# Exponentiation

## Introduction

To understand **exponents** and the **exponential function**, let's first review how multiplication came about. Let's say we wanted to add 3 ten times. We could write this out as but this gets burdensome if we wanted to add, for example, 3 three hundred times. Thus, we define the multiplication function (usually denoted or ) such that where there are 300 threes in the sum. For integers, it is straightforward how multiplaction works. We can then extend the notion of multiplication to non-integers.

Similarly, the exponential function is defined as the repitition 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.

Before we proceed, we define 3 terms:

- exponent or power - in the exponent is 6; this tells us how many times we multiply the 4
- base - in the base is 10; this tells us what we will be multiplying 9 times

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. So we want to do the opposite of multiplication four times. In other words, we want to divide by 2 four times. Therefore

It is also possible to extend the exponential function to all non-integers.

## Basic Properties

Listed below are some important properties of exponents:

- (if . is undefined.)

Here are explanations of the properties listed above:

- On both sides we are multiplying
**b**together**x+y**times. Thus, they are equivalent. - This is described in the previous section.
- This results from using the previous two properties.
- We are multiplying by itself
**y**times which is the same as multiplying**b**by itself**xy**times. - After multiplying
**ab**by itself**x**times we can collect**a**and**b**terms thus establishing the property. - Hoping that property #1 will be true when , we see that should (hopefully) be equal to . Thus, we
*define*to be equal to in order to make this be true.

## Fractional exponents

If is a number and each of and is a positive integer, then as explained above (property 1) . For example, .

How could we make sense of an expression like ""? Well, hoping that property 1 will remain true when , we see that should (hopefully) be equal to . For that reason, we *define* , in order to make that be true. (And we only make this definition in the case where . We choose to leave undefined.)

We can make sense of an expression like "" in a similar way. Hoping that property 1 will remain true even if or is negative, we see that should (hopefully) be equal to . Thus, we *define* to be , in order to make this be true. Similarly, if is a positive integer, we define to be . (This depends on having . Otherwise we'd be dividing by .)

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.