Foundations for Algebra Chapter 1 PDF

Foundations for Algebra 1A The Language of Algebra 1-1 Variables and Expressions Lab Create a Table to Evaluate Expressions 1-2 Adding and Subtracting Real Numbers 1-3 Multiplying and Dividing Real Numbers 1-4 Powers and Exponents 1-5 Roots and Real Numbers 1B The Tools of Algebra 1-6 O...

Foundations for Algebra 1A The Language of Algebra 1-1 Variables and Expressions Lab Create a Table to Evaluate Expressions 1-2 Adding and Subtracting Real Numbers 1-3 Multiplying and Dividing Real Numbers 1-4 Powers and Exponents 1-5 Roots and Real Numbers 1B The Tools of Algebra 1-6 Order of Operations 1-7 Simplifying Expressions 1-8 Introduction to Functions • Solve problems with real numbers. • Make connections between verbal and algebraic representations. Discovering the “Magic” You can use the operations and properties in this chapter to complete a magic square. KEYWORD: MA7 ChProj 2 Chapter 1 Vocabulary Match each term on the left with a definition on the right. A. the distance around a figure 1. difference B. a number that is multiplied by another number to form a product 2. factor 3. perimeter C. a result of division 4. area D. the number of square units a figure covers E. a result of subtraction Whole Number Operations Add, subtract, multiply, or divide. 5. 23 + 6 6. 156 ÷ 12 7. 18 × 96 8. 85 - 62 Add and Subtract Decimals Add or subtract. 9. 2.18 + 6.9 10. 0.32 - 0.18 11. 29.34 + 0.27 12. 4 - 1.82 15. 1.5 × 1.5 16. 3.04 × 0.12 19. 7.2 ÷ 0.4 20. 92.7 ÷ 0.3 Multiply Decimals Multiply. 13. 0.7 × 0.6 14. 2.5 × 0.1 Divide Decimals Divide. 17. 6.15 ÷ 3 18. 8.64 ÷ 2 Multiply and Divide Fractions Multiply or divide. Give your answer in simplest form. 3 ×_ 7 ×_ 1 2 ÷_ 1 4 21. _ 22. _ 23. _ 5 2 3 6 8 7 2 24. 4 ÷ _ 3 Add and Subtract Fractions Add or subtract. Give your answer in simplest form. 3 -_ 2 +_ 2 1 1 +_ 1 25. _ 26. _ 27. _ 5 5 8 8 2 4 2 -_ 4 28. _ 3 9 Foundations for Algebra 3 Key Vocabulary/Vocabulario Previously, you • learned words related to • • • mathematical operations. identified numbers on a real number line. performed operations on whole numbers, decimals, and fractions. plotted points in the coordinate plane. You will study • how to evaluate and simplify • • • expressions. properties of the real number system. the order of operations. patterns formed by points plotted in the coordinate plane. additive inverse inverso aditivo coefficient coeficiente constant constante coordinate plane plano cartesiano irrational numbers números irracionales like terms términos semejantes origin origen rational numbers números racionales variable variable Vocabulary Connections To become familiar with some of the vocabulary terms in the chapter, consider the following. You may refer to the chapter, the glossary, or a dictionary if you like. 1. The word variable comes from the word vary. What does vary mean? Which of the key vocabulary terms above has the opposite meaning? 2. Another word for inverse is reverse. The word additive relates to the operation of addition. What do you think an additive inverse is? You can use the skills in this chapter • to form a solid foundation for • • 4 Chapter 1 the rest of this algebra course. in other classes, such as Biology, History, and Physics. to determine final costs, stock values, and profit. 3. The prefix ir- means “not.” What relationship do you think rational numbers and irrational numbers may have? 4. To originate means “to begin at.” What do you think the origin of a coordinate plane is? Reading Strategy: Use Your Book for Success Understanding how your textbook is organized will help you locate and use helpful information. Pay attention to the margin notes. Know-It Note icons point out key information. Writing Math notes, Helpful Hints, and Caution notes help you understand concepts and avoid common mistakes. / iÃiÊiÝ«ÀiÃȜ˜ÃÊ “i>˜ÊºÓÊ̈“iÃÊÞ»\ ÓÞÊ Þ ÓÊÊ ÊÞ The Glossary is found in the back of your textbook. Use it as a resource when you need the definition of an unfamiliar word or property. The Index is located at the end of your textbook. Use it to locate the page where a particular concept is taught. Glossary/Glos Index A A ENGLISH absolute value (p. 14) The absolute value of x is the distance from zero to x on a number line, d d
