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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.

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HINT:

$$a^x\cdot b^x= (a\cdot b)^x$$ 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

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