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How can you get $10^{n+1}$ from $9\cdot 10^n+10^n$? This is part of a proof I am working on.

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  • $\begingroup$ A basket has $9$ apples. We add an apple. $\endgroup$ – André Nicolas Mar 1 '12 at 22:50
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    $\begingroup$ Hint $\ $ Multiply $\: 10\ =\ 9 + 1\: $ by $\:10^n$ $\endgroup$ – Bill Dubuque Mar 1 '12 at 22:57
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Start with the distributive law.

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  • $\begingroup$ Wow I totally did not think of that. I thought it would be much more difficult. Thanks! $\endgroup$ – Jared Mar 1 '12 at 22:40
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$$\begin{aligned} 9 \cdot 10^n + 10^n &= (10-1) \cdot 10^n + 10^n \\ &= 10 \cdot 10^n - 10^n + 10^n \\ &= 10^{n+1} \end{aligned}$$

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  • $\begingroup$ Another way is $9 \cdot 10^n + 10^n = (9 + 1) \cdot 10^n = 10 \cdot 10^n = 10^{n+1}$. $\endgroup$ – Ilmari Karonen Mar 2 '12 at 1:11

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