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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 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.
ANSWER BOX:

RESULTS BOX:


2. Write 54 in standard form.
ANSWER BOX:

RESULTS BOX:


3. What is 500,000,000 raised to the zero power?
ANSWER BOX:

RESULTS BOX:


4. What is 237 raised to the first power?
ANSWER BOX:

RESULTS BOX:


5. The number 81 is 3 raised to which power?
ANSWER BOX:

RESULTS BOX:



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

Elementary Math Lessons
Factors and GCF
Multiples and LCM
Primes and Composites
Divisibility Rules
Exponents
Patterns and Exponents
Practice Exercises
Challenge Exercises
Solutions

Related Activities
Exponents and Scientific Notation
Factor Tree Game
Number Theory WebQuest
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