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.
|
1.
|
Write 45 in standard form.
|
|
2.
|
Write 54 in standard form.
|
|
3.
|
What is 500,000,000 raised to the zero power?
|
|
4.
|
What is 237 raised to the first power?
|
|
5.
|
The number 81 is 3 raised to which power?
|