ÊÀi«>Vi“i˜ÌÊÃiÌ >ÊÃiÌʜvʘՓLiÀÃÊÌ V>˜ÊLiÊÃÕLÃ̈ÌÕÌi` vœÀ > Û>Àˆ>Li Aaron, Hank, 42 Absolute error, S55 Absolute value, 14, 148 equations, 148–149 functions, 366–367 ˜ÊÌ iÊiÝ«ÀiÃȜ˜ÊÊ xÊÓÊ Ê]ÊxʈÃÊÌ iÊL>Ãi LiV>ÕÃiÊÌ iʘi}>Ì Ãˆ}˜ ˆÃ ˜œÌ ˆ˜ The Problem-Solving Handbook is found in the back of your textbook. These pages review strategies that can help you solve realworld problems. Problem Sol Draw a Diagram You can draw a diagram t the words of a problem are d Try This Use your textbook for the following problems. 1. Use the index to find the page where each term is defined: algebraic expression, like terms, ordered pair, real numbers. 2. What mnemonic device is taught in a Helpful Hint in Lesson 1-6, Order of Operations? 3. Use the glossary to find the definition of each term: additive inverse, constant, perfect square, reciprocal. Foundations for Algebra 5 1-1 Variables and Expressions Objectives Translate between words and algebra. Why learn this? Variables and expressions can be used to determine how many plastic drink bottles must be recycled to make enough carpet for a house. Evaluate algebraic expressions. Vocabulary variable constant numerical expression algebraic expression evaluate A home that is “green built” uses many recycled products, including carpet made from recycled plastic drink bottles. You can determine how many square feet of carpet can be made from a certain number of plastic drink bottles by using variables, constants, and expressions. A variable is a letter or symbol used to represent a value that can change. Container City, in East London, UK, is a development of buildings made from recycled shipping containers. A constant is a value that does not change. A numerical expression may contain only constants and/or operations. An algebraic expression may contain variables, constants, and/or operations. You will need to translate between algebraic expressions and words to be successful in math. The diagram below shows some of the ways to write mathematical operations with words. Plus, sum, increased by EXAMPLE 1 Minus, difference, Times, product, less than equal groups of Divided by, quotient Translating from Algebraic Symbols to Words Give two ways to write each algebraic expression in words. A x+3 These expressions all mean “2 times y”: 2y 2 ( y) 2·y (2)(y) 2 × y (2) y B m-7 the sum of x and 3 x increased by 3 the difference of m and 7 7 less than m C 2·y D k÷5 2 times y the product of 2 and y k divided by 5 the quotient of k and 5 Give two ways to write each algebraic expression in words. 1a. 4 - n 6 Chapter 1 Foundations for Algebra t 1b. _ 5 1c. 9 + q 1d. 3(h) To translate words into algebraic expressions, look for words that indicate the action that is taking place. Put together, combine EXAMPLE 2 Find how much more or less Put together equal groups Separate into equal groups Translating from Words to Algebraic Symbols A Eve reads 25 pages per hour. Write an expression for the number of pages she reads in h hours. h represents the number of hours that Eve reads. 25 · h or 25h Think: h groups of 25 pages. B Sam is 2 years younger than Sue, who is y years old. Write an expression for Sam’s age. y represents Sue’s age. y-2 Think: “younger than” means “less than.” C William runs a mile in 12 minutes. Write an expression for the number of miles that William runs in m minutes. m represents the total time William runs. m _ Think: How many groups of 12 are in m? 12 2a. Lou drives at 65 mi/h. Write an expression for the number of miles that Lou drives in t hours. 