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Chapter 5—Powers, Exponents, and Roots 54 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.
 Terminology for powers and exponential notation.   The powerof10 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 powerof10 notation is always 10 Your success with powerof10 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:  10^{2} = 10 • 10 = 100 (2 tens multiplied together)
 10^{3} = 10 • 10 • 10 = 1000 (3 tens multiplied together)
 10^{12} = ten multiplied by itself 12 times, or 1,000,000,000,000 (12 tens multiplied together)
Next, there are two special cases Finally, there are the negative exponents:  10^{1} = 1/10 = 0.1
 10^{2} = 1/10^{2} = 1/100 = 0.01
 10^{3} = 1/10^{3} = 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.  Examples and Exercises Powerof10 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 10^{3} = 1000. This can be stated the other way around: 1000 = 10^{3}. Also:  1 = 1 x 10^{0}
 10 = 1 x 10^{1}
 0.1 = 1 x 10^{1}
 0.0001 = 1 x 10^{4}
Examples & Exercises Rewrite these decimals as powers of ten.   Combining Coefficients with Powersof10 Powerof10 notation becomes a lot more useful when combined with coefficients. Definition The main parts of poweroften 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 powerof10 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 10^{3}  1.5 x 10^{2}  45 x 10^{6}  Coefficient: 2  Coefficient: 1.5  Coefficient: 45  Base: 10  Base: 10  Base: 10  Exponent: 3  Exponent: 2  Exponent: 6  Powerof10 Value: 10^{3 }= 1000  Powerof10 Value: 10^{2} = 0.01  Powerof10 Value: 10^{6} = 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 powerof10 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 powerof10 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 "Fortythreethousand twohundred twentynine" 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 10^{4}. 43229 = 4.3222 x 10^{4}  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 powerof10 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 powerof10 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 10^{4} 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: 10^{4 + 2} = 10^{6} This is the same as multiplying the base and exponent by 100: 10^{4} x 100 = 10^{4} x 10^{2} = 10^{6} But as you already know, it is not proper to change the value one part of a math term without compensating with an oppositebutequal 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 10^{4} = 0.0864 x 10^{6}  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 powerof10 notation: 0.006428 x 10^{6} change the value of the exponent from 6 to 2. The change in the exponent looks like this: 10^{2} = 10^{64} 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 10^{6} = 0.006428 (10000) x 10^{64 =} 64.28 x 10^{2} 0.006428 x 10^{6 = }64.28 x 10^{2}  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 powerof10 notation after changing the value of the exponent.  
