Euler's identity extended into quaternions is:
$q = a + bi + cj + dk$ with a,b,c,d real numbers
for the below: $\sqrt{b^2+ c^2 + d^2} = r > 0$, and $\frac{bi+cj+dk}{r}$ = $\sqrt{-1}$,
$e^q = e^{a + r\sqrt{-1}} = e^ae^{r\sqrt{-1}} = e^a(\cos(r) + \sqrt{-1} \sin(r)) = e^a(\cos(r) + \frac{\sin(r)}{r}(bi + cj + dk))$
Therefore, what is the higher power of $[e^{a} * (cos(r) + \frac{sin(r)}{r} (bi+cj+dk)) ]^ 2$ ?
I found this site on the powers of quaternions, but it doesn't address quaternions in Euler's form.
http://www.euclideanspace.com/maths/algebra/realNormedAlgebra/quaternions/functions/power/index.htm