Haven't got a 'mnemonic' as such but this is what I do:
A x B = B x A etc
At rush hour people from town A get in their cars and drive to work in town B. And people in town B get in their cars and drive to work in town A. At the end of the day, they all swap back. I believe that's why they are called commuters; because they swap over.
If you've got three numbers to multiply / add / divide etc, which ones do you associate together first?
A # B # C ?
Is it A # B, get the answer and then do (answer # C)
Or is it B # C, get the answer and then do (A # answer).
So it's all to do with the ones you associate together first.
# is an associative operation if
(A # B) # C = A # (B # C)
A # (B @ C) where # and @ are different operations you can do with 2 numbers.
So A has to be distributed to the contents of the bracket; B gets it and C gets it:
(A # B) @ ( A # C)
You are probably learning which operations are commutative, associative and distributive over another operation.
What is correct in each case is much more important than remembering the correct word, so I shouldn't worry too much about them.
Hope that helps,