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Chapter 5—Powers, Exponents, and Roots

5-4 Powers of Ten

You should already be familiar with basic exponent notation:

Definition

Power notation—the method for indicating the power of a number—has two parts:
  • The base indicates the number to be multiplied.
  • The exponent indicates the number of times the base is to be multiplied.

fig0701_01.jpg (5380 bytes)

Terminology for powers and exponential notation.

The power-of-10 notation introduced in this lesson is a special version or ordinary exponent notation. The special item is the base -- the base always 10.

The base for power-of-10
notation is always 10

Your success with power-of-10 notation depends entirely upon your understanding of exponents that have a base of 10.

First, recall that powers are simply convenient ways to express how a number is multiplied by itself, over and over again:

  • 102 = 10 10 = 100 (2 tens multiplied together)
  • 103 = 10 10 10 = 1000 (3 tens multiplied together)
  • 1012 = ten multiplied by itself 12 times, or 1,000,000,000,000 (12 tens multiplied together)

Next, there are two special cases

  • 101 = 10
  • 100 = 1
Finally, there are the negative exponents:
  • 10-1 = 1/10 = 0.1
  • 10-2 = 1/102 = 1/100 = 0.01
  • 10-3 = 1/103 = 1/1000 = 0.001
  • 10-12 = 1/1000000000000 = 0.000000000001

 Important

A negative exponent does not mean the decimal value is negative. It means the decimal value is a fraction a value less than 1.

 

Use this button to open a table of powers of ten. You will find a wide range of powers-of-10 expressed in terms of their fraction and decimal values. Make sure you understand exactly what this table means. If you know what it means, you don't have to memorize it ... you will know how to figure out the values for yourself.

Examples and Exercises

Power-of-10 to Decimal

Run through these exercises until you can make the conversions quickly and with no errors.

Rewriting Decimals as Powers of Ten

Earlier in this lesson, you saw that 103 = 1000. This can be stated the other way around: 1000 = 103. Also:

  • 1 = 1 x 100
  • 10 = 1 x 101
  • 0.1 = 1 x 10-1
  • 0.0001 = 1 x 10-4

Examples & Exercises

Rewrite these decimals as powers of ten.

Combining Coefficients with Powers-of-10

Power-of-10 notation becomes a lot more useful when combined with coefficients.

Definition

The main parts of power-of-ten notation are:

  • The coefficient —  the decimal part
  • The base always 10
  • The exponent for the base

Examples

Here are three examples of numbers presented with power-of-10 notation. In each case, identify the coefficient,  base, and exponent. Then use the information to determine the decimal value.

Example 1

Example 2

Example 3

2 x 103

1.5 x 10-2

45 x 106

Coefficient: 2
Coefficient: 1.5
Coefficient: 45
Base: 10
Base: 10
Base: 10
Exponent: 3
Exponent: -2
Exponent: 6
Power-of-10 Value:
103 = 1000
Power-of-10 Value:
10-2 = 0.01
Power-of-10 Value:
106 = 1000000
Decimal Value:
2 x 1000 = 2000
Decimal Value
1.5 x 0.01 = 0.015
Decimal Value:
45 x 1000000 = 45,000,000

Examples & Exercises

Rewrite the given power-of-10 notation as a decimal value.

"Moving" the Decimal Point

The next unit of study for this course introduces the four basic arithmetic operations for powers of ten: adding, subtracting, multiplying and dividing values expressed in power-of-10 notation. As with fractions, however, the values sometimes have to be rewritten in a different form before the arithmetic can be completed. When adding or subtracting fractions, for example you often have to adjust the values of the denominators in order to make them equal -- in preparation for doing the adding or subtracting. In the case of powers of ten, it is often necessary to adjust the values of the coefficient and exponent to meet certain requirements. In  this section, you will learn how to change the location of the decimal  point in the coefficient. In the section that  follows, you well see how you can change the value of the exponent.

NOTE


Moving the Decimal Point  to the Left

Tutorial Example

Example: Part 1

Suppose you are starting with an ordinary decimal value:

43229    "Forty-three-thousand two-hundred twenty-nine"

For reasons to be discussed later, you want to place a decimal  point between the 2 and the 9:

        4322.9

But introducing the decimal point changes the value of the number, and that is illegal unless you do something to restore the original value, but leave the decimal point where it is.

Changing 43229 to 4322.9 is the same as dividing the original number by 10. So to maintain equality, it is necessary to multiply the result by 10:

    43229 = 4322.2  x 10

Equality is maintained, and it is perfectly legal to rewrite 43229 as 4322.2  x 10.

