Which of the following is an appropriate symbol for the unit of a quantity found by dividing an area by a volume? Meters cubed, meters squared, meters, meters to the negative one power, meters to the negative two power.
Let’s first recall that we can manipulate units like algebraic variables. So to find the overall units when we multiply or divide two quantities, we can multiply or divide the individual units from each quantity. In this particular question, the units of an area divided by a volume are exactly the units of area divided by the units of volume.
Looking at our answer choices, we see they’re all expressed in terms of the base unit meters. So we need to express a unit of area and a unit of volume in terms of the base unit meters. Now we recall that the meter is a unit of length. And an area is in general a product of two lengths, as in the case of a rectangle, where the area is the product of the length of the two sides. Now since meters are an appropriate unit for the length of each of these two sides, appropriate units for area will be meters times meters, or meters square.
Similarly, a volume is generally a product of three lengths. For example, we could add a third dimension to our rectangle. Now we have a box, and the volume of this box is the product of the length of all three of its sides. Now since meters are an appropriate unit for each of these three lengths, the appropriate units for volume are meters times meters times meters, which is meters cubed. Now we have an appropriate unit symbol in terms of the base unit meters for both an area and a volume. All that’s left is to divide meters squared by meters cubed to find an appropriate unit symbol for an area divided by a volume.
One convenient way to evaluate meters squared divided by meters cubed is to observe that meters cubed is meters times meters squared. We can now see immediately that meters squared in the numerator divided by meters squared in the denominator is just one, which leaves us with one divided by meters. Now we recall that one divided by anything is that thing raised to the negative one power. So the appropriate unit symbol for an area divided by a volume using meters as the base unit is meters to the negative one power. And this is the answer that we are looking for. Just as an aside, we often read meters to the negative one power as per meter.