quaternion $z=a+ib+jc+kd=a+\mathbf{v}$ can be write in polar form as:$ z=|z| e^ {\mathbf{x} \theta} = |z|\left(\cos \theta +\mathbf{n}\sin \theta \right)= e^{\log |z| + \mathbf{x} \theta}$, where :
$$
\begin{split}
& |z|= \sqrt{a^2+b^2+c^2+d^2}\\
& \cos \theta=\dfrac{a}{|z|} \qquad \sin \theta=\dfrac{|\mathbf{v}|}{|z|}\\
& \mathbf{x}=\dfrac{\mathbf{v}}{|z|\sin \theta}
\end{split}
$$
so we can define a ''principal'' logarithm as:
$$
\log z= \log |z| + \theta \mathbf{x} 
$$
and we have $z=e^{\log z}$
Now it seems to me that we can well define: $q^p = \left(e^{\log q}\right)^p=e^{p\log |z|}e^{\theta\, \mathbf{x}\,p}$
The ''problem'' is that $e^{\theta\, \mathbf{x}\,p} \ne e^{\theta\,p \mathbf{x}}$. But is this really a problem or simply a property of quaternon exponential?
