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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.

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    $\begingroup$ $9^{x+2} = 9^x \cdot 9^2 = 9^x \cdot 81$ $\endgroup$
    – Will Jagy
    Commented Jan 21, 2019 at 1:46
  • $\begingroup$ Oh wow I overthought that WAY too much. Thank you $\endgroup$
    – Bill
    Commented Jan 21, 2019 at 1:49

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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.

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  • $\begingroup$ Appreciate the clarity in this response, helped me. $\endgroup$ Commented Jan 15, 2022 at 22:38

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