# Properties of Exponents

When working with exponential notation, it is useful to understand some of the basic properties of exponents. This is probably review for many of you, but I will post this here now for those who are just learning it for the first time or need a bit more help to understand it.

First, let's examine the form of exponents.

If we have a real number, a, and a natural number, n, we can define the exponential expression an as:
an = a x a x a ... a (n factors of a),
where we say that a is the base and n is the exponent. The exponent is also frequently referred to as the power.

Now, knowing the basic form of an exponential expression, let's look at some of the properties of exponents that will prove to be very helpful to you in the future!  These will be used over and over again, and will be the basis of future lessons, so it will be very good for you to understand these well.  There are really only four of them, so it shouldn't be too hard to memorize them.

1.  The first property defines how to multiply exponents.  If we have a real number, a, and natural numbers n and m, we can say:

aman = am+n

So, what this basically says is that when you have exponential terms to multiply together, if they have the same base number, you can simply ADD the exponents.  To demonstrate this, we can write out an example.  If we want to simplify a2a3, we can expand it to show [a x a] x [a x a x a], which is a x a x a x a x a, which is a5. The base a was common, so we could just add the 2 and 3 in the exponent.  Same thing applies when the terms are bigger.  Try for yourself to prove that (x+1)2(x+1)5 = (x+1)7.

2.  On the other hand, a similar property exists to divide exponents.  If you want to divide exponential terms, you just subtract the exponents, like this:

a/ an = am-n

Prove to yourself that a/ a= a, in the same way I demonstrated above.

3.  If we want to raise an exponential term to another power, such as with (am)n, we can simplify it to amn.  Again, try it yourself by writing it out like I did above to demonstrate this with the expression (a5)2 = a10.

4.  Finally, similarly to multiplying a coefficient times everything inside a set of brackets, if we want to raise a product or quotient expression by a power, the exponent can be written for each term, like so:

(ab)m = ambm, or
(a/b)m = am / bm

In this case, it should be simple to see that this means something like (ab)2 = a2b2. If you want to start combining these properties, you can see something like (a5b2)2 = a10b4.

Hopefully with these examples you can see that the properties of exponents are not that complicated to work with, and once you get the hang of the basics and can start combining them, they are actually very easy to use!