The long form of my previous post...

The original question in the thread was, "Why isn't '20 cu ft' the same thing as 'a cube with sides of length 20 ft'?" The subsequent question about the number of cubic inches in a cubic foot is essentially a rehash of the same issue, and the two can be addressed as one problem.

To begin, consider the general case of the expression (x * y) ^ n. Standard rules for the order of operations dictate that this can be expanded as (x ^ n) * (y ^ n); expanding it as x * (y ^ n) is not correct.

If we have a cube with sides of length (x * y), then the volume of that cube will be calculated as (x * y) ^ 3, or (x ^ 3) * (y ^ 3). Now take our cube with sides of length 20 ft, whose volume is calculated as 20 ft * 20 ft * 20 ft, or (20 ft) ^ 3. Next, we expand the expression 20 ft to 20 * 1 ft, and substitute it back into our volume calculation, which now becomes (20 * 1 ft) ^ 3. Referring back to the general case, the proper expansion of this expression is (20 ^ 3) * (1 ft ^ 3); and since (1 ft) ^ 3 = 1 cu ft, this gives us our volume as 8000 cu ft. (Similarly, a cube with sides of 12 in has a volume of 1728 cu in, demonstrating that the relation 1 ft = 12 in is only valid for linear measurements; it does not hold for measurements of either area or volume.)

The argument in the original post, that '20 cu ft' (which can also be written as 20 (ft ^ 3)) should be the same as 'a 20 ft X 20 ft X 20 ft cube', is only achievable as the result of using the incorrect expansion for the general case as stated above, as it requires you to cube only one of the factors in the expression.

That said, there is one exceptional case: when the length of a side is 1 ft (or indeed one of whatever unit you happen to be using), you can get away with doing things the wrong way around. Basically, because 1 * 1 = 1, you still get the correct volume even if you forget to cube the first term - if and only if the length of a side is exactly 1. But it is important to remember that this is a special case, not the general case.

BOOM! Mathematician'd!