TITLE: Simplify $x ^ y + x ^ z$ to a formula with only one $x$
QUESTION [5 upvotes]: Is there a way to simplify $x ^ y + x ^ z$ to a formula with only one $x$?
I know $(x ^ y)(x ^ z) = (x ^ {y + z})$, but how can it change in addition?

REPLY [2 votes]: Addition doesn't behave nicely with exponentiation, like your multiplication example shows. If you want a single operation, you can always define $\text{expadd}_{y,z}(x):=x^y+x^z$, then you have a single formula:
$$\text{expadd}_{y,z}(x)$$
Somehow, I doubt that's what you wanted.