When trying to explain multiplying variables with exponents, the students could not remember whether to add or multiply the exponents. I could understand their frustration. So I used the Brady Bunch as an example.
I told them that in the Land of Quadrants (my imaginery algebraic world), the variables with exponents were like match.com. People didn't use their real names, just user names. They went searching for a match. But, in my land, you were required by law to only make a match with someone who had the same initials as yourself.
In the example:
c 3 x d5
this couple would not be a match. A Carol, Cathi,or Cindy with three kids could not marry a David, Doug, or Dilbert with five kids because you can not multiply unlike variables.
In the Land of Quadrants a match could be made with
b 2 x b4 = b6
A Barbara, Brenda, or Bethany with twi kids can match with a Billy, Bob, or Bud with four kids. But, the law requires them, in my imaginary land, to take on a new name with the same initials-- such as Brady. In this example, the Brady's combined or added their kids together (exponents) to make a new family of six kids.
It is important to remember, that unlike variables (letters) can not be multiplied together in algebra. This rule comes in handy when distributing and simplifying complex expressions.