2b. Miriam is 5 cm taller than her sister, who is m cm tall. Write an expression for Miriam’s height in centimeters. 2c. Elaine earns $32 per day. Write an expression for the amount that she earns in d days. To evaluate an expression is to find its value. To evaluate an algebraic expression, substitute numbers for the variables in the expression and then simplify the expression. EXAMPLE 3 Evaluating Algebraic Expressions Evaluate each expression for x = 8, y = 5, and z = 4. A x+y x+y=8+5 = 13 B Substitute 8 for x and 5 for y. Simplify. _x z x =_ 8 _ z 4 =2 Substitute 8 for x and 4 for z. Simplify. Evaluate each expression for m = 3, n = 2, and p = 9. 3a. mn 3b. p - n 3c. p ÷ m 1- 1 Variables and Expressions 7 EXAMPLE 4 Recycling Application Approximately fourteen 20-ounce plastic drink bottles must be recycled to produce 1 square foot of carpet. a. Write an expression for the number of bottles needed to make c square feet of carpet. The expression 14c models the number of bottles needed to make c square feet of carpet. A replacement set is a set of numbers that can be substituted for a variable. The replacement set in Example 4 is {40, 120, 224}. b. Find the number of bottles needed to make 40, 120, and 224 square feet of carpet. Evaluate 14c for c = 40, 120, and 224. c 14 c 40 14 (40) = 560 120 14 (120) = 1680 224 14 (224) = 3136 To make 40 ft 2 of carpet, 560 bottles are needed. To make 120 ft 2 of carpet, 1680 bottles are needed. To make 224 ft 2 of carpet, 3136 bottles are needed. 4. To make one sweater, sixty-three 20-ounce plastic drink bottles must be recycled. a. Write an expression for the number of bottles needed to make s sweaters. b. Find the number of bottles needed to make 12, 25, and 50 sweaters. THINK AND DISCUSS 1. Write two ways to suggest each of the following, using words or phrases: addition, subtraction, multiplication, division. 2. Explain the difference between a numerical expression and an algebraic expression. 3. GET ORGANIZED Copy and complete the graphic organizer. Next to each operation, write a word phrase in the left box and its corresponding algebraic expression in the right box. 8 Chapter 1 Foundations for Algebra Words Algebra Addition Subtraction Multiplication Division 1-1 Exercises KEYWORD: MA7 1-1 KEYWORD: MA7 Parent GUIDED PRACTICE 1. Vocabulary A(n) constant, or variable) SEE EXAMPLE 1 p. 6 SEE EXAMPLE 2 p. 7 is a value that can change. (algebraic expression, ? Give two ways to write each algebraic expression in words. 2. n - 5 f 3. _ 3 4. c + 15 5. 9 - y x 6. _ 12 7. t + 12 8. 8x 9. x - 3 10. George drives at 45 mi/h. Write an expression for the number of miles George travels in h hours. 11. The length of a rectangle is 4 units greater than its width w. Write an expression for the length of the rectangle. SEE EXAMPLE 3 p. 7 SEE EXAMPLE 4 p. 8 Evaluate each expression for a = 3, b = 4, and c = 2. 12. a - c 13. ab 14. b ÷ c 15. ac 16. Brianna practices the piano 30 minutes each day. a. Write an expression for the number of hours she practices in d days. b. Find the number of hours Brianna practices in 2, 4, and 10 days. PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 17–24 25–26 27–30 31 1 2 3 4 Extra Practice Skills Practice p. S4 Application Practice p. S28 Give two ways to write each algebraic expression in words. 17. 5p 18. 4 - y 19. 3 + x 20. 3y 21. -3s 22. r ÷ 5 23. 14 - t 24. x + 0.5 25. Friday’s temperature was 20° warmer than Monday’s temperature t. Write an expression for Friday’s temperature. 26. Ann sleeps 8 hours per night. Write an expression for the number of hours Ann sleeps in n nights. Evaluate each expression for r = 6, s = 5, and t = 3. 