# Units of measure & converion factors for higher powers

Take the following expression:

$$\frac{45000cm^3}{1}*\frac{1m^3}{100cm^3}$$

This is how I initially solved it:

$$\frac{45000}{1}*\frac{1m^3}{100}$$

$$\frac{45000}{100}*\frac{1m^3}{1}$$

$$450m^3$$

Now, that is obviously wrong. I had assumed that since the cubic in $$cm^3$$ cancel out that I wouldn't need to raise $$100$$ to the 3rd power, but I was wrong. Although the cm^3 did cancel out, I still had to raise the 100 to the 3rd power:

$$\frac{45000}{1}*\frac{1m^3}{100}$$

$$\frac{45000}{100^3}*\frac{1m^3}{1}$$

$$0.045m^3$$

Now, my questions:

1. Why do we still have to raise the 100 to the 3rd power ? I guess I am treating them like normal exponents, but it appears when dealing with units the exponents are treated differently. It appears that $$100cm^3$$ should in fact be interpreted as $$100^3cm^3$$.

2. If the hunch in my first question is valid ($$100^3cm^3$$) then how come no one writes it like that ? and wouldn't that cause ambiguity ? i.e. not knowing whether a number has already been cubed or not in this case ?

EDIT: Here is the problem for reference: https://youtu.be/b2JCZDeLGF4?t=1174

• What you did first is obviously right, and what you did afterwards is obviously wrong. Who did tell you otherwise? Jul 11, 2020 at 15:42
• @NeitherNor youtu.be/b2JCZDeLGF4?t=1174 Jul 11, 2020 at 15:45
• When in the video does your formula occur? Jul 11, 2020 at 15:54
• @NeitherNor the link is already timestamped to that problem, just click on it and play. Jul 11, 2020 at 15:56
• Your equation is not the one appearing in the video. You probably made the classic error to assume that $1m^3=100cm^3$. This is not the case. $1m^3=(100cm)^3=10^6cm^3$. Jul 11, 2020 at 16:05

There is really nothing special when calculating with units. Think of it as calculating with variables: when you have that $$y=100x$$, then you have that $$y^3=(y)^3=(100x)^3=100^3x^3$$. Now replace y by meters and x by centimeters, and you are done.
I think your problem is not the math, but that you think that $$1m^3$$ should be $$100cm^3$$. What you need is intuition. So $$1m^3=1m*1m*1m$$, that's a washing machine. And $$1cm^3=1cm*1cm*1cm$$, that's a small dice. How many small dice can you put into a washing machine? Surely more than 100!
Your cancellation computation is correct, but you didn't tell us what you wanted to achieve originally. I suppose you want to express a volume of $$45000\,\text{cm}^3$$ in $$\text{m}^3$$. To this end, note that $$1\,\text m=100\,\text{cm}$$, hence $$1\,\text{m}^3=(100\,\text{cm})^3=1000000\,\text{cm}^3.$$