
Exponents 

Unit 3
> Lesson 5 of 9 
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 timeconsuming 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?

This lesson is by Gisele Glosser. You can find me on Google.
