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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?

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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 $\text{dm}^3$.

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    $\begingroup$ A useful addition may be that $\frac{1}{L} = L^{-1}$ and $(L^{-1})^3 = L^{-3}$. And +1 for litres :) $\endgroup$ – Felipe S. S. Schneider Jan 21 '17 at 17:25
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The unit of volume is $\mathrm{dm}^{3}$. When you write moles per cubic decimeter (concentration), it can be expressed as

$${\mathrm{mol}\over\mathrm{dm}^{3}}$$

which is equivalent to

$$\mathrm{mol}\cdot{1\over\mathrm{dm}^{3}}$$

and because

$${1\over\mathrm{dm}^{3}} = \mathrm{dm}^{-3}$$

it can therefore be expressed as

$$\mathrm{mol}\cdot\mathrm{dm}^{-3}$$

In summary, it's simply because the volume term is in the denominator and the sign of the exponent changes when the product of it and $\mathrm{mol}$ is written instead of the fraction.

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