I have trouble understanding simple equality passage

Question

I don't know why I always have trouble with these things, there is something that must do 'click' in my head.

Knowing that 1000000 cm³ = 1 m³, how do you get that

    cm³
-----------
 0,000001 m³`? 

It is pretty obvious visually that 1 cm³ = 0,000001 m³, but how to do it matematically? It is for unit conversion.

1000000 cm³ = 1 m³

   1 cm³ = 1 m³
          ------
           1000000

then

1 =    1 m³
      ------      Now i'm lost!
       1000000
    ---------- 
        cm³

Answer 1

Problem is 1000 cm³ = 1 m³ is FALSE.

1 m = 100 cm

so

1 m³ = (100 cm)³ = 1000000 cm³