Exponential manipulation: is $8^x-2^x=(8 - 2)^x$ valid?

Question

Given: $8^x-2^x$. I saw someone manipulate this as $(8 - 2)^x$. I wasn't sure if this was correct.

If it is, what exponential rule exactly allows one to do this?

I understand $(8^x)/(2^x) = (8/2)^x$, but not the first technique. Thanks.

Answer 1

I always like to try an example or two to see whether or not a statement is true. Does $$8^x-2^x=(8-2)^x$$ when $x=2$?

$$ \begin{align*} \color{#0a0}{8^x-2^x}&=\color{#c00}{(8-2)^x}\\ \color{#0a0}{8^2-2^2}&=\color{#c00}{(8-2)^2}\\ \color{#0a0}{64-4}&=\color{#c00}{6^2}\\ \color{#0a0}{60}&\ne \color{#c00}{36} \end{align*} $$ So, we know the statement is certainly not true in general.