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.
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:
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.
|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.
|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:
|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.
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?|
Create an unlimited supply of worksheets for practicing exponents and powers. Students can solve simple expressions involving exponents, such as 33, (1/2)4, (-5)0, or 8-2, or write multiplication expressions using an exponent. The worksheets can be made in html or PDF format (both are easy to print).
Options include negative and zero exponents, and using fractions, decimals, or negative numbers as bases. You can also make worksheets that have one other operation besides exponentiation (add/subtract/multiply/divide powers).
These worksheets are most useful in 6th, 7th, and 8th grade, when exponents are introduced and practiced. Note: variables with exponents are not included (such as practiced in an algebra course).
Basic instructions for the worksheets
Each worksheet is randomly generated and thus unique. The answer key is automatically generated and is placed on the second page of the file.
You can generate the worksheets either in html or PDF format — both are easy to print. To get the PDF worksheet, simply push the button titled "Create PDF" or "Make PDF worksheet". To get the worksheet in html format, push the button "View in browser" or "Make html worksheet". This has the advantage that you can save the worksheet directly from your browser (choose File → Save) and then edit it in Word or other word processing program.
Sometimes the generated worksheet is not exactly what you want. Just try again! To get a different worksheet using the same options:
- PDF format: come back to this page and push the button again.
- Html format: simply refresh the worksheet page in your browser window.
Below you will find some common worksheet types both in html and PDF format. They are randomly generated so unique each time. The answer key is automatically included on the second page. To get a different worksheet using the same options, press 'refresh' in the browser window (only when viewed in browser).
Scroll down the page to the generator if you want to customize the worksheets yourself.
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