Background : I was reviewing some practice problems for a small local math competition, and I don't understand how the give solution works.
I don't know how this, $$9^{x+2} - 9^x=240$$ is the same as this $$9^x(81-1)=240$$
I don't know if there is something simple I am missing, or if there is a property I have never learned that is being used in this solution. I have looked around for a solution but haven't found one that fits my needs, or at least I haven't figured out how to apply it to this problem.
Will Jagy is correct, but to expand just a little bit take what he said and then let $9^x=a$. If we do that we get
$$81a-a=240$$.
From here we could just take one $a$ away and express $80a=240$, but the authors chose to express it as $a(81-1)$. If we replace $a$ with $9^x$ we have
$$9^x(81-1) = 240$$.
I hope that helps anyone else looking at the thread.