27. r - s 28. s + t 29. r ÷ t 30. sr 31. Jim is paid for overtime when he works more than 40 hours per week. a. Write an expression for the number of hours he works overtime when he works h hours. b. Find the number of hours Jim works overtime when he works 40, 44, 48, and 52 hours. 32. Write About It Write a paragraph that explains to another student how to evaluate an expression. Write an algebraic expression for each verbal expression. Then write a real-world situation that could be modeled by the expression. 33. the product of 2 and x 34. b less than 17 35. 10 more than y 1- 1 Variables and Expressions 9 36. This problem will prepare you for the Multi-Step Test Prep on page 38. The air around you puts pressure on your body equal to 14.7 pounds per square inch (psi). When you are underwater, the water exerts additional pressure on your body. For each foot you are below the surface of the water, the pressure increases by 0.445 psi. a. What does 14.7 represent in the expression 14.7 + 0.445d? b. What does d represent in the expression? c. What is the total pressure exerted on a person’s body when d = 8 ft? 37. Geometry The length of a rectangle is 9 inches. Write an expression for the area of the rectangle if the width is w inches. Find the area of the rectangle when the width is 1, 8, 9, and 11 inches. 38. Geometry The perimeter of any rectangle is the sum of its lengths and widths. The area of any rectangle is the length  times the width w. a. Write an expression for the perimeter of a rectangle. b. Find the perimeter of the rectangle shown. c. Write an expression for the area of a rectangle. d. Find the area of the rectangle shown. ŰÊÊ£{ÊV“ ÜÊÊnÊV“ Complete each table. Evaluate the expression for each value of x. 39. Astronomy x x + 12 40. x 41. 10 x x÷2 x 1 1 12 2 5 20 3 10 26 4 15 30 42. Astronomy An object’s weight on Mars can be found by multiplying 0.38 by the object’s weight on Earth. a. An object weighs p pounds on Earth. Write an expression for its weight on Mars. b. Dana weighs 120 pounds, and her bicycle weighs 44 pounds. How much would Dana and her bicycle together weigh on Mars? Average Annual Precipitation 58.53 Rainfall (in.) 60 47.84 37.07 40 16.84 20 8.29 D Ta S co ea m ttl a, e– W A M ia m i, FL N ar ck , sm Bi H ou st on x, A ,T X Z 0 oe ni 43. Meteorology Use the bar graph to write an expression for the average annual precipitation in New York, New York. a. The average annual precipitation in New York is m inches more than the average annual precipitation in Houston, Texas. b. The average annual precipitation in New York is s inches less than the average annual precipitation in Miami, Florida. Ph A crater on Canada’s Devon Island is geologically similar to the surface of Mars. However, the temperature on Devon Island is about 37 °F in summer, and the average summer temperature on Mars is -85 °F. City 10 Chapter 1 Foundations for Algebra 44. Critical Thinking Compare algebraic expressions and numerical expressions. Give examples of each. Write an algebraic expression for each verbal expression. Then evaluate the algebraic expression for the given values of x. Verbal x reduced by 5 45. 7 more than x 46. The quotient of x and 2 47. The sum of x and 3 Algebraic x = 12 x = 14 x-5 12 - 5 = 7 14 - 5 = 9 48. Claire has had her driver’s license for 3 years. Bill has had his license for b fewer years than Claire. Which expression can be used to show the number of years Bill has had his driver’s license? 3-b b-3 3+b b+3 49. Which expression represents x? 12 - 5 12(5) 12 + 5 12 ÷ 5 Ý x £Ó 50. Which situation is best modeled by the expression 25 - x? George places x more video games on a shelf with 25 games. Sarah has driven x miles of a 25-mile trip. Amelia paid 25 dollars of an x dollar lunch that she shared with Ariel. Jorge has 25 boxes full of x baseball cards each. CHALLENGE AND EXTEND Evaluate each expression for the given values of the variables. 