# How to simplify $\frac{1}{3^{-2}}$ properly?

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

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

$\frac{1}{3^{-2}}=\frac{1}{3^{-2}}\cdot 1 = \frac{1}{3^{-2}}\cdot\frac{3^2}{3^2}=\frac{1\cdot 3^2}{3^{-2}\cdot 3^2}=\frac{3^2}{3^{-2+2}}=\frac{3^2}{3^0}=\frac{3^2}{1}=3^2$
In general, you can skip many of these steps and use the result $x^n=\frac{1}{x^{-n}}$ for any nonzero $x$.
• In general, $a^{-b}= \frac{1}{a^b}$ so $\frac{1}{a^{-b}}= \frac{1}{\frac{1}{a^b}}= a^b$ Commented Aug 24, 2017 at 20:56
• 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 $\frac {1}{3^{-2}}\cdot 1.$ Commented Aug 25, 2017 at 11:41
$\frac{1}{3^{-2}}=3^{-(-2)}=3^2=3\times3=9$