Factor Analysis is the use of units as algebraic quantities to decide easily how to make conversions...

Example 1: Suppose we want the Earth-Moon distance in miles. From the front cover of many textbooks, the distance is 3.84 x 10⁸ m. Various conversions are printed on the back cover but I won't use those since they would make the problem very easy; I'll use only the conversions 5280 ft/mi, 12 in/ft, and 2.54 cm/in, as well as the decimal relationship between metric quantities:

Do you see how this first step is done? I set up the units so that they divided out, yielding miles. The next step is to insert numbers for each of the empty brackets (the brackets are empty of numbers, not units). For example, since there are 2.54 cm to 1 inch, for the second bracket I put 1 inch on top and 2.54 cm on the bottom, as suggested by the units which were put in that set of brackets first. If you use this method (called factor analysis), you never have to decide whether to multiply or divide by a conversion factor since the units tell you how to do it!

The next example involves powers. Remember that, when you cube a number, you also cube the units!

Example 2: The lattice parameter for iron at room temperature is 2.866 Angstroms. Determine the volume of an iron atom in cubic inches.

Solution: The sysmbol usually used for the "lattice parameter" is "a", which is the length of one side of the unit cell. It is determined by X-ray analysis. An Angstrom (A with a little circle above it, but I'll use just "A" because of display difficulties), from the back cover, is 10^-10 m. The body-centered-cubic (bcc) structure has 2 atoms per unit cell. The packing factor is 0.680. Therefore, since the volume of the cell is a³ = (2.866 A)³, 0.680 times (2.866 A)³ is the volume of the iron atoms in the unit cell. The volume of one atom is this value divided by 2 atoms:

Now I use factor analysis, setting up the units first:

Notice that cubing the numbers also means that the units must be cubed?