51. 2ab ; a = 6, b = 3 52. 2x + y ; x = 4, y = 5 53. 3x ÷ 6 y ; x = 6, y = 3 54. Multi-Step An Internet service provider charges $9.95/month for the first 20 hours and $0.50 for each additional hour. Write an expression representing the charges for h hours of use in one month when h is more than 20 hours. What is the charge for 35 hours? SPIRAL REVIEW The sum of the angle measures in a triangle is 180°. Find the measure of the third angle given the other two angle measures. (Previous course) 55. 45° and 90° 56. 120° and 20° 57. 30° and 60° Write an equivalent fraction for each percent. (Previous course) 58. 25% 59. 50% 60. 75% 61. 100% Find a pattern and use it to give the next three numbers. (Previous course) 62. 4, 12, 20, 28, … 63. 3, 9, 27, 81, 243, … 64. 2, 3, 5, 8, 12, … 1- 1 Variables and Expressions 11 1-1 Create a Table to Evaluate Expressions You can use a graphing calculator to quickly evaluate expressions for many values of the variable. Use with Lesson 1-1 Activity 1 KEYWORD: MA7 Lab1 Evaluate 2x + 7 for x = 25, 125, 225, 325, and 425. 1 Press and enter 2X+7 for Y1. 2 Determine a pattern for the values of x. The x-values start with 25 and increase by 100. to view the Table Setup window. 3 Press Enter 25 as the starting value in TblStart=. Enter 100 as the amount by which x changes in Tbl=. to create a table of values. 4 Press The first column shows values of x starting with 25 and increasing by 100. The second column shows values of the expression 2x + 7 when x is equal to the value in the first column. You can use the arrow keys to view the table when x is greater than 625. Try This 1. Use the table feature of a graphing calculator to evaluate 5x - 7 for x = 4, 6, 8, 10, and 12. a. What value did you enter in TblStart=? b. What value did you enter in Tbl=? 2. Use the table feature of a graphing calculator to evaluate 3x + 4 for x = -5, -1, 3, 7, and 11. a. What value did you enter in TblStart=? b. What value did you enter in Tbl=? 12 Chapter 1 Foundations for Algebra You can also use a spreadsheet program to evaluate expressions. Activity 2 Evaluate 2x + 7 for x = 3, 5, 7, 9, and 11. 1 In the first column, enter the values 3, 5, 7, 9, and 11. 2 Enter the expression in cell B1. To do this, type the following: = 2 * A1 + 7 3 Press Enter. The value of 2x + 7 when x = 3 appears in cell B1. 4 Copy the formula into cells B2, B3, B4, and B5. Use the mouse to click on the lower right corner of cell B1. Hold down the mouse button and drag the cursor through cell B5. For each row in column B, the number that is substituted for x is the value in the same row of column A. You can continue the table by entering more values in column A and copying the formula from B1 into more cells in column B. Try This 3. Use a spreadsheet program to evaluate -2x + 9 for x = -5, -2, 1, 4, and 7. a. What values did you enter in column A? b. What did you type in cell B1? 4. Use a spreadsheet program to evaluate 7x - 10 for x = 2, 7, 12, 17, and 22. a. What values did you enter in column A? b. What did you type in cell B1? 1-1 Technology Lab 13 1-2 Adding and Subtracting Real Numbers Objectives Add real numbers. Why learn this? The total length of a penguin’s dive can be determined by adding real numbers. (See Example 4.) Subtract real numbers. Vocabulary real numbers absolute value opposites additive inverse The set of all numbers that can be represented on a number line are called real numbers . You can use a number line to model addition and subtraction of real numbers. Addition To model addition of a positive number, move right. To model addition of a negative number, move left. Subtraction To model subtraction of a positive number, move left. To model subtraction of a negative number, move right. EXAMPLE 1 Adding and Subtracting Numbers on a Number Line Add or subtract using a number line. A -3 + 6 È Î Start at 0. Move left to -3. x { Î Ó £ ä £ Ó Î { x To add 6, move right 6 units. -3 + 6 = 3 B -2 - (-9) -(-9) -2 -3 -2 -1 Start at 0. Move left to -2. 