I see the answer and what you are telling me works in my head like so:
However if, for example, I only had :
then when I try to simplify I get
and my answer ends up being
Where am I going wrong?
I'm guessing that you are going wrong in a terrible way.
Symbolically, you could (COULD, not SHOULD!) see the C^2 on top and likewise a C^2 on bottom, and literally just erase them from the page. Same with A^2 and B^2.
In that case, you would have ABC(eraser mark) + ABC(eraser mark) + ABC(eraser mark), all divided by (eraser mark + eraser mark + eraser mark)
And maybe you're thinking that since you have ABC three times, that this is 3ABC.
This is terribly wrong.
"Cancel factors, not terms" is a concise way to avoid this problem. I'll let someone else fill in the distinction.