Exponents

Exponents

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:

21,000,000?

read as?two raised to the millionth power


exponentsExample 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:?

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