In this sections, we review the rules of exponents and look at ways to simplify expressions using positive and negative exponents and rational exponents. We will use these skills throughout the course. This is a new section unique to this custom edition. It was written by Katie Hall.
We have seen that Using this rule, we can see several properties of exponents. First, let's explore what we get if we have an exponent of 0 or a negative number. We will start by looking at the values of for several values of .
Notice that if we start in the middle of the table, we have to multiply by 2 to move up a row and divide by 2 to move down. Using this division principle, we can fill in several more rows of the table.
By considering the definition of an expressions like or , we can start to derive more rules of exponents. For example, if we have this is the same as since we end up with five s all multiplied together. This sort of analysis leads to the first five rules below.
You may have seen in an algebra class that . From the third rule, we know that . This allows us to conclude that is the same as the number which when raised to the th power becomes , which is the defintion of . One way to think about fraction powers, like is the the numerator gives the power on and the denominator tells which root it is. When dealing with roots, it is important to remember that we can't take an even root of a negative number.
We can use these rules to simplify complicated expressions. When we simplify (for now) we want to have positive and negative powers but no radicals and division. For example, to simplify