I am trying to understand quaterions division.
Imagine I have the following equation, where every member is a quaternion:
$$Q = (qq_1)(qq_2)...(qq_n)$$
I suppose that, if I maintain the order of division exactly reverse to the order of multiplication, i get (by division I mean multiplication by a reciprocal of a given $q_n$, this notion is taken from wikipedia):
$$q^n = \frac{Q}{(q_nq_{n-1}...q_1)}$$
Which leads to
$$q = \sqrt[n]{\frac{Q}{(q_nq_{n-1}...q_1)}}$$
Do these relations hold for quaternions? Will this give me back the original value of $q$ as one of the roots?
UPDATE
As Jason mentioned in the comments, there're two ways to divide: left and right. This is an important detail. Will this obscurity prevent me from receiving the original quaternion, or is it possible to pick the correct (left or right) division to actually get the correct result?