Transcript Ch. 1.2 power point

Chapter 1
Section 2
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1.2
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Exponents, Order of Operations,
and Inequality
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.
Interpret data in a bar graph.
Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Objective 1
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 81  3  3  3  3 ,
the factor 3 appears four times, so the product is written as 34 and
is read “3 to the fourth power.”
For this exponential expression, 3 is the base, and 4 is the
exponent, or power.
A number raised to the first power is simply that number.
Example: 51  5
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
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EXAMPLE 1
Evaluating Exponential
Expressions
Find the value of the exponential expression.
1
 
2
4
Solution:
1 1 1 1
1
   

2 2 2 2
16
Notice on your calculator the power (xy) key. Refer to Appendix A,
“An Introduction to Calculators.”
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Objective 2
Use the rules for order of
operations.
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Use the rules for order of operations.
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 5   2  3 , which equals 5  6, or 11.
If the addition is to be performed first, the expression
can be written  5  2  3 , which equals 7  3 , or 21.
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Use the rules for order of operations. (cont’d)
Other grouping symbols include [ ], { }, and
fraction bars.
82
For example, in
, the expression 8  2 is
3
considered to be grouped in the numerator.
To work problems with more than one operation,
we use the following order of operations.
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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.
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.
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.
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EXAMPLE 2
Using the Rules for Order of
Operations
Find the value of each expression.
Solution:
10  6  2  10  3  7
7  6  3 8  1  7  6  3  9   42  27  15
2  32  5  2  9  5  11  5  6
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Objective 3
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 2 8  3  6  5  , can be confusing. For clarity,
we often use brackets , [ ], in place of one pair of
parentheses.
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EXAMPLE 3
Using Brackets and Fraction
Bars as Grouping Symbols
Simplify each expression.
Solution:
9  4  8   3  9 12  3  9  9   81
2 15   2 30  2 28
2  7  8  2



or 2
3  5 1
15  1 14
3  5 1
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Objective 4
Know the meanings of ≠, , ,
≤, and ≥.
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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  8. 7 is not equal to 8.
The symbol  represents “is less than,” so
7  8. 7 is less than 8.
The symbol  means “is greater than.” For example
8  2 . 8 is greater than 2.
Remember 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  9. 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  5. 9 is greater than or equal to 5.
The slash ( / ) can also be used to indicate “not” with the inequality
symbols. </ (is not less than) or >/ (is not greater than).
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EXAMPLE 4
Using Inequality Symbols
Determine whether each statement is true or false.
Solution:
12  6
True
28  4  7
False
21  21
True
1 1

3 4
False
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Objective 5
Translate word statements to
symbols.
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EXAMPLE 5
Translating from Words to
Symbols
Write in symbols:
Nine is equal to eleven minus two.
Solution: 9  11  2
Fourteen is greater than twelve.
  
Two is greater than or equal to two.

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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.
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,
Interchange numbers.
6  10
becomes
10  .
Reverse symbol.
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EXAMPLE 6
Converting between Inequality
Symbols
Write the statement with the inequality symbol reversed.
Solution:
9  15
15  9
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:
410 gives a relationship; 4+10 tells you how to operate on 4 and
10 to get the number 14.
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Objective 7
Interpret data in a bar graph.
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Slide 1.2- 23
EXAMPLE 7
Subtracting Fractions
In what years were the outlays less than 260 billion
dollars?
Solution:
1996, 1997, and 1998
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