# Why are exponents not associative?

I ran into something that seemed odd to me today: exponents are not associative. The following equation sums that up:

$$10 * 2^{5x} \not\equiv 20^{5x}$$

Why is this the case? Is there some way to combine the "10" and the "2"?

Goal: The reason I ask this question is I am interested in reducing the amount of operations a program will have to perform in order to compute the answer.

$$a^x\cdot b^x= (a\cdot b)^x$$ for positive real $a,b$ and real $x$
but $$a\cdot b^x\ne (a\cdot b)^x$$ in general
In fact if finite $a\cdot b\ne0$, $$a\cdot b^x= (a\cdot b)^x\implies a^{x-1}=1$$
$\displaystyle\implies$ either $a=1,$ or $x-1=0$ or $a=-1,x-1$ is even