Section 7.5 Order of Operations with Real Numbers HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Objectives o Use the rules for order of operations to evaluate expressions. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Using the Rules for Order of Operations 1. 2. 3. 4. Rules for Order of Operations Simplify within grouping symbols, such as parentheses ( ), brackets [ ], and braces { }, working from the innermost grouping outward. Find any powers indicated by exponents. Moving from left to right, perform any multiplications or divisions in the order they appear. Moving from left to right, perform any additions or subtractions in the order they appear. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Using the Rules for Order of Operations Notes Other grouping symbols are the absolute value bars 47 (such as 3 5 ), the fraction bar (as in ), and the 10 1 square root symbol (such as 5 11 ). HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Using the Rules for Order of Operations Notes Even though the mnemonic PEMDAS is helpful, remember that multiplication and division are performed as they appear, left to right. Also, addition and subtraction are performed as they appear, left to right. For example 12 3 4 4 4 16, but 12 3 4 36 4 9. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Order of Operations with Integers Use the rules for order of operations to evaluate each of the following expressions. a. 36 4 6 22 Solution 36 4 6 2 36 4 6 4 9 24 15 2 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. exponents. Divide and multiply, left to right. Subtract. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Order of Operations with Integers (cont.) b. 2 32 1 3 23 Solution 2 32 1 3 23 2 9 1 3 8 2 8 3 8 16 24 8 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. exponents Subtract inside the parentheses. Multiply. Subtract (or add algebraically). Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Order of Operations with Integers (cont.) c. 9 2 3 5 72 2 22 Solution 9 2 3 5 72 2 22 9 2 3 5 49 2 4 9 2 15 49 2 4 9 2 34 2 4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. exponents Multiply inside the parentheses. Subtract inside theparentheses. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 1: Order of Operations with Integers (cont.) 9 2 17 4 9 2 13 9 26 35 Divide inside the brackets. Add inside the brackets. Multiply. Add. Note: Because of the rules for order of operations at no time did we even subtract 9 − 2. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Completion Example 2: Order of Operations with Integers Evaluate the expression 6 52 42 2 33. Solution 6 5 4 23 2 2 3 27 25 ____ 16 2 ____ 6 ____ exponents 9 2 27 6 ____ 54 ____ 54 ____ Multiply. 0 ____ Subtract. HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Simplify within parentheses. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Order of Operations with Fractions Use the rules for order of operations to evaluate each of the following expression. 1 1 1 a. 2 3 4 3 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Order of Operations with Fractions (cont.) 1 1 1 Solution 2 3 4 3 7 3 4 3 12 12 7 7 3 12 Change the mixed number to an improper fraction and find the LCD in the parentheses. Add within parentheses. 4 7 12 3 7 Divide and reduce. 4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Order of Operations with Fractions (cont.) 3 5 1 5 b. 5 6 4 2 Solution 2 3 5 1 5 5 6 4 2 2 3 5 1 25 5 6 4 4 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. exponents Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Example 3: Order of Operations with Fractions (cont.) 3 5 1 4 5 6 4 25 Divide, then multiply and reduce. 1 1 2 25 25 2 50 50 27 50 Find the LCD. 2 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Add. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems Use the rules for order of operations to evaluate each expression. 1. 36 2.6 2. 8 2 4 2 42 3. 9 3 2 5 2 3 3 4 2 2 5. 4 8 5 5 3 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. 2 7 5 1 4. 6 12 8 1 1 6. 15 3 10 Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problems (cont.) Simplify. 2 16 2 b 7 7. 5a 3a 7 5b HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved. Practice Problem Answers 1. 38.6 2. 8 3. 27 97 4. 168 13 7. 40 2 5. 3 450 6. 13 HAWKES LEARNING SYSTEMS Students Matter. Success Counts. Copyright © 2013 by Hawkes Learning Systems/Quant Systems, Inc. All rights reserved.