An exponent is something that raises a number or a variable to a power. It is a process of repeated multiplication. For example, the expression means to multiply 2 times itself 3 times or . In , the 2 is called the base and the 3 is called the exponent. Both the base and the exponent can be either a number or a variable.

Each of the following is an example of an exponential expression.

When both the base and the exponent are numbers, we can evaluate the expression as we did with . If either the base or the exponent is a variable, we need to be given additional information in order to make a numerical evaluation.

One type of exponent that was not used in the previous list of exponential expressions was a negative exponent. When dealing with a negative exponent, we have a rule to follow. The rule says:

In other words, when there is a negative exponent, we need to create a fraction and put the exponential expression in the denominator and make the exponent positive. For example,

But working with negative exponents is just rule of exponents that we need to be able to use when working with exponential expressions.

Rules of Exponents:

If the bases of the exponential expressions that are multiplied are the same, then you can combine them into one expression by adding the exponents.

This makes sense when you look at

If the bases of the exponential expressions that are divided are the same, then you can combine them into one expression by subtracting the exponents.

This makes sense when you look at

When you have an exponential expression raised to a power, you have to multiply the two exponents.

This makes sense when you look at

Notice that we had to use another rule of exponents to help us make sense of this rule. This is a common occurrence. Many times you will use more than one rule of exponents when working problems.

 

http://www.algebralab.org/lessons/lesson.aspx?file=Algebra_ExponentsRules.xml

http://www.math.temple.edu/~ludwick/teach/spring01/exponents/

A number that indicates how many times a given quantity, called the base, is to be multiplied by itself, usually denoted by a superscript number or symbol immediately after the quantity concerned, eg 64 = 6 ′ 6 ′ 6 ′ 6. Also called power, index.

 

http://www.allwords.com/query.php?SearchType=3&Keyword=Exponent&goquery=Find+it!&Language=ENG

The last lesson explained how to simplify exponents of numbers by multiplying as shown below. You know that 3 squared is the same as 1 * 3 * 3.

Exponents of variables work the same way - the exponent indicates how many times 1 is multiplied by the base of the exponent. Take a look at the example below.

The first problem we will work on is below. It doesn't contain a variable, but it will help us to learn how to simplify a similar problem with a varable in place of the first 3.

Normally, you would simplify this problem by simplifying the inside of the parentheses first:

Then, simplify the exponent outside the parentheses.

This method gives a correct answer, but there is an easier way.

We will be solving the same problem again:

This time, instead simplifying inside of the parentheses first, we will "distribute" the exponent of the parentheses to the inside of the parentheses.

Now the only thing left to do is simplify the exponent that is left.

As you can see this method also gives an answer of 729.

The first example with variables is

We will try simplifying it the first way, by simplifying the inside of the parentheses followed by simplifying the exponent on the outside.

Now that the inside is simplified, the exponent on the parentheses indicates that the expression is equivalent to a 1 multiplied by the parentheses, three times. As you can see x is being multiplied 6 times, hence the answer x to the sixth power.

Again, the problem we are working is

As with the second number example earlier in this lesson, simply multiply the two exponents:

Then remove the parentheses, and as you can see the answer is the same.

http://www.jiskha.com/mathematics/algebra/exponents_of_variables.html