CCBC Math 081 Order of Operations Section 1.7. Step 2: Exponents and Roots Simplify any numbers being raised to a power and any numbers under the

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1 CCBC Math 081 Order of Operations Order of Operations Now you know how to perform all the operations addition, subtraction, multiplication, division, exponents, and roots. But what if we have a problem that contains more than one operation? For instance, what if we have the problem 5 4? What operation should we perform first the addition, the exponent, or the multiplication? It is important that everyone do the problem the same way in order to get the same answer. For this reason, mathematicians developed a set of rules for evaluating problems that involve more than one arithmetic operation. The rules, called the Order of Operations, specify the order in which the computations should be performed. The Order of Operations is given below. It is important to follow these rules, one step at a time, in the order in which they are presented. ORDER OF OPERATIONS Step 1: Parentheses If there are any operations in parentheses, those computations should be performed first. Step : Exponents and Roots Simplify any numbers being raised to a power and any numbers under the symbol. Step : Multiplication and Division Do these two operations in the order in which they appear, working from left to right. Step 4: Addition and Subtraction Do these two operations in the order in which they appear, working from left to right. To help remember the Order of Operations, try using the phrase in the box on the left below. Please Excuse My Dear Parentheses Exponents and Roots Multiplication and Division... working from left to right Aunt Sally Addition and Subtraction... working from left to right 61

2 CCBC Math 081 Order of Operations 1.7 Before we present some examples using the Order of Operations, keep in mind that the sign of a number always precedes the number. Consider the following problems ( ) ( ) 54 9 Example 1: Evaluate Step 1 Parentheses. There are none. 5 4 Step Exponents and Roots. We have an exponent, gives Step Multiplication and Division. We multiply 9 4 and get Step 4 Addition and Subtraction. We add 5 6 and get The answer is 41. Practice 1: Evaluate 0 5. Answer: 14 Example : Evaluate 8 5 (4 9). 8 5 (49) Step 1 Parentheses. We do the multiplication in the parentheses, 4 9 gives Step Exponents and Roots. There are none Step Multiplication and Division. There are none Step 4 Addition and Subtraction. Since the addition comes before the subtraction, we add 8 5 and get Step 4 Addition and Subtraction (continued). We change the subtraction problem to an addition problem by adding the opposite Step 4 Addition and Subtraction (continued). Now we add. 9 The answer is 9. 6

3 CCBC Math 081 Order of Operations 1.7 Practice : Evaluate 7 4 (8 ). Answer: -1 Example : Evaluate 80 4( ) 9. Remember that the 4( ) means multiplication. 80 4( ) 9 Step 1 Parentheses. There are no operations in parentheses. 80 4( ) 9 Step Exponents and Roots. We have a root, 9 is ( ) Step Multiplication and Division. Working left to right, the division comes first. We divide 80 4 and get 0. 0 (-) Step Multiplication and Division (continued). We multiply 0( ) and get Step 4 Addition and Subtraction. Add ( 40) 7 The answer is 7. Practice : Evaluate 45 5( ) 5. Answer: - and get 7. Example 4: Evaluate 4 ( 14) ( ). 4 (-14) ( ) Step 1 Parentheses. There is a subtraction in the parentheses. Change the subtraction problem 14 to an addition problem. 4 (+-14) ( ) Step 1 Parentheses (continued). Do the addition problem in the parentheses. Add ( 14) to get ( 1) ( ) Step Exponents and Roots. Evaluate negative sign in front. We get (-1) (-) and keep the Step Multiplication and Division. We divide ( 1) ( ) which gives 4. Step 4 Addition and Subtraction. We will change the subtraction problem to an addition problem (-4) Step 4 Addition and Subtraction (continued). Now add. 0 The answer is 0. 6

4 CCBC Math 081 Order of Operations 1.7 Practice 4: Evaluate 5 (5 1) ( ). Answer: Example 5: Evaluate (6 4 64) ( 5). 64 (6 4 ) ( 5) Step 1 Parentheses. There is a subtraction, an addition, and a root in the parentheses. Do the root first, 64 is Step 1 Parentheses (continued). There is a subtraction and ( 8) ( 5) ( 5) an addition in the parentheses. We do the subtraction since it comes first. Subtract 6 4 to get. ( +8 ) Step 1 Parentheses (continued). There is an addition left in the parentheses. Add 8 which is 10. (10) ( 5) (10) (-5) 9 Step Exponents and Roots. Evaluate to get 9. Step Multiplication and Division. Do the multiplication since it comes before the division. So, 10 ( 5) gives ( -50 ) Step Multiplication and Division (continued). Divide 9 ( -5 ) ( 50) to get 5. - Step 4 Addition and Subtraction. We will change the subtraction problem to an addition problem. 9+5 Step 4 Addition and Subtraction (continued). Now add. 4 The answer is 4. Practice 5: Evaluate 16 (8 5 ) ( ). Answer:

