To understand how exponents arise, let's first review how we can build multiplication from addition. Let's say we wanted to capture the notion of "the amount equal to 3, ten times." We could write this out as , but this gets burdensome quickly: if we wanted to capture the idea of "the amount equal to two hundred 3s." Thus, we define the multiplication function, usually denoted or , such that where there are 200 threes in the sum. This process (actually an inductive definition) defines the operation of "multiplication by positive integers." We can then extend the notion of multiplication to non-integers.
Similarly, the exponentiation is defined as the repetition of multiplication. For example, writing out can get boring fast, so we define the exponential function to express this in a much more compact form so that the preceeding example can be written as (read 3 to the 5th or 3 to the 5 power). What this means is that we are multiplying 3 by itself 5 times. The base is 3 (what is repeatedly multiplied) and the exponent (or power) is 5 (the number of times to repeat multiplication).
Our definition of exponentiation makes sense if the exponent is a positive integer. How about negative integers such as ? How do we multiply 2 by itself -4 times!? Let's think about what a negative sign means a little more. When we append a negative sign to a number (say 4, for example), we are basically saying go four units in the opposite direction. We want to do the opposite of multiplication four times. In other words, we want to divide by 2 four times. Therefore,
Listed below are some important properties of exponents (with explanations and notes):
- We are multiplying together times and then times, so in total, we are multiplying together times.
- From the property of same base multiplication, , so dividing both sides by results in .
- is undefined.
- From the property of same base multiplication and zero exponent, , so dividing both sides by results in .
- This results from using the property of same base multiplication and negative exponent property.
- We are multiplying by itself times, and because of the first property, we are multiplying by itself times.
- We are multiplying together times, which is the same as multiplying together times and together times.
So far we discussed expressions with integer exponents. However, it is also possible to extend the exponential function to all non-integers.
How could we make sense of an expression like ? If you don't already know the answer, this is a good exercise; I recommend puzzling over it for awhile.
Answer: Hoping that property 1 will remain true when or is a fraction, we see that should (hopefully) be equal to . Thus, we define to be , in order to make this be true.
For the time being, how to deal with other fractions in the exponent can be an exercise for the reader.
Hint: What would be? What about ? Do you notice anything?
Try to figure out -- how does it relate to ?
- Practice Problems on Alcumus
- General Positive Exponents (Prealgebra)
- Negative and Zero Exponents (Prealgebra)