Question

How to divide continued fractions in abbreviated notation?

Answer 1

Sorry, but there is no known simple way to do the basic 4 operations with continued fractions.  They are very useful for revealing approximation properties of real numbers, but they are a *terrible* way to represent numbers for the purpose of making computations with multiple numbers.

If you look online then you can find out about Gosper's algorithm, but I do not believe this is the kind of "nice" process you're hoping exists.