Simplify $x ^ y + x ^ z$ to a formula with only one $x$

Question

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?

Answer 1

You can't rewrite $x^y+x^z$ to something with only one $x$.

Assume that $x^y+x^z=x^{f(y,z)}$.

Then $$ 24=8+16=2^3+2^4=2^{f(3,4)} $$ giving us $f(3,4)=\log_2(24)=4,584\ldots$

and $$ 108=27+81=3^3+3^4=3^{f(3,4)} $$ giving us $f(3,4)=\log_3(108)=4,2618\ldots$

That's a contradction.

Answer 2

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.