1. Exponents and Order of Operations
1.2
Exponents and Order of Operations
Use exponents.
Use the rules for order of operations.
Use more than one grouping symbol.
Know the meanings of ≠, <, >, ≤, and ≥. 2 3 4

2. Objective 1
Use exponents.
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3. 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.”
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:
Squaring, or raising a number to the second power, is NOT the same as doubling the number.
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4. 92 = 9 • 9 = 81 EXAMPLE 1 Evaluating Exponential Expressions
Find the value of each exponential expression.
Solution: 92
= 9 • 9
= 81
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5. Use the rules for order of operations.
Objective 2
Use the rules for order of operations.
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6. 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
If the multiplication is to be performed first, it can be written , which equals , or 11.
If the addition is to be performed first, it can be written , which equals , or 21.
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7. 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.
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8. Use the rules for order of operations. (cont’d)
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.
If no grouping symbols are present, start with Step 1.
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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9. Using the Rules for Order of Operations
EXAMPLE 2
Using the Rules for Order of Operations
Find the value of each expression.
Solution:
In expressions such as 3(7) or (─5)(─4),multiplication is understood.
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10. Use more than one grouping symbol.
Objective 3
Use more than one grouping symbol.
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11. Simplify each expression.
EXAMPLE 3
Using Brackets and Fraction Bars as Grouping Symbols
Simplify each expression.
Solution: or
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12. 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.
The expression can be written as the quotient below, which shows
that the fraction bar “groups” the numerator and denominator separately.
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13. Know the meanings of ≠, <, >, ≤, and ≥.
Objective 4
Know the meanings of ≠, <, >, ≤, and ≥.
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14. 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.
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.
Remember that the “arrowhead” always points to the lesser number.
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15. Know the meanings of ≠, <, >, ≤, and ≥. (cont’d)
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.
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16. True False True False EXAMPLE 4 Using Inequality Symbols
Determine whether each statement is true or false.
Solution:
True
False
True
False
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