How do I convert meters cubed to cubic centimeters?

[Wanting to Learn]

This comes from the Intro Homework to my AP Physics 1 class.
The teacher said that it is more important to figure out how to use the website than get the correct answers to the questions. However, I want to know how to get the correct answer, especially since I took chemistry last year and think I should know this.
I'm not certain, but I might just be missing a small step in the calculations.

Problem Statement, Variables and Given Data:
What is the volume V of a sample of 2.70 mol of copper?
The atomic mass of copper (Cu) is 63.5 g/mol.
The density of copper is 8.92×10³kg/m³.
Express your answer in cubic centimeters to 3 significant figures.

Attempt at the Solution:
I think I need to multiply and cancel units, however I am not sure what I'm missing.
(I do know how to use the proper number of significant figures.)
2.70mol * 63.5g/mol = 171.45g
8.92×10³kg * 1000g/1kg = 8.92 * 10⁶g
(Should I convert the 171.45g to kg here instead?)

*171.45g * 8.92 * 10⁶g/m³ = 1,529,334,000g²/m³
*I am not sure where to go from here in order to get to just meters cubed, assuming that is what I should be looking for.
*update of the two lines above:
I just realized that if I put m³/g then the grams would cancel, and got this:
171.45g * 1m³/8.92 *10⁶g = 0.00001922m³
Now I need to convert that to cm³. I always get confused when doing this, because of the ³. How exactly do I do this last step? I know that I should know this more basic part but I want understand exactly what to do with it.
I tried and the answer I got was one decimal place off; I got 1.92 instead of 19.2cm³.

I did put "volume of 2.70mol Cu" into Wolfram Alpha and got an answer, however I want to know how to get it on my own. Did I miss or do something wrong in the calculations I did, or am I missing something not expressly stated in the question?

Thank you in advance for your help.

 

[scottdave]

If you are going to put mass in the denominator (10³ kg or 10⁶ grams), then you also need to put the corresponding value (8.92) in the denominator.

 

[Wanting to Learn]

If you are going to put mass in the denominator (10³ kg or 10⁶ grams), then you also need to put the corresponding value (8.92) in the denominator.

Oops, I did that correctly in my calculation (on a different piece of paper) but typed it in wrong, I just fixed it (edited my original post).
Thank you for pointing that out.

 

[Bandersnatch]

I just realized that if I put m3/g then the grams would cancel, and got this:

It's good that you followed the units. They're one of the most reliable signs telling you if you're on the right path or not.
It'd be even better if you first manipulated algebraic expressions before trying to plug in the numbers.

Density is mass over volume:
$\rho = \frac{m}{V}$
So if you need to find volume:
$V = \frac{m}{\rho}$

Now I need to convert that to cm³. I always get confused when doing this, because of the ³. How exactly do I do this last step?

You want to convert metres to third power into centimetres to the same power. Just follow these three steps.
1. How many centimetres are there in a metre? 100
2. What is the power? 3
3. Raise 100 to the power of 3
Answer 1 m^3= 10^6 cm^3

Conceptually, imagine you have a physical cube 1 m long, 1 m wide, and 1 m tall. You're trying to figure out how many 1 cm^3 cubes (1 cm long, wide, and tall) are needed to build that 1 m^3 cube.
So you take the 1 cm^3 cube and stack 100 of those next to another (since there is 100 cm in a metre) - this gives you a line 1 metre long, but still 1 cm wide and tall. So you stack 100 such lines on top of another, to get a wall of 1 m length and height, but still 1 cm wide (by this point you've used 100x100 small cubes). You need 100 such walls next to one another to get a full 1m^3 cube. You've used 100x100x100 of the 1 cm^3 cubes, i.e. there's 10^6 cm^3 in 1 m^3.

And if you were to convert something like centimetres squared into metres squared:
1. How many metres in a centimetre? 0.01
2. What is the power? 2
3. Raise 0.01 to the power of 2
Answer 1 cm^2 = 10^-4 m^2

 

[scottdave]

Oops, I did that correctly in my calculation (on a different piece of paper) but typed it in wrong, I just fixed it (edited my original post).
Thank you for pointing that out.

Great. Now to convert meters³ to cm³.
Think of it like this: 1 m = 100 cm. So we can make a conversion factor by dividing both sides by (1 m), and we get this:
$$1 = \frac {100 ~ cm} {1 ~m}$$
You could also divide both sides by (100 cm) and get this: $$\frac {1 ~ m} {100 ~cm} = 1$$

Note that 1 is dimensionless. You can multiply or divide anything by dimensionless 1, without changing the value.

So if you have something that is in meters³, just multiply by (100 cm) / (1 m), and you need to do that three times to cancel out m³. Then you will have cm³.