What is the difference between $\dfrac{1}{3^{-2}}$ and $3^{-2}$

Question

Stupid question but how in the world does 1/3^-2 not be the same thing as 3^-2? The first answer I got is 9, the second one I got is 0.111... Isn't the negative power just 1/3/3? What difference does it make when I do 1/1/3/3?

Answer 1

Brackets are your friend. Without brackets it is difficult to determine the meaning of the symbols. You have $$(1/3)^{(-2)} = (3^{-1})^{(-2)} = 3^{(-1)(-2)} = 3^2 = 9$$ whereas $$3^{(-2)} = 3^{(2)(-1)}= (3^2)^{(-1)} = 9^{-1}.$$ But this demonstration is pointless unless you are clear on the meaning of the symbols $a^{-1}$ and $a^b$. This may be a topic for another question.

Answer 2

Dividing by a fraction is the same as multiplying by its inverse.

\begin{equation} 3^{-2}=\frac{1}{3^2}=\frac19\approx 0.111111111\\ \\ \frac{1}{3^{-2}} =\dfrac{1}{\frac{1}{3^2}} =1\times\frac{3^2}{1}=3^2=9 \end{equation}