1.6 EXPONENTS AND THE ORDER OF OPERATIONS

Transcription

1 1.6 EXPONENTS AND THE ORDER OF OPERATIONS Writing Whole Numbers and Variables in Exponent Form Student Learning # # # # = 24 Objectives the number is called # # # # Recall that in the multiplication problem a factor. We can write the repeated multiplication using a shorter notation, 5, because there are five factors of in the repeated multiplication. We say that 5 is written in exponent form. 5 is read three to the fifth power. After studying this section, you will be able to: Write whole numbers and variables in exponent form. EXPONENT FORM The small number 5 is called an exponent. Whole number exponents, except zero, tell us how many factors are in the repeated multiplication. The number is called the base. The base is the number that is multiplied. Evaluate numerical and algebraic expressions in exponent form. Use symbols and key words for expressing exponents. # # # # = 5 appears as a factor 5 times. The base is. The exponent is 5. Follow the order of operations. We do not multiply the base by the exponent 5. The 5 just tells us how many s are in the repeated multiplication. If a whole number or variable does not have an exponent visible, the exponent is understood to be 1. 9 = 9 1 and x = x 1 EXAMPLE 1 Write in exponent form. 2 # 2 # 2 # 2 # 2 # 2 4 # 4 # 4 # x # x (c) 7 (d) y # y # y # # # # 4 # 4 # 4 # x # x = 4 2 # 2 # 2 # 2 # 2 # 2 = 2 6 # x 2 or 4 x 2 (c) (d) y # y # y # # # # = y 7 = 7 1 # 4, or 4 y Note, it is standard to write the number before the variable in a term. Thus y 4 is written 4 y. Practice Problem 1 Write in exponent form. n 6 # 6 # y # y # y # y (c) 5 # 5 # 5 # 5 # 5 # 5 # 5 # 5 (d) x # x # 8 # 8 # 8 EXAMPLE 2 Write as a repeated multiplication. n 6 5 n = n # n # n 6 5 = 6 # 6 # 6 # 6 # 6 Practice Problem 2 Write as a repeated multiplication. x

2 56 Chapter 1 Whole Numbers and Introduction to Algebra Evaluating Numerical and Algebraic Expressions in Exponent Form To evaluate, or find the value of, an expression in exponent form, we first write the expression as repeated multiplication, then multiply the factors. EXAMPLE Evaluate each expression. (c) (c) = # # = = 1 We do not need to write out this multiplication because repeated multiplication of 1 will always equal = 2 # 2 # 2 # 2 = 16 NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Practice Problem Evaluate each expression (c) 10 2 Sometimes we are asked to express an answer in exponent form and other times to find the value of (evaluate) an expression. Therefore, it is important that you read the question carefully and express the answer in the correct form. Write 5 # 5 # 5 in exponent form: 5 # 5 # 5 = 5. Evaluate 5 : 5 = 5 # 5 # 5 = 125. Large numbers are often expressed using a number in exponent form that has a base of 10: 10 1, 10 2, 10, 10 4 and so on. Let s look for a pattern to find an easy way to evaluate an expression when the base is = = (10)(10)(10) = = (10)(10) = = (10)(10)(10)(10) = 1 0,000 Notice that when the exponent is 1 there is 1 trailing zero; when the exponent is 2 there are 2 trailing zeros; when it is there are trailing zeros; and so on. Thus to calculate a power of 10, we write 1 and attach the number of trailing zeros named by the exponent. EXAMPLE 4 Evaluate ,000,000 Write =10,000,000 The exponent is 7; attach 7 trailing zeros. Practice Problem 4 Find the value of To evaluate the expression when x is equal to 4, we replace the variable x with the number 4 and find the value of 4 2 : 4 2 = 4 # 4 = 16. We can write the statement x is equal to 4 using math symbols x = 4. x 2

3 Section 1.6 Exponents and the Order of Operations 57 EXAMPLE 5 Evaluate x for x =. x () # # = 27 Replace x with. Write as repeated multiplication, then multiply. When x =, x is equal to 27. Practice Problem 5 Evaluate y 2 for y = 8. Using Symbols and Key Words for Expressing Exponents How do you say 10 2 or 5? We can say 10 raised to the power 2, or 5 raised to the power, but the following phrases are more commonly used. If the value of the exponent is 2, we say the base is squared. 6 2 is read six squared. If the value of the exponent is, we say the base is cubed. 6 is read six cubed. If the value of the exponent is greater than, we say that the base is raised to the (exponent)th power. 6 5 is read six to the fifth power. EXAMPLE 6 Translate using symbols. Five cubed Seven squared (c) y to the eighth power Five cubed = 5 Seven squared = 7 2 (c) y to the eighth power = y 8 Practice Problem 6 Translate using symbols. Four to the sixth power x cubed (c) Ten squared Following the Order of Operations It is often necessary to perform more than one operation to solve a problem. For example, if you bought one pair of socks for $ and 4 undershirts for $5 each, you would multiply first and then add to find the total cost. In other words, the order in which we performed the operations (order of operations) was multiply first, then add. However, the order of operations may not be as clear when dealing with a math statement. When we see the problem written as understanding what to do can be tricky. Do we add, then multiply, or do we multiply before adding? Let s work this calculation both ways. Add First Multiply First = 7152 = 5 Wrong! = + 20 = 2 Correct Since can be written = + 20, 2 is correct. Thus we see that the order of operations makes a difference. The following rule tells which operations to do first: the correct order of operations. We call this a list of priorities.