5 CCBC Math 081 Order of Operations 1.7 Example 6: Evaluate 4 5. This problem contains the absolute value symbol. Where does this fit in the order of operations? If you look back, you will find that it is not specifically mentioned in the Order of Operations steps. However, in 1. of this chapter we learned that the absolute value symbol is treated as parentheses for the purpose of order of operations. This is because it functions as a grouping symbol just as parentheses do. Therefore, absolute value is included in Step 1 of the Order of Operations Step 1 Parentheses. There are no parentheses, but there is an absolute value symbol. It is treated as parentheses for the purpose of order of operations. The absolute value acts as a grouping symbol, so the operation within the absolute value symbol must be performed before we apply the definition of absolute value. In this case, we must do the subtraction problem 4 first. How? By changing it to an addition problem Step 1 Parentheses (continued). Do the addition problem inside the absolute value symbol. Add 4 to get Step 1 Parentheses (continued). Since there is just one number inside the absolute value symbol, we now apply the definition of absolute value. We know that the absolute value of any non-zero number is positive. So, 7 7 Step Exponents and Roots. There are no exponents or roots. Step Multiplication and Division. There is nothing to multiply or divide. 7+5 Step 4 Addition and Subtraction. We do the addition problem and get 1. = 1 The answer is 1. Practice 6: Evaluate 5 6. Answer: 8 In all these example problems, we performed one operation at a time, and we were careful to bring down the rest of the problem to the next step. It is very important for you to do the same as you complete the practice exercises that follow. Watch All: 65

6 CCBC Math 081 Order of Operations Order of Operations Exercises Evaluate each of the following expressions. Remember that multiplication may be represented with parentheses, or symbols ( ). 15 ( ) ( 7) ( 6) 19. 4( ) 15 ( 4) ( 1) 7. 8 (15 ) 4 8. (8 10) ( ) ( 4)( ) 9. 6 (1 5). 0 5 ( ) 10. 6( 8 4) ( 6 ) (8) 4. 4 ( 8) (5 5) ( 9) 4( 7 ) ( ) ( 7) (17 1) (16 9) 8. 66

7 CCBC Math 081 Order of Operations Order of Operation Exercises Answers

8 CCBC Math 081 Chapter 1 Summary CHAPTER 1 SUMMARY Integers 1.1 Natural Numbers {1,,, 4, 5 } Whole Numbers {0, 1,,, 4, 5 } Integers {, -4, -, -, -1, 0, 1,,, 4, } Negative Positive 1. The Absolute Value of 0 is The Absolute Value of all other real numbers is positive Addition SSS Method Same signs of #s are the same Sum add absolute values Same answer has same sign as original #s DDD Method Different signs of #s are different Difference subtract absolute values Dominant answer has sign of bigger # ( 6) ( 6) 1.4 Subtraction use the Add the Opposite method Subtracting a Positive becomes Adding a Negative: a b a ( b) 58 5 ( 8) Subtracting a Negative becomes Adding a Positive: a ( b) a b 5 ( 8) 58 Multiplication and Division 1.5 Same Signs Positive answer ( ) ( 4) 1 ( 1) ( ) 4 Different Signs Negative answer ( 4) Use of Zero n 0 0 0n 0 n 0 undefined 68

9 CCBC Math 081 Chapter 1 Summary 1.6 Exponents n a - the exponent (n) tells how many times the base (a) is multiplied with itself ( 6) ( 6) ( 6) 6 Roots n a = the number that, when raised to the power n, gives a 8 means. The answer is because (6 6) 6 Note: a Order of Operations Please Excuse My Dear Aunt Sally Parentheses Exponents and Roots Multiplication and Division (working left to right) Addition and Subtraction (working left to right) 4 (9 7)

10 CCBC Math 081 Chapter 1 Review CHAPTER 1 Chapter Review 1. Circle each number system that the number 4 belongs to: Natural Numbers Whole Numbers Integers. Circle all the positive integers in the list of numbers:. What is the value of each of the points named by the capital letters on the number line? 4. Place the numbers 4 and 1 on the number line: A B ( 14) ( 4) ( 9) ( 7) ( ) ( 4) 17. 5( 8)(4) ( 1). ( 5) (1 4)

11 CCBC Math 081 Chapter 1 Review C h a p t e r 1 R e v i e w A n s w e r s 1. Natural Numbers Whole Numbers Integers A = B =