Raising a dual quaternion to a real number power

Question

I'm having trouble finding the power of a dual quaternion ($Q=q_r+q_dε$) raised to some real number $n$: $Q^n=(q_r+q_dε)^n=?$ It is known that for a quaternion $q=e^{\vec{u}}=\cos(|\vec{u}|)+\frac{\vec{u}}{|\vec{u}|}\sin(|\vec{u}|)$, its power to a number $n$ would come to be $q^n=e^{n\vec{u}}=\cos(n|\vec{u}|)+\frac{\vec{u}}{|\vec{u}|}\sin(n|\vec{u}|)$. Something similar happens for dual numbers $(a+bε)^n=a^n+nba^{n-1}ε$. However, this is true for numbers whose multiplication is commutative, since for the case $n=2$ we have $(a+bε)^2=a^2+(ab+ba)ε=a^2+2baε$ given that $ab=ba$. Nevertheless, this is not true for quaternions.