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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

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  • $\begingroup$ What you did first is obviously right, and what you did afterwards is obviously wrong. Who did tell you otherwise? $\endgroup$
    – NeitherNor
    Jul 11, 2020 at 15:42
  • $\begingroup$ @NeitherNor youtu.be/b2JCZDeLGF4?t=1174 $\endgroup$
    – Rijndael
    Jul 11, 2020 at 15:45
  • $\begingroup$ When in the video does your formula occur? $\endgroup$
    – NeitherNor
    Jul 11, 2020 at 15:54
  • $\begingroup$ @NeitherNor the link is already timestamped to that problem, just click on it and play. $\endgroup$
    – Rijndael
    Jul 11, 2020 at 15:56
  • $\begingroup$ 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. $1m^3=(100cm)^3=10^6cm^3$. $\endgroup$
    – NeitherNor
    Jul 11, 2020 at 16:05

2 Answers 2

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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!

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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 $$1\,\text{m}^3=(100\,\text{cm})^3=1000000\,\text{cm}^3.$$

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