Volume in decimetres cubed dimensional analysis
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 $\text{dm}^3$.