Difference between a positive cubic meter and an inverse cubic meter?

Question

In science I commonly see the notation $m^{3}$ and $m^{-3}$, especially when discussing the amount of molecules in a volume.
What is the difference between those two units? For example, does $1.5\times 10^{10}\ m^{-3}$ translate into $10^{10}$ molecules per $m^3$ of air?

Answer 1

$m^3$ is units of cubic meters. $m^{-3}$ is inverse cubic meters (not negative), meaning something per meter. So if I say the number of grains of sand on the beach is $10^{10}m^{-3}$, what I am really saying is that every cubic meter contains $10^{10}$ grains of sand, or that the density of the sand is $10^{10}$grains$/m^3$.

Full disclosure: I know nothing about sand. That number is probably way off. I would count them, but not even I can count that much.

Note, negative cubic meters would just be $-m^3$.

Answer 2

The negative in the exponent means it is per cubic meter. You can think of it placing the cubic meters in the denominator.

Answer 3

The relation between these units is more or less the same as for two different measures of fuel consumption cars:

• In the US, it is typically measured as "Miles per Gallon".
• In Germany, it is measured as "Litre per 100 km".

If you get rid of the different units, it is "km per litre" vs. "litre per km".