Converting Units with ExponentsReference > Science > Unit Conversions
We've now covered almost everything you need to know in order to be an expert unit-converter. But there's one more scenario we should talk about: what happens if your units have exponents?
"What? Units can have exponents?"
Sure! Units of area are things like ft2 and in2. Volume uses cubes (meter3), and when you deal with physics, you'll get some really scary units like kilogram x meter/second2!
Don't panic, though; it's really not all that difficult...
Example One - Convert 12 ft2 to inches2.
We start off as we normally do, by multiplying by our conversion factor (12 inches)/(1 foot):
(12 ft2) x (12 inches)/(1 foot) = 144 foot x inches.
You see what happened? One of the feet in feet2canceled, leaving us with a unit of foot x inch.
WHAT? That's a ridiculous looking unit - who ever heard of a foot-inch? What went wrong? Nothing went wrong. We just need to multiply our conversion factor TWICE:
(144 foot x inches) x (12 inches)/(1 foot) = 1728 inches2
If a unit has an exponent of 2, you need to daisy chain the conversion factor twice to get rid of the unit!
In fact, we can generalize that: If a unit has an exponent of n, we need to daisy chain the conversion factor n times to get rid of the unit.
Example Two - Convert 81 feet3 to yards3
Answer - Our conversion factor is (1 yard)/(3 feet), and we need to daisy chain it 3 times. But daisy chaining the conversion factor 3 times is the same as cubing it, right? So let's cube that conversion factor:
(13 yard3)/(33 feet3) = (1 yard3)/(27 feet3).
Now let's use that conversion factor:
(81 feet3) x (1 yard3)/(27 feet3) = 3 yard3.