Learn about the rules of exponents with the following examples and interactive exercises.
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.
Base^{Exponent}
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 = 2^{5 }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 |
2^{2} = | 2 x 2 = | 4 |
2^{3} = | 2 x 2 x 2 = | 8 |
2^{4} = | 2 x 2 x 2 x 2 = | 16 |
2^{5} = | 2 x 2 x 2 x 2 x 2 = | 32 |
2^{6} = | 2 x 2 x 2 x 2 x 2 x 2 = | 64 |
2^{7} = | 2 x 2 x 2 x 2 x 2 x 2 x 2 = | 128 |
2^{8} = | 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 = 3^{4 }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 = 6^{5 }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 = 8^{7 }8 raised to the seventh power
Example 5: Write 10^{3}, 3^{6}, and 1^{8} in factor form and in standard form.
Solution:
Exponential Form |
Factor Form |
Standard Form |
10^{3} | 10 x 10 x 10 | 1,000 |
3^{6} | 3 x 3 x 3 x 3 x 3 x 3 | 729 |
1^{8} | 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. | 149^{0} = 1 |
Any number raised to the first power is always equal to itself. | 8^{1} = 8 |
If a number is raised to the second power, we say it is squared. | 3^{2} is read as three squared |
If a number is raised to the third power, we say it is cubed. | 4^{3} 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 4^{5} in standard form. |
2. | Write 5^{4} 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? |