Example: Part 2

Problem: Rewrite 43229 to have the decimal point between the 4 and the 3:  4.3229

In order to put the decimal  point  in that position, you must compensate by multiplying the original value by 10,000, or 104.

43229 = 4.3222  x 104

This animation shows how moving the decimal point to the left is the same as dividing the number by a factor of 10 with each location:

Examples & Exercises

Determine the value of the exponent required relocating the decimal point.

 


Moving the Decimal Point  to the Right

Tutorial Example

Here is a decimal  value where the decimal point appears between the 1 and the 2.

1.2486

Moving the decimal  point one place to the right looks like this 12.486. That is the same as multiplying the original value by a factor of 10. But, again, you should remind yourself that you should not change the value of a number without taking additional action to restore the original  value. Going from 1.2486 to 12.486 is like multiplying by 10 To keep the change legal, the result must be divided by 10. So:

1.2486 = 12.486 x 10-1

Moving the decimal  point another digit to the right is the same as multiplying the value by 10, you must divide  by 100 to maintain the value:

1.2486 = 124.86 x 10-2

 

Examples & Exercises

Determine the value of the exponent required relocating the decimal point.

Moving the Decimal Point  Left or Right

Procedures

For each place you move the decimal point  to the left in the coefficient, you must add 1 to the exponent.

For each place you move the decimal point  to the right in the coefficient, you must subtract 1 from the exponent.

Note

The process of "moving" a decimal point is not a true mathematical principle, but rather a convenient technique for indicating that multiplication or division by 10 has taken place.

 

More Examples

More Examples & Exercises

Rewrite the value given in power-of-10 notation after "moving" the decimal point a given number of places.

 

Remember Why You're Doing This

It is easy to lose sight of the purpose for studying the kinds of principles and procedures presented in this section. For  many people, it is a difficult task; and so it is nice to be reminded it has a purpose that will be come very obvious in later lessons.

Changing the Value of the Exponent

In the previous section of this lesson, you learned  how to adjust the location of the decimal point in the coefficient of power-of-10 notation. This sort of operation is necessary for carrying out operations of arithmetic and algebra with powers of  ten. But that's only one side of the story. In this section, you will learn about how to adjust the exponent to a desired  value, then compensate for the change in value by making an appropriate change in the location of the decimal  point  in the coefficient -- the reverse of the operation demonstrated in the previous section of  this lesson.


Increasing the Value of the Exponent

Tutorial Example

Suppose you are given this value:

8.64 x 104

For some reason (and there will be good reasons), you need to  change the exponent from  4  to 6 -- you need to increase the value of the exponent by 2. The exponent notation looks like this:

104 + 2 = 106

This is the same as multiplying the base and exponent by 100:

        104 x 100 = 104 x 102 = 106

But as you already know,  it is not proper to change the value one part of a math term  without compensating with an opposite-but-equal change in the second part of the term. In this example, it isn't enough to simply increase the value of the exponent from 4 to 6. It is absolutely necessary to make a corresponding adjustment in the coefficient.

Increasing the value of the exponent by 2 is the same thing as multiplying the value by 100. So what must you do with the coefficient? Divide it  by 100.

8.64 x 104 = 0.0864 x 106

Examples & Exercises

Determine the value of the coefficient upon increasing the value of the exponent.

Decreasing the Value of the Exponent

Tutorial Example

Given this value in power-of-10 notation:

0.006428 x 106

change the value of the exponent from 6 to 2.

The change in the exponent looks like this:

102 = 106-4

Decreasing the value of the exponent from 6 to 2 is a matter of dividing by 10000. Dividing the power of ten by 10000 makes it necessary to multiply the coefficient by that same amount -- 10000:

0.006428 x 106 = 0.006428 (10000) x 106-4  = 64.28 x 102

          0.006428 x 106 = 64.28 x 102

Examples & Exercises

Determine the value of the coefficient upon decreasing the value of the exponent.

Increasing and Decreasing the Value of the Exponent

Procedure

Whenever you increase the value of the exponent (multiply by 10), move the decimal  point in the coefficient the same number of places to the left (divide by 10).

Whenever you decrease the value of the exponent, move the decimal  point in the coefficient the same number of places to the right.

Note

Remember: The process of "moving" a decimal point is not a true mathematical principle, but rather a convenient technique for indicating that multiplication or division by 10 has taken place.

Examples & Exercises

Rewrite the value provided in power-of-10 notation after changing the value of the exponent.

 

David L. Heiserman, Editor

Copyright   SweetHaven Publishing Services
All Rights Reserved

Revised: June 06, 2015