chapter 1

Chapter 1 Section 2

Objectives

1

Copyright © 2012, 2008, 2004 Pearson Education, Inc.

Use exponents.

Use the rules for order of operations.

Use more than one grouping symbol.

Know the meanings of ≠, <, >, ≤, and ≥.

Translate word statements to symbols.

Write statements that change the direction of inequality symbols.

1.2 Exponents, Order of Operations, and Inequality

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3

4

5

6


Objective 1

Use exponents.

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Use exponents.

Repeated factors are written with an exponent. For example, in the prime factored form of 81, written , the factor 3 appears four times, so the product is written as 34, read “3 to the fourth power.”

A number raised to the first power is simply that number.

Example:

For this exponential expression, 3 is the base, and 4 is the exponent, or power.

Squaring, or raising a number to the second power, is NOT the same as doubling the number.

15 5

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81 3 3 3 3


Find the value of each exponential expression.

Evaluating Exponential Expressions

41

2

Solution:

1 1 1 1

2 2 2 2

1

16

EXAMPLE 1

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92 = 9 • 9 = 81


Objective 2


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If the addition is to be performed first, it can be written , which equals , or 21.

5 2 3 7 3

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Many problems involve more than one operation. To indicate the order in which the operations should be performed, we often use grouping symbols.

Consider the expression . 5 2 3

If the multiplication is to be performed first, it can be written , which equals , or 11.

5 2 3 5 6


Use the rules for order of operations. (cont’d)

Other grouping symbols include [ ], { }, and fraction bars.

For example, in , the expression is considered to be

grouped in the numerator.

To work problems with more than one operation, we use the following order of operations.

8 2

3

8 2

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Order of Operations

If grouping symbols are present, simplify within them, innermost first (and above and below fraction bars separately), in the following order:

Step 1: Apply all exponents.

Use the memory device “Please Excuse My Dear Aunt Sally” to help remember the rules for order of operations: Parentheses, Exponents, Multiply, Divide, Add, Subtract.

Step 2: Do any multiplications or divisions in the order in which they occur, working from left to right.

Step 3: Do any additions or subtractions in the order in which they occur, working from left to right.

If no grouping symbols are present, start with Step 1.

Slide 1.2-9

Use the rules for order of operations. (cont’d)


EXAMPLE 2

Find the value of each expression.

Solution:

Using the Rules for Order of Operations

10 6 2

7 6 3 8 1

22 3 5 2

10 3 7

7 6 3 9 42 27 15

2 9 10 11 10 1

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In expressions such as 3(7) or (─5)(─4),multiplication is understood.


Objective 3


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EXAMPLE 3

Simplify each expression.

Solution:

Using Brackets and Fraction Bars as Grouping Symbols

9 4 8 3

2 7 8 2

3 5 1

2 15 2

3 5 1

30 2

15 1

32

16 2

9 12 3 9 9

or

81

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An expression with double (or nested) parentheses, such

as , can be confusing. For clarity, we often use brackets , [ ], in place of one pair of parentheses.

2 8 3 6 5

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The expression can be written as the quotient below, which shows

that the fraction bar “groups” the numerator and denominator separately.

4(5 3) 3

2(3) 1

4(5 3) 3 2(3) 1


Objective 4

Know the meanings of

≠, <, >, ≤, and ≥.

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Know the meanings of ≠, <, >, ≤, and ≥.

The symbols ≠, , , ≤, and ≥ are used to express an inequality, a statement that two expressions may not be equal. The equality symbol (=) with a slash though it means “is not equal to.”

For example, 7 is not equal to 8.7 8The symbol represents “is less than,” so

7 is less than 8.7 8The symbol means “is greater than.” For example

8 is greater than 2.8 2

Remember that the “arrowhead” always points to the lesser number.

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Two other symbols, ≤ and ≥, also represent the idea of inequality. The symbol ≤ means “less than or equal to,” so

5 is less than or equal to 9.

Note: If either the part or the = part is true, then the inequality ≤ is true.

The ≥ means “is greater than or equal to.” Again

9 is greater than or equal to 5.

5 9

9 5

Know the meanings of ≠, <, >, ≤, and ≥. (cont’d)

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EXAMPLE 4

Determine whether each statement is true or false.

Solution:

Using Inequality Symbols

12 6

28 4 7

21 21

1 1

3 4

True

False

True

False

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Objective 5

Translate word statements to symbols.

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EXAMPLE 5

Write each word statement in symbols.

Nine is equal to eleven minus two.

Fourteen is greater than twelve.

Two is greater than or equal to two.

Solution:

Translating from Words to Symbols

9 11 2

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Objective 6

Write statements that change the direction of inequality symbols.

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Write statements that change the direction of the inequality symbols.Any statement with can be converted to one with >, and any statement with > can be converted to one with . We do this by reversing the order of the numbers and the direction of the symbol.

For example,

6 10 10 becomes .

Interchange numbers.

Reverse symbol.

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Write the statement as another true statement with the inequality symbol reversed.

EXAMPLE 6 Converting between Inequality Symbols

9 15 15 9

Solution:

Equality and inequality symbols are used to write mathematical sentences, while operations symbols (+, -, ·, and ÷) are used to write mathematical expressions. Compare the following:

Sentence: 4 10 gives a relationship between 4 and 10

Expression: 4 + 10 tells how to operate on 4 and 10 to get 14

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