Copyright © 2013, 2009, 2005 Pearson Education, Inc. Section 1.3 Integer Exponents.

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Section 1.3

Integer Exponents

Copyright © 2013, 2009, 2005 Pearson Education, Inc.

Objectives

• Bases and Positive Exponents

• Zero and Negative Exponents

• Product, Quotient, and Power Rules

• Order of Operations

• Scientific Notation

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Bases and Positive Exponents

The expression 82 is an exponential expression with base 8 and exponent 2.

28Exponent

Base

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Example

Using the given base, write each number as an exponential expression.

a. 100,000 (base 10) b. 128 (base 2)

Solutiona. 100,000

b. 128

510 10 10 10 10 10

72 2 2 2 2 2 2 2

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Let a and b be nonzero real numbers and m and n be positive integers. Then

1.

2.

3.

4. 5.

INTEGER EXPONENTS

... ( factors of )na a a a a n a

0 01 (Note: 0 is undefined.)a

1 1 and = n n

n na a

a a

=n m

m n

a b

b a

=

n na b

b a

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Example

Simplify each expression.a. b. c. d.

Solutiona.

b.

c.

524

1

3

33

4

2

3

3

4t

52

5

1

2

1

2 2 2 2 2

1

32

4

1

343 3 3 3 3 81

33

4

34

3

4 4 4

3 3 3

64

27

d.2

3

3

4t

3

23 4

t

3

36

t

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, 0 and 1,xf x a a a

For any number a and integers m and n,

THE PRODUCT RULE

.m n m na a a

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Example

Multiply and simplify. Use positive exponents.a. b. c. d.Solutiona.

b.

c.

d.

3 610 104 28 8 4 5 6x x x 3 54 3y y

3 610 10 3 610 910 1,000,000,000

4 28 8 4 ( 2)8 28 64

4 5 6x x x 4 ( 5) 6x 5x

3 54 3y y 3 54 3 y y 3 ( 5)12y 212y 2

12

y

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, 0 and 1,xf x a a a

For any nonzero number a and integers m and n,

THE QUOTIENT RULE

.m

m nn

aa

a

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Example

Simplify each expression. Use positive exponents.a. b. c. d.

Solutiona.

b.

c.

d.

3

6

10

10

7

3

x

x

2 4

6

24

6

x y

x y

3 6

5 4

2

6

a b

a b

3

6

10

103 610 310

3

1

10

7 3x 4x2 4

6

24

6

x y

x y

2 4

6

24

6

x y

x y 2 6 4 14x y

3 6

5 4

2

6

a b

a b

4 6

5 3

2

6

b b

a a

10

83

b

a

1

1000

7

3

x

x

4 34x y3

4

4y

x

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, 0 and 1,xf x a a a

For any real number a and integers m and n,

RAISING POWERS TO POWERS

.nm mna a

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Example

Simplify each expression. Use positive exponents.a. b.

c. d.

2 3(6 ) 3 3(2 )

2 36 66

3( 3)2 92

9

1

2

46,6561

512

5 4( )a

5 4a 20a

20

1

a

4 2

3 2

( )

( )

x

x

8

6

x

x

6

8

x

x

2

1

x

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, 0 and 1,xf x a a a

For any real numbers a and b and integer n,

RAISING PRODUCTS TO POWERS

.n n nab a b

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Example

Simplify each expression. Use positive exponents.a. b.

c. d.

3(6 )a 3 3( )x y

3 36 a3216a 33

1

x y

3 3 3

1

x y9 3

1

x y

5 4(2 )ab4 4 5 42 a b

4 2016a b

2 4 2

4 3

(3 )

9( )

w y

wy

2 2 2 4 2

3 4 3

3

9

w y

w y

4 8

3 12

9

9

w y

w y

4wy

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, 0 and 1,xf x a a a

For nonzero numbers a and b and any integer n,

RAISING QUOTIENTS TO POWERS

.n n

n

a a

b b

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Example

Simplify each expression. Use positive exponents.a. b.

c. d.

34

x

2

2

1

3

3

3

4

x

3

64

x

223

1

4

2

3

1 43 81

32

3

2x

w

3 2 3

3 3

2 x

w

6

9

8x

w

6 9

8

x w

23

3 2

4

2

x

y z

2 3 2

2 3 2 2 2

4

2

x

y z

6

6 4

16

4

x

y z

6

6 4

4x

y z

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Using the following order of operations, first perform all calculations within parentheses and absolute values, or above and below the fraction bar. Then use the same order of operations to perform the remaining calculations.

1. Evaluate all exponential expressions. Do any negations after evaluating exponents.

2. Do all multiplication and division from left to right.

3. Do all addition and subtraction from left to right.

ORDER OF OPERATIONS

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Example

Evaluate each expression. a. b. 8 3 2 (5 6) 2 6 9

43 2

18 13 2

8 116

2 6 94

3 2

2 154

5

1516

5

6 31

12 1

9

13

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, 0 and 1,xf x a a a A real number a is in scientific notation when a is written as b 10n , where 1 ≤ |b| < 10 and n is an integer.

SCIENTIFIC NOTATION

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, 0 and 1,xf x a a a 1. Move the decimal point in a number a until it represents

a number b such that 1 ≤ b < 10.

2. Count the number of decimal places that the decimal point was moved. Let this positive integer be n. (If the decimal point is not moved, then a = a 100.)

3. If the decimal point was moved to the left, then a = b 10n.

If the decimal point was moved to the right, then a = b 10-n.

WRITING A POSITIVE NUMBER IN SCIENTIFIC NOTATION

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Important Powers of 10

Number 10-3 10-2 10-1 103 106 109 1012

Value Thousandth Hundredth Tenth Thousand Million Billion Trillion

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Example

Write each number in scientific notation. a. 475,000 b. 0.00000325

Solution

a. 475,000 b.Move the decimal point 5 places to the left.

54.75 10

0.00000325

63.25 10

Move the decimal point 6 places to the right.

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Example

Write each number in standard form. a. b.

Solution

63 10 36.4 10

0.0064

Move the decimal point 6 places to the right since the exponent is positive.

3,000,000

Move the decimal point 3 places to the left since the exponent is negative.

6a. 3 10 3b. 6.4 10