1.6 EXPONENTS AND THE ORDER OF OPERATIONS

Writing Whole Numbers and Variables in Exponent Form Student Learning Objectives Recall that in the multiplication problem 3 # 3 # 3 # 3 # 3 = 243 the number 3 is called a factor. We can write the repeated multiplication 3 # 3 # 3 # 3 # 3 using a shorter nota- After studying this section, you will tion, 35, because there are five factors of 3 in the repeated multiplication. We say be able to: 5 5 that 3 is written in exponent form. 3 is read “three to the fifth power.” Write whole numbers and variables in exponent form.

Evaluate numerical and EXPONENT FORM algebraic expressions in The small number 5 is called an exponent. Whole number exponents, except exponent form. zero, tell us how many factors are in the repeated multiplication. The number Use symbols and key words 3 is called the base. The base is the number that is multiplied. for expressing exponents.

Follow the order of operations. 3 # 3 # 3 # 3 # 3 = 35 The exponent is 5.

3 appears as a factor 5 times. The base is 3.

We do not multiply the base 3 by the exponent 5. The 5 just tells us how many 3’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 = 91 and x = x1

EXAMPLE 1 Write in exponent form.

(a)2 # 2 # 2 # 2 # 2 # 2 (b)4 # 4 # 4 # x # x (c) 7 (d) y # y # y # 3 # 3 # 3 # 3 Solution # # # # # = 6 # # # x # x = 3 # x2 3x2 (a)2 2 2 2 2 2 2 (b) 4 4 4 4 or 4 (c)7 = 71 (d) y # y # y # 3 # 3 # 3 # 3 = y3 # 34, or 34y3 Note, it is standard to write the number before the variable in a term. Thus y334 is written 34y3.

Practice Problem 1 Write in exponent form. (a) n (b) 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.

(a)n3 (b) 65 Solution (a)n3 = n # n # n (b) 65 = 6 # 6 # 6 # 6 # 6

Practice Problem 2 Write as a repeated multiplication. (a)x6 (b) 17

55 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 3 Evaluate each expression.

(a)33 (b)19 (c) 24 Solution (a) 33 = 3 # 3 # 3 = 27 (b) 19 = 1 We do not need to write out this multiplication because repeated multiplica- tion of 1 will always equal 1. (c) 24 = 2 # 2 # 2 # 2 = 16

NOTE TO STUDENT: Fully worked-out Practice Problem 3 Evaluate each expression. solutions to all of the Practice Problems can be found at the back of the text (a)43 (b)81 (c) 102 starting at page SP-1

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 = 53. Evaluate 53: 53 = 5 # 5 # 5 = 125. Large numbers are often expressed using a number in exponent form that has a base of 10: 101, 102, 103, 104 and so on. Let’s look for a pattern to find an easy way to evaluate an expression when the base is 10. 10 1 = 1 0 10 3 = (10)(10)(10) = 1 000 10 2 = (10)(10) = 1 00 10 4 = (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 3 there are 3 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 107. Solution

Write 1. 10,000,000

The exponent is 7; attach 7 trailing zeros. 10‡=10,000,000

Practice Problem 4 Find the value of 105.

To evaluate the expression x2 when x is equal to 4, we replace the variable x with the number 4 and find the value of 42: 42 = 4 # 4 = 16. We can write the state- ment “x is equal to 4” using math symbols “x = 4.” Section 1.6 Exponents and the Order of Operations 57

EXAMPLE 5 Evaluate x3 for x = 3. Solution

x 3 (3)3 Replace x with 3. # # 3 3 3 = 27 Write as repeated multiplication, then multiply.

When x = 3, x3 is equal to 27.

Practice Problem 5 Evaluate y2 for y = 8.

Using Symbols and Key Words for Expressing Exponents 2 3 How do you say 10 or 5 ? We can say “10 raised to the power 2,” or “5 raised to the power 3,” but the following phrases are more commonly used.

If the value of the exponent is 2, we say the base is squared. 62 is read “six squared.” If the value of the exponent is 3, we say the base is cubed. 63 is read “six cubed.” If the value of the exponent is greater than 3, we say that the base is raised to the (exponent)th power. 65 is read “six to the fifth power.”

EXAMPLE 6 Translate using symbols.

(a) Five cubed (b) Seven squared (c) y to the eighth power Solution (a)Five cubed = 53 (b)Seven squared = 72 (c) y to the eighth power = y8

Practice Problem 6 Translate using symbols.

(a) Four to the sixth power (b) 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 $3 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 3 + 4152 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 3 + 4152 = 7152 = 35 Wrong!3 + 4152 = 3 + 20 = 23 Correct

Since 3 + 4152 can be written 3 + 15 + 5 + 5 + 52 = 3 + 20, 23 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. 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. 3. 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 3 + 4152, 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. 3. Replace the operation with your result.

EXAMPLE 7 Evaluate. 23 - 6 + 4 Solution

2 3 - 6 + 4 = 8 - 6 + 4 Identify: The highest priority is exponents. Calculate: 2 ## 2 2=8. Replace: 2‹ with 8. = 8 - 6 + 4 Identify: Subtraction has the highest priority. Calculate: 8-6=2. Replace: 8-6 with 2. = 2 + 4 Identify: Addition is last. Calculate: 2+4=6. 2 3 - 6 + 4 = 6 Replace: 2+4 with 6.

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 NOTE TO STUDENT: Fully worked-out first, so we subtract before we add. solutions to all of the Practice Problems can be found at the back of the text Practice Problem 7 Evaluate. 32 + 2 - 5 starting at page SP-1

EXAMPLE 8 Evaluate. 2 # 32 Solution # # 2 32 = 2 9 Identify: The highest priority is exponents. Calculate: 3 # 3=9. 3¤ # Replace: with 9. = 2 9 Identify: Multiplication is last. Calculate: 2 # 9=18. Replace: 2 # 9 with 18. 2 # 32 = 18

CAUTION: 2 # 32 does not equal 62! We must follow the rules for the order of operations and simplify the exponent 32 before we multiply; otherwise, we will get the wrong answer.

Practice Problem 8 Evaluate. 4 # 23 Section 1.6 Exponents and the Order of Operations 59

EXAMPLE 9 Evaluate. 4 + 316 - 222 - 7

Solution We always perform the calculations inside the parentheses first. Once inside the parentheses, we proceed using the order of operations. 4 + 316 - 22 2 - 7 Within the parentheses, exponents have the highest priority: 22 = 4. = 4 + 316 - 4 2 - 7 We must finish all operations inside the = 4 + 31 2 2 - 7 parentheses, so we subtract: 6 - 4 = 2. # = 4 + 6 - 7 The highest priority is multiplication: 3 2 = 6. = 10 - 7 Add first: 4 + 6 = 10. = 3 Subtract last: 10 - 7 = 3. 4 + 316 - 222 - 7 = 3

Practice Problem 9 Evaluate. 2 + 7110 - 3 # 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.

16 + 6 , 32 EXAMPLE 10 Evaluate. 15 - 12 Solution We rewrite the problem as division and then follow the order of operations. 16 + 6 , 32 , 15 - 12 We perform operations inside parentheses first. 16 + 2 2 , 4 6 , 3 = 2; 5 - 1 = 4. 8 , 4 = 2 Divide.

Practice Problem 10 Evaluate. 14 + 8 , 22 17 - 32

Reviewing for an Exam Reviewing for an exam enables you to connect concepts you 3. Read the Chapter Organizer and solve some of the review learned over several classes. Your review activities should problems at the end of the chapter. Check your answers cover all the components of the learning cycle. and redo problems you got wrong. 4. After you finish the exercises in Section 1.6, complete the The Learning Cycle " How Am I Doing? Sections 1.1–1.6. Complete this as if it Reading Writing were the real exam. Do not refer to notes or to the text c T while completing the exercises. Then check your answers. Seeing ; Verbalizing ; Listening The problems you missed are the type of problems that you should get help with and review before the exam. 1. Reread your textbook. Make 3-by-5 study cards as follows. 5. Start reviewing several days before the exam so that you • Write the name of the new term or rule on the front of have time to get help if you need it. the card. Then write the definition of the term or the rule on the back. 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 • Write sample examples on the front of the card and the start the process early so that you are done at least three days solutions on the back. before the exam. • Periodically use these cards as flash cards and quiz yourself, or study with a classmate. Exercise 2. Reread your notes. Study returned homework and quizzes 1. Can you think of other ways of preparing for an exam that and redo problems you got wrong. include activities in the learning cycle?