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

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?

• What do you even mean by 1/3/3 and 1/1/3/3? – Saucy O'Path Apr 6 at 22:22
• To be clear, do you want to compare $$\frac1{3^{-2}} \quad\text{and}\quad 3^{-2}$$ or do you mean $$\left(\frac{1}{3}\right)^{-2} \quad\text{and}\quad 3^{-2}$$ – Blue Apr 6 at 22:22
• $a \neq \frac{1}{a}$ unless $a$ is $\pm 1$ – randomgirl Apr 6 at 22:24
• First, by "$1/3^{-2}$" do you mean $1/(3^{-2})$ or $(1/3)^{-2}$. What you wrote should be interpreted as the first, but I have known it to be written when the second was intended. Assuming you really mean $1/(3^{-2})$ then $3^{-2}= \frac{1}{9}$ so $1/(3^{-2})= \frac{1}{\frac{1}{9}}= 9$. – user247327 Apr 6 at 22:26
• 1/3^-2. The 3 is to the power of -2. No parentheses involved. Also how do you do the power sub character? – Solomon Mao Apr 6 at 22:30

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.