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Saw how it was simplified and I was wondering what allows you to rewrite this as $3^2 = 9$?

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  • $\begingroup$ To the power of a negative number, is the same as taking the inverse. So $1/3=3^{-1}$ $\endgroup$ Commented Aug 24, 2017 at 21:09
  • $\begingroup$ And the inverse function is an involution which means the inverse of the inverse takes you back to the starting value. $\endgroup$ Commented Aug 25, 2017 at 3:46

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$\frac{1}{3^{-2}}=\frac{1}{3^{-2}}\cdot 1 = \frac{1}{3^{-2}}\cdot\frac{3^2}{3^2}=\frac{1\cdot 3^2}{3^{-2}\cdot 3^2}=\frac{3^2}{3^{-2+2}}=\frac{3^2}{3^0}=\frac{3^2}{1}=3^2$

In general, you can skip many of these steps and use the result $x^n=\frac{1}{x^{-n}}$ for any nonzero $x$.

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    $\begingroup$ In general, $a^{-b}= \frac{1}{a^b}$ so $\frac{1}{a^{-b}}= \frac{1}{\frac{1}{a^b}}= a^b$ $\endgroup$
    – user247327
    Commented Aug 24, 2017 at 20:56
  • $\begingroup$ I recall trying (!) to tutor a younger cousin, who was so intent on simplifying that if we got to something like $\frac {1}{3^{-2}}\cdot \frac {3^2}{3^2}$ he would insist on simplifying it back to $\frac {1}{3^{-2}}\cdot 1.$ $\endgroup$ Commented Aug 25, 2017 at 11:41
  • $\begingroup$ To that I would respond with a metaphor that to move a chess piece like a rook from one corner of the board to the other, we can choose to go vertically before horizontally or vice versa and both are equally correct and help us accomplish our goal. In the same way, while simplifying we have multiple options. There are many times multiple ways in which we can simplify, some of them will help us get to our goal, others may take us back where we started. Keep your eye on the goal and what might help. Its like a puzzle some times. $\endgroup$
    – JMoravitz
    Commented Aug 25, 2017 at 15:01
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$\frac{1}{3^{-2}}=3^{-(-2)}=3^2=3\times3=9$

Take a look here to know about laws of indices

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