In the table below, the number 2 is written as a
factor repeatedly.
The product of factors is also displayed in this table. Suppose that your teacher asked you to Write 2 as a factor one million times for homework. How long do you think that would take?
Answer
Factors
Product of Factors
Description
2 x 2 =
4
2 is a factor 2 times
2 x 2 x 2 =
8
2 is a factor 3 times
2 x 2 x 2 x 2 =
16
2 is a factor 4 times
2 x 2 x 2 x 2 x 2 =
32
2 is a factor 5 times
2 x 2 x 2 x 2 x 2 x 2 =
64
2 is a factor 6 times
2 x 2 x 2 x 2 x 2 x 2 x 2 =
128
2 is a factor 7 times
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 =
256
2 is a factor 8 times
Writing 2 as a factor one million times would be a very time-consuming and tedious task.
A better way to approach this is to use exponents. Exponential notation is an easier way to write a
number as a product of many factors.
BaseExponent
The exponent tells us how many times the base is used as a factor.
For example, to write 2 as a factor one million times, the base is 2, and the exponent is 1,000,000.
We write this number in exponential form as follows:
2
1,000,000
read as two raised to the millionth power
Example 1:
Write 2 x 2 x 2 x 2 x 2 using exponents, then read your answer aloud.
Solution:
2 x 2 x 2 x 2 x 2 = 25
2 raised to the fifth power
Let us take another look at the table from above to see how exponents work.
Exponential Form
Factor Form
Standard Form
22 =
2 x 2 =
4
23 =
2 x 2 x 2 =
8
24 =
2 x 2 x 2 x 2 =
16
25 =
2 x 2 x 2 x 2 x 2 =
32
26 =
2 x 2 x 2 x 2 x 2 x 2 =
64
27 =
2 x 2 x 2 x 2 x 2 x 2 x 2 =
128
28 =
2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 =
256
So far we have only examined numbers with a base of 2. Let's look at some examples of writing
exponents where the base is a number other than 2.
Example 2:
Write 3 x 3 x 3 x 3 using exponents, then read your answer aloud.
Solution:
3 x 3 x 3 x 3 = 34
3 raised to the fourth power
Example 3:
Write 6 x 6 x 6 x 6 x 6 using exponents, then read your answer aloud.
Solution:
6 x 6 x 6 x 6 x 6 = 65
6 raised to the fifth power
Example 4:
Write 8 x 8 x 8 x 8 x 8 x 8 x 8 using exponents, then read your answer aloud.
Solution:
8 x 8 x 8 x 8 x 8 x 8 x 8 = 87
8 raised to the seventh power
Example 5:
Write 103, 36, and 18 in factor form and in standard form.
Solution:
Exponential Form
Factor Form
Standard Form
103
10 x 10 x 10
1,000
36
3 x 3 x 3 x 3 x 3 x 3
729
18
1 x 1 x 1 x 1 x 1 x 1 x 1 x 1
1
The following rules apply to numbers with exponents of 0, 1, 2 and 3:
Rule
Example
Any number (except 0) raised to the zero power is equal to 1.
1490 = 1
Any number raised to the first power is always equal to itself.
81 = 8
If a number is raised to the second power, we say it is squared.
32 is read as three squared
If a number is raised to the third power, we say it is cubed.
43 is read as four cubed
Summary:
Whole numbers can be expressed in standard form, in factor form and in exponential form.
Exponential notation makes it easier to write a number as a factor repeatedly.
A number written in exponential form is a base raised to an exponent. The exponent
tells us how many times the base is used as a factor.
Exercises
Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER.
Do not use commas in your answers, just digits.
After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect.
To start over, click CLEAR.
