Volume in decimetres cubed dimensional analysis In chemistry class, we were learning about concentration, which is the number of moles of a substance per decimeter cubed of water, expressed as $\mathrm{mol} \cdot \mathrm{dm}^{-3}$.

However, I was confused when my teacher wrote decimeters cubed as $\mathrm{dm}^{-3}$ instead of what I presumed would be $\mathrm{dm}^3$. I asked why and he didn't know. Why do chemists write it like this?
This touches a very powerful analytical technique called unit analysis (aka dimensional analysis). 

Moles per decimetre cubed is $\dfrac{\text{moles}}{\text{dm}^3} = \text{moles}\cdot \text{dm}^{-3} \ne \text{moles}\cdot \text{dm}^3$

So the -3 indicates that the unit is in the denominator not the numerator. 

PS - I learned chemistry when there were only four elements - earth, wind, water and air. I think of liters not $\text{dm}^3$.