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Using the Order of Operations Mathematicians have established an order of operations to evaluate an expression involving more than one operation. Finally, do additions and subtractions from left to right. Start with operations within grouping symbols. Then evaluate powers. Then do multiplications and divisions from left to right.

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Evaluating Without Grouping Symbols Evaluate the expression when x = 4. 3x 2 + 1 Substitute 4 for x. 3x 2 + 1 Evaluate power. = 3 16 + 1 Evaluate product. = 48 + 1 Evaluate sum. = 49 SOLUTION = 3 4 2 + 1

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Evaluating Without Grouping Symbols Evaluate the expression when x = 4. 32 x 2 – 1 Substitute 4 for x. Evaluate power. = 32 16 – 1 Evaluate quotient. = 2 – 1 Evaluate difference. = 1 SOLUTION 32 x 2 – 1 = 32 4 2 – 1

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INVESTIGATING GROUPING SYMBOLS Without grouping symbols, the expression 3 4 2 + 8 4 has a value of 50. ACTIVITY Developing Concepts 3 (4 2 + 8) 4 Insert grouping symbols in the expression 3 4 2 + 8 4 to produce the indicated values. Using the Order of Operations When you want to change the established order of operations for an expression, you must use parentheses or other grouping symbols. 146 5438 You can insert grouping symbols to produce a different value. For example: = 3 (16 + 8) 4= 3 24 4= 72 4 = 18

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THE LEFT-TO-RIGHT RULE Operations that have the same priority, such as multiplication and division or addition and subtraction, are performed using the left-to-right rule, as shown below. Using the Left-to-Right Rule Work from left to right. = (24 – 8) – 6 = 16 – 6 = 10 24 – 8 – 6

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THE LEFT-TO-RIGHT RULE Operations that have the same priority, such as multiplication and division or addition and subtraction, are performed using the left-to-right rule, as shown below. Using the Left-to-Right Rule Work from left to right. = (24 – 8) – 6 = 16 – 6 = 10 Work from left to right. = (15 2) 6 = 30 6 = 5 24 – 8 – 6 15 2 6

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ORDER OF OPERATIONS First do operations that occur within grouping symbols. Using the Order of Operations

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ORDER OF OPERATIONS First do operations that occur within grouping symbols. Using the Order of Operations Then evaluate powers.

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ORDER OF OPERATIONS First do operations that occur within grouping symbols. Using the Order of Operations Then evaluate powers. Then do multiplications and divisions from left to right.

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ORDER OF OPERATIONS First do operations that occur within grouping symbols. Using the Order of Operations Then evaluate powers. Then do multiplications and divisions from left to right. Finally, do additions and subtractions from left to right.

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Using a Fraction Bar Evaluate power. Simplify the numerator. Work from left to right. Subtract. Simplify. 28 56 = 1212 = 7 4 8 + 7 2 – 1 = 7 4 8 + 49 – 1 28 8 + 49 – 1 = 28 57 – 1 = A fraction bar can act as a grouping symbol: (1 + 2) (4 – 1) = 1 + 2 4 – 1

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Evaluating Expressions with a Calculator Many calculators use the established order of operations, but some do not. When you enter the following in your calculator, does the calculator display 6. 1 or 0. 6 ? – Enter – 1.41.4 SOLUTION 10. 5 – 6. 3 ÷ 2. 1 – 1. 4= 10. 5 – (6. 3 ÷ 2. 1) – 1. 4 = 10.5 – (3) – 1.4 = 6.1 10. 56.36.32.12.1 If your calculator uses order of operations, it will display. 6.16.1

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Evaluating Expressions with a Calculator Many calculators use the established order of operations, but some do not. When you enter the following in your calculator, does the calculator display 6. 1 or 0. 6 ? – Enter – 1.41.4 SOLUTION 10. 5 – 6. 3 ÷ 2. 1 – 1. 4= 10. 5 – (6. 3 ÷ 2. 1) – 1. 4 = 10.5 – (3) – 1.4 = 6.1 If your calculator displays, it performs the operations as they are entered. 0.60.6 SOLUTION [(10. 5 – 6. 3) ÷ 2. 1] – 1. 4= ( 4. 2 ÷ 2. 1) – 1. 4 = 2 – 1. 4 = 0. 6 10. 56.36.32.12.1 If your calculator uses order of operations, it will display. 6.16.1

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