4 58 Chapter 1 Whole Numbers and Introduction to Algebra ORDER OF OPERATIONS Follow this order of operations. Do first 1. Perform operations inside parentheses. 2. Simplify any expression with exponents.. Multiply or divide from left to right. Do last 4. Add or subtract from left to right. parentheses : exponents : multiply or divide : add or subtract Now, following the order of operations, we can clearly see that to find , we multiply and then add. You will find it easier to follow the order of operations if you keep your work neat and organized, perform one operation at a time, and follow the sequence identify, calculate, replace. 1. Identify the operation that has the highest priority. 2. Calculate this operation.. Replace the operation with your result. EXAMPLE = Identify: The highest priority is exponents. Calculate: 2 # 2 # 2=8. Replace: 2 with = 6 = = Identify: Subtraction has the highest priority. Calculate: 8-6=2. Replace: 8-6 with 2. Identify: Addition is last. Calculate: 2+4=6. Replace: 2+4 with 6. NOTE TO STUDENT: Fully worked-out solutions to all of the Practice Problems can be found at the back of the text starting at page SP-1 Note that addition and subtraction have equal priority. We do the operations as they appear, reading from left to right. In Example 7 the subtraction appears first, so we subtract before we add. Practice Problem EXAMPLE 8 2 # 2 2 # 2 = 2 # 9 Identify: The highest priority is exponents. Calculate: # =9. = 2 # 9 Replace: with 9. Identify: Multiplication is last. Calculate: 2 # 9=18. # 2 # 2 = 18 Replace: 2 9 with 18. CAUTION: 2 # 2 does not equal 6 2! We must follow the rules for the order of operations and simplify the exponent 2 before we multiply; otherwise, we will get the wrong answer. Practice Problem 8 4 # 2

5 Section 1.6 Exponents and the Order of Operations 59 EXAMPLE We always perform the calculations inside the parentheses first. Once inside the parentheses, we proceed using the order of operations = = = = 10-7 = = Within the parentheses, exponents have the highest priority: 2 2 = 4. We must finish all operations inside the parentheses, so we subtract: 6-4 = 2. The highest priority is multiplication: # 2 = 6. Add first: = 10. Subtract last: 10-7 =. Practice Problem # 22-4 As we stated earlier, it is easier to follow the order of operations if we keep our work neat and organized, perform one operation at a time, and follow the sequence: identify, calculate, replace. EXAMPLE 10 operations. We rewrite the problem as division and then follow the order of , 2, , 4 8, 4 = , We perform operations inside parentheses first. 6, = 2; 5-1 = 4. Divide. Practice Problem , Reviewing for an Exam Reviewing for an exam enables you to connect concepts you learned over several classes. Your review activities should cover all the components of the learning cycle. The Learning Cycle Reading " Writing c T Seeing ; Verbalizing ; Listening 1. Reread your textbook. Make -by-5 study cards as follows. Write the name of the new term or rule on the front of the card. Then write the definition of the term or the rule on the back. Write sample examples on the front of the card and the solutions on the back. Periodically use these cards as flash cards and quiz yourself, or study with a classmate. 2. Reread your notes. Study returned homework and quizzes and redo problems you got wrong.. Read the Chapter Organizer and solve some of the review problems at the end of the chapter. Check your answers and redo problems you got wrong. 4. After you finish the exercises in Section 1.6, complete the How Am I Doing? Sections Complete this as if it were the real exam. Do not refer to notes or to the text while completing the exercises. Then check your answers. The problems you missed are the type of problems that you should get help with and review before the exam. 5. Start reviewing several days before the exam so that you have time to get help if you need it. It is not a good idea to complete all six steps at one time. For best results, complete each step at a separate sitting and start the process early so that you are done at least three days before the exam. Exercise 1. Can you think of other ways of preparing for an exam that include activities in the learning cycle?