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on October 22, 2007 at 6:05:22 pm


What is an Exponent?



The representation of a number in the form:
X y

where x and y are integers is called an exponential representation.

The integer X is called the base and Y is called the exponent.
Exponents are used as a short way to represent a number. The exponent is the number of times the base is multiplied by itself. Sometimes the operator ^ is used to represent an exponent.
X y = X ^ y

Here are some examples:

  • 27 can be represented as 33

    Did you know that this is same as 3 multiplied by itself 3 times?



    27 = 33 = 3 x 3 x 3

  • 32 can be represented as 25

    You guessed right, this can also be represented as 2 multiplied by itself 5 times.



    32 = 25 = 2 x 2 x 2 x 2 x 2

  • 1,000,000 can be represented as 10 6

    That is a short way to represent 10 multiplied by itself 6 times.

    10 6 is really 1,000,000. Remember:

    1,000,000 = 106 = 10 x 10 x 10 x 10 x 10 x 10

10 6 = 1,000,000


An exponent is a number that tells how many times the base number is used as a factor. For example, 42 indicates that the base number 4 is used as a factor 2 times. To determine the value of 42, multiply 4*4 which would give the result 16. Squares indicate that the exponent has a value of two. The term square comes from the geometrical shape that has the same width and length. To find the area of a square you would multiply the width times the length.

Exponents are written as a superscript number (e.g. 42).

Some facts about exponents:

  • Zero squared is zero (e.g. 02 = 0)
  • One squared is one (e.g. 12 = 1)





Exponents are shorthand for multiplication: (5)(5) = 52, (5)(5)(5) = 53. The "exponent" stands for however many times the thing is being multiplied. The thing that's being multiplied is called the "base". This process of using exponents is called "raising to a power", where the exponent is the "power". "53" is "five, raised to the third power". When we deal with numbers, we usually just simplify; we'd rather deal with "27" than with "33". But with variables, we need the exponents, because we'd rather deal with "x6" than with "xxxxxx".






Rule 1: To multiply identical bases, add the exponents.



Example 1: tex2html_wrap_inline49 means




which in turn can be written tex2html_wrap_inline53 . According to Rule 1, you can get to the answer directly by adding the exponents.




Rule 2: To divide identical bases, subtract the exponents.


Example 1: tex2html_wrap_inline90 can be written


which can be written as







The later can also be written tex2html_wrap_inline98 . According to Rule 2, you can get to the answer directly by subtracting the exponents



Rule 3: When there are two or more exponents and only one base, multiply the exponents.


Example 1: tex2html_wrap_inline128 can be written tex2html_wrap_inline130 . According to Rule 1, we can add the exponents. tex2html_wrap_inline130 can now be written tex2html_wrap_inline134 . According to Rule 3, we could have gone directly to the answer by multiplying the exponents






Rules of Exponents


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.



$displaystyle (10^{20})^{5} = 10^{(20times 5)} = 10^{100}   ,   (2^{-5})^{-2} = 2^{-5times (-2)} = 2^{10}$ 


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.





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.








An exponent refers to the number of times a number is multiplied by itself. For example, 2 to the 3rd (written like this: 23) means:

2 x 2 x 2 = 8.

23 is not the same as 2 x 3 = 6.

Remember that a number raised to the power of 1 is itself. For example,

a1 = a

51 = 5.

There are some special cases:

1. a0 = 1

When an exponent is zero, as in 60, the expression is always equal to 1.

a0 = 1

60 = 1

14,3560 = 1

2. a-m = 1 / am

When an exponent is a negative number, the result is always a fraction. Fractions consist of a numerator over a denominator. In this instance, the numerator is always 1. To find the denominator, pretend that the negative exponent is positive, and raise the number to that power, like this:

a-m = 1 / am

6-3 = 1 / 63

You can have a variable to a given power, such as a3, which would mean a x a x a. You can also have a number to a variable power, such as 2m, which would mean 2 multiplied by itself m times. We will deal with that in a little while.




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