Dimensional Analysis with volume?

[Titandwedebil]

Can you work units cubed the same way you can work units that aren't cubed in dimensional analysis?

Like...does 1m³ equal 250cm³ like it would if they weren't cubed?

 

[Mute]

I'm not sure where you're getting 1 m^3 = 250 cm^3, but here's how things work:

If you have 1 m^3 and you want to convert to cm cubed, you can use the fact that 1 m = 100 cm. Since 1 m^3 is really just (1 m)^3, I can write (1 m)^3 = (100 cm)^3. The power of 3 now has to distribute over both the unit and the number, so we get that

1 m^3 = 100^3 cm^3 = 10^6 cm^3.

The key point to take away is that a relation between units like 1 m = 100 cm is an equality, so you can always replace a unit with one of these equalities, and then distribute the powers over both the unit and the number, as done in this example here.