© 2010 Pearson Prentice Hall. All rights reserved.

CHAPTER 5

Number Theory and the Real Number System

© 2010 Pearson Prentice Hall. All rights reserved. 2

5.6

Exponents and Scientific Notation

© 2010 Pearson Prentice Hall. All rights reserved. 3

Objectives

1. Use properties of exponents.

2. Convert from scientific to decimal notation.

3. Convert from decimal to scientific notation.

4. Perform computations using scientific notation.

5. Solve applied problems using scientific notation.

© 2010 Pearson Prentice Hall. All rights reserved. 4

Properties of Exponents

Property Meaning Examples

The Product Rule

bm · bn = bm + n

When multiplying exponential expressions with the same base, add the exponents. Use this sum as the exponent of the common base.

96 · 912 = 96 + 12

= 918

The Power Rule

(bm)n = bmn

When an exponential expression is raised to a power, multiply the exponents. Place the product of the exponents on the base and remove the parentheses.

(34)5 = 34·5 = 320

(53)8 = 53·8 = 524

The Quotient Rule When dividing exponential expressions with the same base, subtract the exponent in the denominator from the exponent in the numerator. Use this difference as the exponent of the common base.

nmn

m

bb

b 35540

5

40

84124

12

999

9

555

5

© 2010 Pearson Prentice Hall. All rights reserved. 5

• If b is any real number other than 0, b0 = 1.

The Zero Exponent Rule

© 2010 Pearson Prentice Hall. All rights reserved. 6

Use the zero exponent rule to simplify:

a. 70=1

b.

c. (5)0 = 1

d. 50 = 1

Example 1: Using the Zero Exponent Rule

0 1

© 2010 Pearson Prentice Hall. All rights reserved. 7

• If b is any real number other than 0 and m is a natural number,

The Negative Exponent Rule

.1m

m

bb

© 2010 Pearson Prentice Hall. All rights reserved. 8

Use the negative exponent rule to simplify:

a.

b.

c.

Example 2: Using the Negative Exponent Rule

64

1

88

1

8

18

22

125

1

555

1

5

15

33

7

1

7

17

11

© 2010 Pearson Prentice Hall. All rights reserved. 9

Powers of Ten

1. A positive exponent tells how many zeros follow the 1. For example, 109, is a 1 followed by 9 zeros: 1,000,000,000.

2. A negative exponent tells how many places there are to the right of the decimal point. For example, 10-9 has nine places to the right of the decimal point.

10-9 = 0.000000001

© 2010 Pearson Prentice Hall. All rights reserved. 10

Scientific Notation

A positive number is written in scientific notation when it is expressed in the form

a 10n ,

where a is a number greater than or equal to 1 and less than 10 (1 ≤ a < 10), and n is an integer.

© 2010 Pearson Prentice Hall. All rights reserved. 11

• If n is positive, move the decimal point in a to the right n places.

• If n is negative, move the decimal point in a to the left |n| places.

Convert Scientific Notation to Decimal Notation

© 2010 Pearson Prentice Hall. All rights reserved. 12

Write each number in decimal notation:

a. 2.6 107 b. 1.1 10-4

Solution:

Example 3: Converting from Scientific to Decimal Notation

© 2010 Pearson Prentice Hall. All rights reserved. 13

Converting From Decimal to Scientific Notation

To write the number in the form a 10n:• Determine a, the numerical factor. Move the decimal

point in the given number to obtain a number greater than or equal to 1 and less than 10.

• Determine n, the exponent on 10n. The absolute value of n is the number of places the decimal point was moved. The exponent n is positive if the given number is greater than or equal to 10 and negative if the given number is between 0 and 1.

© 2010 Pearson Prentice Hall. All rights reserved. 14

Write each number in scientific notation:

a. 4,600,000 b. 0.000023

Solution:

Example 4: Converting from Decimal Notation to Scientific Notation

© 2010 Pearson Prentice Hall. All rights reserved. 15

Computations with Scientific Notation

• We use the product rule for exponents to multiply numbers in scientific notation:

(a 10n) (b 10m) = (a b) 10n+m

Add the exponents on 10 and multiply the other parts of the numbers separately.

© 2010 Pearson Prentice Hall. All rights reserved. 16

Example 6: Multiplying Numbers in Scientific Notation

Multiply: (3.4 109)(2 10-5). Write the product in decimal notation.

Solution: (3.4 109)(2 10-5) = (3.4 2)(109 10-5)

= 6.8 109+(-5)

= 6.8 104

= 68,000

© 2010 Pearson Prentice Hall. All rights reserved. 17

• We use the quotient rule for exponents to divide numbers in scientific notation:

Subtract the exponents on 10 and divide the other parts of the numbers separately.

Computations with Scientific Notation

mnm

n

b

a

b

a

1010

10

© 2010 Pearson Prentice Hall. All rights reserved. 18

Divide: . Write the quotient in decimal notation.

Solution:

Example 7: Dividing Numbers In Scientific Notation

4

7

104

104.8

0.0021

102.1

101.2

10

10

4

4.8

104

104.8

3-

)4(7

4

7

4

7

Regroup factors.

Subtract the exponents.

Write the quotient in decimal notation.

© 2010 Pearson Prentice Hall. All rights reserved. 19

Example 9: The National Debt

As of December 2008, the national debt was $10.8 trillion, or 10.8 1012 dollars. At that time, the U.S. population was approximately 306,000,000, or 3.06 108. If the national debt was evenly divided among every individual in the United States, how much would each citizen have to pay?

Solution: The amount each citizen would have to pay is the total debt, 1.08 1013, divided among the number of citizens, 3.06 108.

© 2010 Pearson Prentice Hall. All rights reserved. 20

Every citizen would have to pay approximately $35,300 to the federal government to pay off the national debt.

13 13

8 8

13 8

5

4

1.08 10 1.08 10

3.06 10 3.06 10

0.353 10

0.353 10

3.53 10

35,300

Example 9: The National Debt continued