Chapter 1 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

Chapter Chapter 11Section Section 22

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley


Exponents, Order of Operations, and Inequality

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33

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1.21.21.21.2Use 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.Interpret data in a bar graph.77


Objective 11

Use exponents.

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Use exponents.

In algebra, 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 and is read “3 to the fourth power.”

81 3 3 3 3

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. For example, 32 means 3·3, not 2·3

15 5

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EXAMPLE 1

Find the value of the exponential expression.

Evaluating Exponential Expressions

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2

Solution:

1 1 1 1

2 2 2 2

1

16

Notice on your calculator the power (xy) key. Refer to Appendix A, “An Introduction to Calculators.”

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Objective 22

Use the rules for order of operations.

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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 .

Use the rules for order of operations.

5 2 3

5 2 3 5 6

If the addition is to be performed first, the expression can be written , which equals , or 21. 5 2 3 7 3

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

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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.

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

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.

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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

10 3 7

7 6 3 9 42 27 15

2 9 5 11 5 6

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Objective 33

Use more than one grouping symbol.

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Use more than one grouping symbol.

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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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

28

14 2

9 12 3 9 9

or

81

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Objective 44

Know the meanings of ≠, , , ≤, and ≥.

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The symbol represents “is less than,” so

. 7 is less than 8.

The symbol means “is greater than.” For example

. 8 is greater than 2.

Know the meanings of ≠, , , ≤, and ≥. The symbols ≠, , , ≤, and ≥ are used to express

inequality, a statement that two expressions are not equal. The equality symbol (=) with a slash though it means “is not equal to.” For example,

. 7 is not equal to 8.7 8

7 8

8 2Remember that the “arrowhead” always points to the lesser number.

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Know the meanings of ≠, , , ≤, and ≥. (con’t)

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

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The slash ( / ) can also be used to indicate “not” with the inequality symbols. < (is not less than) or > (is not greater than). //


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 55

Translate word statements to symbols.

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EXAMPLE 5

Write 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 toSymbols

9 11 2

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Objective 66

Write statements that change the direction of inequality symbols.

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Write statements that change the direction of the inequality.

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

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becomes .

Interchange numbers.

Reverse symbol.


Write the statement with the inequality symbol reversed.

EXAMPLE 6Converting between InequalitySymbols

9 15 15 9Solution:

The equality and inequality symbols are used to write mathematical sentences. They are different from symbols of operations (+, -, ·, and ÷) which are used for mathematical expressions. For example compare:

410 gives a relationship; 4+10 tells you how to operate on 4 and 10 to get the number 14.

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Objective 77

Interpret data in a bar graph.

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EXAMPLE 7 Subtracting FractionsEXAMPLE 7

In what years were the outlays less than 260 billion dollars?

Solution:

1996, 1997, and 1998

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Chapter 1 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.

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Transcript of Chapter 1 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley.