Exponents, Roots, and Order of Operations. OBJECTIVE 1 Use exponents. In algebra, w e use exponents as a w ay of writing

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1 SECTION 1. Exponents, Roots, and Order of Operations Solve each problem. 91. The highest temperature ever recorded in Juneau, Alaska, was 90 F. The lowest temperature ever recorded there w as -22 F. What is the dif ference between these tw o temperatures? (Source: World Almanac and Book of Facts.) 92. On August 10, 196, a temperature of 120 F w as recorded in P onds, Arkansas. On February 1, 1905, Ozark, Arkansas, recorded a temperature of - 29 F. What is the difference between these two temperatures? (Source: World Almanac and Book of Facts.) 9. Andrew McGinnis has $48.5 in his checking account. He uses his debit card to make purchases of $5.99 and $20.00, w hich overdraws his account. His bank charges his account an overdraft fee of $ He then deposits his paycheck for $66.27 from his part-time job at Arby s. What is the balance in his account? 94. Kayla Koolbeck has $7.50 in her checking account. She uses her debit card to mak e purchases of $25.99 and $19.4, w hich overdraws her account. Her bank char ges her account an overdraft fee of $ She then deposits her pa ycheck for $58.66 from her par t-time job at Subway. What is the balance in her account? 1. Exponents, Roots, and Order of Operations OBJECTIVES 1 Use exponents. 2 Find square roots. Use the order of operations. 4 Evaluate algebraic expressions for given values of variables. Two or more numbers whose product is a third number are factors of that third number. For example, 2 and 6 are f actors of 12, since 2 # 6 = 12. Other integer factors of 12 are 1,, 4, 12, -1, -2, -, -4, -6, and -12. OBJECTIVE 1 Use exponents. In algebra, w e use exponents as a w ay of writing products of repeated factors. For example, the product is written 2 # 2 # 2 # 2 # 2 2 # 2 # 2 # 2 # 2 = factors of 2 The number 5 shows that 2 is used as a f actor 5 times. The number 5 is the exponent, and 2 is the base. 2 5 Exponent Base Read 2 5 as 2 to the f ifth power, or 2 to the f ifth. Multiplying the five 2s gives 2 5 = 2 # 2 # 2 # 2 # 2 = 2. Exponential Expression If a is a real number and n is a natural number, then a n a # a # a #... # a, n factors of a where n is the exponent, a is the base, and Exponents are also called powers. a n is an exponential e xpression.

2 24 CHAPTER 1 Review of the Real Number System EXAMPLE 1 Using Exponential Notation Write using exponents. (a) # Here, 4 is used as a f actor times. 5 5 = a 2 5 b 4 # 4 # 4 = 4 Read as A 5B 2 5 squared. factors of 4 Read 4 as 4 cubed. (c) (-6)(-6)(-6)(-6) = (-6) 4 Read (-6) 4 as -6 to the fourth power, or -6 to the fourth. (d) (0.)(0.)(0.)(0.)(0.) = (0.) 5 (e) x # x # x # x # x # x = x 6 4 # 4 # 4 (b) 2 factors of 5 # (a) = squared, or # # (b) = 6 cubed, or FIGURE Exercises 1, 15, 17, and 19. In parts (a) and (b) of Example 1, we used the terms squared and cubed to refer to powers of 2 and, respecti vely. The term squared comes from the f igure of a square, which has the same measure for both length and width, as shown in Figure 16(a). Similarly, the term cubed comes from the f igure of a cube. As shown in Figure 16(b), the length, width, and height of a cube ha ve the same measure. EXAMPLE 2 Evaluating Exponential Expressions Evaluate. (a) 5 2 = 5 # 5 = 25 5 is used as a factor 2 times. (b) (c) a 2 b 5 2 = 5 # 5, NOT 5 # 2. = 2 # 2 # 2 = = 2 # 2 # 2 # 2 # 2 # 2 = 64 is used as a factor times. Exercises 21 and 27. EXAMPLE Evaluating Exponential Expressions with Negative Signs Evaluate. (a) (-) 5 = (-)(-)(-)(-)(-) = -24 The base is -. (b) (-2) 6 = (-2)(-2)(-2)(-2)(-2)(-2) = 64 The base is -2. (c) -2 6 There are no parentheses. The exponent 6 applies only to the number 2, not to = -(2 # 2 # 2 # 2 # 2 # 2) = -64 The base is 2. Exercises 29, 1, and. Examples (a) and (b) suggest the following generalizations. The product of an odd number of negative factors is negative. The product of an even number of negative factors is positive.

3 SECTION 1. Exponents, Roots, and Order of Operations 25 CAUTION As sho wn in Examples (b) and (c), it is impor tant to distinguish between and -a n (-a) n. -a n = -1(a # a # a #... # a) n factors of a (-a) n = (-a)(-a) #... # (-a) n factors of -a The base is a. The base is -a. Be careful when evaluating an exponential expression with a negative sign. OBJECTIVE 2 Find square roots. As we saw in Example 2(a), 5 2 = 5 # 5 = 25, so 5 squared is 25. The opposite (inverse) of squaring a number is called taking its square root. For example, a square root of 25 is 5. Another square root of 25 is -5, since (-5) 2 = 25. Thus, 25 has two square roots: 5 and -5. We write the positive or principal square root of a number with the symbol 1, called a radical sign. For example, the positive or principal square root of 25 is written The 125 = 5. negative square root of 25 is written = -5. Since the square of any nonzero real number is positive, the square root of a negative number, such as 1 25, is not a real number. EXAMPLE 4 Finding Square Roots Find each square root that is a real number. (a) 16 = 6, since 6 is positive and (b) 10 = 0, since 0 2 = = 6. (c) 9, since a 2 4 b = 9 A 16 = (d) = 0.4, since (0.4) 2 = (e) (f) (g) 1100 = 10, since 10 2 = = -10, since the negative sign is outside the radical sign is not a real number, because the negative sign is inside the radical sign. No real number squared equals Notice the difference among the square roots in parts (e), (f ), and (g). Part (e) is the positive or principal square root of 100, part (f) is the negative square root of 100, and part (g) is the square root of -100, which is not a real number. Exercises 7, 41, 4, and 47. CAUTION The symbol 1 is used onl y for the positive square root, e xcept that 10 = 0. The symbol - 1 is used for the negative square root. OBJECTIVE Use the order of operations. To simplify #, what should we do first add 5 and 2 or multiply 2 and? When an expression involves more than one operation symbol, we use the following order of operations.

4 26 CHAPTER 1 Review of the Real Number System Order of Operations 1. Work separately above and below any fraction bar. 2. If grouping symbols such as parentheses ( ), brackets [ ], or absolute value bars are present, start with the innermost set and work outward.. Evaluate all powers, roots, and absolute values. 4. Multiply or divide in order from left to right. 5. Add or subtract in order from left to right. EXAMPLE 5 Using the Order of Operations Simplify. (a) # = = 11 Add. (b) 24, # Multiplications and divisions are done in the order in which they appear from left to right, so divide first. 24, # = 8 # = = 22 Divide. Add. Exercises 5 and 57. EXAMPLE 6 Using the Order of Operations Simplify. (a) 10, = 10, Subtract inside the absolute value bars. = 10, # 1 Take the absolute value. = Divide; multiply. = 4 Add. (b) Add inside parentheses. Evaluate the power. 2 #, NOT # 2. Add. Subtract. (c) 4 # (2 + 8) = 4 # = 4 # = = 4-10 = 1 2 # 4 + (6, - 7) = 1 2 # 4 + (2-7) Divide inside parentheses. = 1 2 # 4 + (-5) = 2 + (-5) = - Subtract inside parentheses. Add. Exercises 65 and 71.

5 SECTION 1. Exponents, Roots, and Order of Operations 27 EXAMPLE 7 Using the Order of Operations 5 + (-2 )(2) Simplify 6 # 19-9 # 2. Work separately above and below the fraction bar. 5 + (-2 )(2) 6 # 19-9 # 2 = 5 + (-8)(2) 6 # - 9 # 2 = = Evaluate the power and the root. Subtract. Since division by 0 is undef ined, the given expression is undefined. Exercise 75. OBJECTIVE 4 Evaluate algebraic expressions for given values of variables. Any sequence of numbers, v ariables, operation symbols, and/or g rouping symbols for med in accordance with the rules of algebra is called an algebraic expression. 6ab, 5m - 9n, and -2(x 2 + 4y) Algebraic expressions Algebraic expressions have different numerical values for different values of the variables. We evaluate such expressions by substituting given values for the variables. For example, if movie tickets cost $8 each, the amount in dollars you pay for x tickets can be represented by the algebraic expression 8x. We can substitute different numbers of tickets to get the costs of purchasing those tick ets. EXAMPLE 8 Evaluating Algebraic Expressions Evaluate each expression if m = -4, n = 5, p = -6, and q = 25. (a) (b) 5m - 9n = 5(-4) - 9(5) = = -65 m + 2n 4p = (5) 4(-6) = -24 = 6-24 = Use parentheses around substituted values to avoid errors. Substitute; let m = -4 and n = 5. Subtract. Substitute; let m = -4, n = 5, and p = -6. Work separately above and below the fraction bar. Write in lowest terms; also, a - b = - a b.

6 28 CHAPTER 1 Review of the Real Number System (c) -m - n 2 (1q) = -(-4) - (5) 2 (125) Substitute; let m = -4, n = 5, and q = 25. = -(-64) - 25(5) Evaluate the powers and the root. = = 67 Subtract. Notice the careful use of parentheses around substituted v alues. Exercises 79 and Exercises Exercise True or False Decide whether each statement is true or false. If it is false, correct the statement so that it is true = (-4) = (-4) is a positive number # 6 = + (5 # 6) 5. (-2) 7 is a negative number. 6. (-2) 8 is a positive number. 7. The product of 8 positi ve factors and 8 negative factors is positive. 9. In the e xponential e xpression - 5, - is the base. 8. The product of positi ve f actors and negative factors is positive a is positive for all positive numbers a. Concept Check In Exercises 11 and 12, evaluate each exponential expression. 11. (a) 8 2 (b) (a) 4 (b) -4 (c) (-8) 2 (d) -(-8) 2 (c) (-4) (d) -(-4) Write each expression by using exponents. See Example # # # # 10 # 10 # 10 # 10 8 # 8 # # (-9)(-9)(-9) 18. (-4)(-4)(-4)(-4) z # z # z # z # z # z # z a # a # a # a # a Evaluate each expression. See Examples 2 and a 4 4 a 1 4 a b 6 b 5 b 0.91 a 7 10 b 29. (-5) 0. (-2) 5 1. (-2) 8 2. (-) Find each square root. If it is not a real number, say so. See Example A A Matching Match each square root with the appropriate v alue or description. (a) 1144 (b) (c) A. -12 B. 12 C. Not a real number 50. Explain why is not a real number.

7 SECTION 1. Exponents, Roots, and Order of Operations 29 Concept Check In Exercises 51 and 52, a represents a positive number. 51. Is - 1-a positive, negative, or not a real number? 52. Is - 1a positive, negative, or not a real number? Simplify each expression. Use the order of operations. See Examples # # # - 12, # 4-8, , 2 # , # (5) 2 - (-2)(-8) (2) 2 - (-)(-2) # - (-2) # (16) - (-2)(-) , # (-6 - ) -2 -, (-9) - 5 # (164) - (-)(-7) -6-5 (-8) (-8) + 12 # a b, (2 # 6-10) -12a - 4 b - (6 # 5, ) ( )(-2 2 ) (-5) + (-)(-2) a -5-9 b # 7 ( )(- 2 ) -4-1 (-4) + (-5)(-8) (-8) -4a # b - 5(-1-7) (-7) (-8)4 Evaluate each expression if a = -, b = 64, and c = 6. See Example a + 1b a - 1b 81. 1b + c - a 82. 1b - c + a 2c + a c + a a + 2c 84. -a 4 - c b + 6a 2b - 6c Evaluate each expression if w = 4, x = - y = 1 4, 2, and z = See Example wy - 8x 88. wz - 12y 89. xy + y xy - x 2 7x + 9y 7y - 5x 91. -w + 2x + y + z 92. w - 6x + 5y - z w 2w Solve each problem. 95. An appro ximation of the amount in billions of dollars that Americans have spent on their pets from 1998 to 2009 can be obtained b y substituting a given year for x in the expression 2.076x (Source: American P et Products Association.) Approximate the amount spent in each y ear. Round answers to the nearest tenth. (a) 1998 (b) 2005 (c) 2009 (d) How has the amount Americans have spent on their pets changed from 1998 to 2009?

8 0 CHAPTER 1 Review of the Real Number System 96. An approximation of federal spending on education in billions of dollars from 2001 through 2005 can be obtained using the e xpression y = x - 18,071.87, where x represents the year. (Source: U.S. Department of the Treasury.) (a) Use this expression to complete the table. Round answers to the nearest tenth. Year Education Spending (in billions of dollars) (b) How has the amount of federal spending on education changed from 2001 to 2005? 1.4 Properties of Real Numbers OBJECTIVES 1 Use the distributive property. 2 Use the inverse properties. Use the identity properties. 4 Use the commutative and associative properties. 5 Use the multiplication property of Area of left part is 2. = 6. Area of right part is 2. 5 = 10. Area of total rectangle is 2( + 5) = 16. The study of an y object is simplif ied when we know the proper ties of the object. F or example, a property of water is that it freezes when cooled to 0 C. Knowing this helps us to predict the behavior of water. The study of numbers is no different. The basic properties of real numbers studied in this section reflect results that occur consistently in work with numbers, so they have been generalized to apply to expressions with variables as well. OBJECTIVE 1 and so Use the distributive property. Notice that This idea is illustrated by the divided rectangle in Figure 17. Similarly, and so 2( + 5) = 2 # 8 = 16 2 # + 2 # 5 = = 16, 2( + 5) = 2 # + 2 # (-)4 = -4(2) = -8-4(5) + (-4)(-) = = -8, (-)4 = -4(5) + (-4)(-). These examples are generalized to all real numbers as the distributive property of multiplication with respect to addition, or simply the distributive property. Distributive Property For any real numbers a, b, and c, FIGURE 17 a(b c) ab ac and (b c)a ba ca. The distributive property can also be written ab ac a(b c) and ba ca (b c)a