Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.3 - 1

Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.3 - 1


Review of the Real Number System

Chapter 1


1.3

Exponents, Roots, and

Order of Operations

Copyright © 2010 Pearson Education, Inc. All rights reserved. Sec 1.3 - 4

1.3 Exponents, Roots, and Order of Operations

Objectives

1. Use exponents.

2. Identify exponents and bases.

3. Find square roots.

4. Use the order of operations.

5. Evaluate algebraic expressionsfor given values of variables.


Using Exponents

Factors are two or more numbers whose product is a

third number. Exponents are a way of writing products

of repeated factors.


factors of

4

4 3

38 3 31 3 3

Base

Exponent

34, read as “3 to the fourth power”, uses 3 as a factor 4 times and equals 81.


Using Exponents

Exponential Expression

If a is a real number and n is a natural number,


factors of

, a

n

n

a a a a a

where n is the exponent, a is the base, and is an exponential expression. Exponents are also called powers.

na



Using Exponential Notation

Write each expression

Using exponents:Exponential notation:

6 · 6 · 6 · 6 · 6 65

(0.7) (0.7) (0.7) (0.7) (0.7)40

m · m · m m3

(–y) (–y) (–y) (–y) (–y)4

2 2

9 9

22

9



Evaluating Exponential Expressions

Evaluate the expression:

Exponential notation:

72 7 · 7 = 49

(0.2)3 (0.2) (0.2) (0.2)= 0.008

m4 m · m · m · m

(–4)4 (–4) (–4) (–4) (–4) = 256

32

5

2 2 2 8

5 5 5 125



Tips to Remember

The product of an even number of negative factors is positive.

The product of an odd number of negative factors is negative.

To raise a number to a power on a calculator, enter the following:

E.g., 23 2 yx 3 = or 2 xy 3 =



Identifying Exponents and Bases

Identify the Exponent and Base

Exponent Base

112 2 11

–43 3 4

(–4)4

4 –4

–(0.8)5 5 0.8



Be Sure to Identify the Base Correctly

CAUTION

factors of

factors of

The base is 1 .

. The base is

n

n a

n

n a

a a a a a a

a a a a a


Square Roots


Squaring a number and taking its square root are opposites.

2

2

Square Square Root

8 64 64

64 64

8 8 8

8 8 8 8

64 has two square roots: 8 and –8.

Principle (positive) square root of 64 is denoted with

Negative square root of 64 is denoted with

.

.


Principle Square Roots


2

If is the principle (positive) square root of , we write .

This means that must be positive, and .

The square of any nonzero real number is positive; so, must

be positive.

The square roo t o

=

=

y x

x

x

y

y

x

y

f a is nnonne ot a gative num real number ber.

2

not real numb

6 since 6

is a since

no real number s

e

quared

36 36

36

36

r


Finding Square Roots


Each square root is given.

2

2

2

2

since is positive and .

0 since is positive and 0 .

since .

since is positive and

since the negative sign is outside the

radic

49 49 49

0.25 .25 .25

0 0

16 16 16

49 4

7 7

.5 0.5

0 0

4 4

7 9 497 .

121al.

11

no real number squared

equals

is not a real number since the negative sign

is inside the radical and

.

1

21

121


Order of Operations


When an expression involves more than one operation symbol, use the following:

1. Work separately above and below any fraction bar.

2. If grouping symbols such as parentheses ( ), square brackets [ ], or absolute value bars | | are present, start with the innermost set and work outward.

3. Evaluate all powers, roots, and absolute values.

4. Do any multiplications or divisions in order, working from left to right.

5. Do any additions or subtractions in order, working from left to right.


= 6 + 18 ÷ (– 3) • 2

–7 + 4 • 6 = –7 + 4 • 6

Evaluating Expressions


Simplify:

6 + 18 ÷ (– 3) • 2

= –7 + 24

=17

= 6 + (–6) • 2

= 6 + (–12)

= –6


5 • 42 + 10 ÷ ( 8 – 6)



Simplify:

= 5 • 42 + 10 ÷ ( 8 – 6)

= 5 • 42 + 10 ÷ 2

= 5 • 16 + 10 ÷ 2

= 80 + 10 ÷ 2

= 80 + 5

= 85




Simplify:

13

• 12 + (– 18 + 15 ÷ 3) 13

• 12 + (– 18 + 15 ÷ 3)=

13

• 12 + (– 18 + 5)=

13

• 12 + (– 13)=

4 + (– 13)=

–9=


Using Order of Operations


Simplify: 33 7

2 8 4 9

3 7

2 8

3

94

5

8

27

3

7

2 4

1 1

7

2

2 7

6

20

4


Algebraic Expressions


Any collection of numbers, variables, operation symbols, and grouping symbols, such as

26 2 7, ,3 4 xy b c x y

is called an algebraic expression. Algebraic expressions have different numerical values for different values of the variables.

and


Evaluating Expressions for Given Values of Variables


The cost for a season pass to a state park is $12 per person. The amount of dollars a family of x members would pay can be represented by $12x.

Cost per person = $12

Number of persons = xTotal cost = $12x

3 member family 5 member family

Total cost = $12x Total cost = $12x

12 • 3 = $36 12 • 5 = $60




If c = 4 and b = –3, evaluate the expression:

3c – 7b = 3(4) – 7(–3)

= 12 + 21

= 33




2

3

7 1

64

r = –1

s = 64

t = –7

Use parentheses to avoid errors.

Given

Substitute and evaluate.

2

3

t r

s

10

149

83

49

3 8

1

50

5




The price per gallon of gasoline can be approximated for the years 2006 – 2008 by substituting a given year for x in the expression

0.17 x – 338.07

and then evaluating. Approximate the price of a gallon of gas in the year 2007.




The approximate price of a gallon of gas in the year 2007, rounded to the nearest cent is

0.17x – 338.07

= 0.17(2007) – 338.07

= 341.19 – 338.07

= $3.12




We can create a table to show how the price of gas changed during these years.

Year Price Per Gallon

2006 $2.95

2007 $3.12

2008 $3.29

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Copyright © 2010 Pearson Education, Inc. All rights reserved Sec 1.3 - 1