0 1 2 3 4 5 6 7 To subtract -9, move right 9 units. -2 - (-9) = 7 Add or subtract using a number line. 1a. -3 + 7 1b. -3 - 7 1c. -5 - (-6.5) The absolute value of a number is its distance from zero on a number line. The absolute value of 5 is written as ⎪5⎥. 5 units -6 -5 -4 -3 -2 -1 14 Chapter 1 Foundations for Algebra 5 units 0 1 2 3 ⎪5⎥ = 5 4 5 6 ⎪-5⎥ = 5 Adding Real Numbers WORDS NUMBERS Adding Numbers with the Same Sign Add the absolute values and use the sign of the numbers. 3+6 -2 + (-9) 9 -11 -8 + 12 3 + (-15) 4 -12 Adding Numbers with Different Signs Subtract the absolute values and use the sign of the number with the greater absolute value. EXAMPLE 2 Adding Real Numbers Add. A -3 + (-16) Same signs: add the absolute values. (3 + 16 = 19) -19 Both numbers are negative, so the sum is negative. B -13 + 7 Different signs: subtract the absolute values. (13 - 7 = 6) -6 Use the sign of the number with the greater absolute value. C 6.2 + (-4.9) Different signs: subtract the absolute values. (6.2 - 4.9 = 1.3) 1.3 Use the sign of the number with the greater absolute value. Add. 2a. -5 + (-7) 2b. -13.5 + (-22.3) 2c. 52 + (-68) Two numbers are opposites if their sum is 0. A number and its opposite are additive inverses and are the same distance from zero. They have the same absolute value. Inverse Property of Addition Because adding 0 to a number does not change the number’s value, 0 is called the additive identity. Two numbers are additive inverses if their sum is the additive identity. WORDS The sum of a real number and its opposite is 0. NUMBERS ALGEBRA 6 + (-6) = (-6) + 6 = 0 For any real number a, a + (-a) = (-a) + a = 0 To subtract signed numbers, you can use additive inverses. Subtracting a number is the same as adding the opposite of the number. Subtracting Real Numbers WORDS To subtract a number, add its opposite. Then follow the rules for adding signed numbers. NUMBERS 3 - 8 = 3 + (-8) = -5 ALGEBRA a - b = a + (-b) 1- 2 Adding and Subtracting Real Numbers 15 EXAMPLE 3 Subtracting Real Numbers Subtract. A 7 - 10 7 - 10 = 7 + (-10) (10 - 7 = 3) -3 To subtract 10, add -10. Different signs: subtract absolute values. Use the sign of the number with the greater absolute value. B -3 - (-12) -3 - (-12) = -3 + 12 On many scientific and graphing calculators, there is one button to express the opposite of a number and a different button to express subtraction. To subtract -12, add 12. (12 - 3 = 9) 9 Different signs: subtract absolute values. Use the sign of the number with the greater absolute value. 3 1 -_ C -_ 8 8 3 1 1 + -3 -_- =-_ 8 8 8 8 3 1 4 1 _+_=_=_ 8 8 8 2 1 -_ 2 _ ( ) ( _) 3 , add - _ 3 . To subtract _ 8 8 Same signs: add absolute values. Both numbers are negative, so the sum is negative. D 22.5 - (-4) To subtract -4, add 4. 22.5 - (-4) = 22.5 + 4 (22.5 + 4 = 26.5) 26.5 Same signs: add absolute values. Both numbers are positive, so the sum is positive. Subtract. 3a. 13 - 21 EXAMPLE 4 ( ) 1 - -3 _ 1 3b. _ 2 2 3c. -14 - (-12) Biology Application 10 ft An emperor penguin stands on an iceberg that extends 10 feet above the water. Then the penguin dives to an elevation of -67 feet to catch a fish. What is the total length of the penguin’s dive? 0 ft elevation of iceberg minus elevation of fish - 10 10 - (-67) = 10 + 67 = 77 -67 To subtract -67, add 67. Same signs: add absolute values. –67 ft The total length of the penguin’s dive is 77 feet. 4. What if…? The tallest known iceberg in the North Atlantic rose 550 feet above the ocean’s surface. How many feet would it be from the top of the tallest iceberg to the wreckage of the Titanic, which is at an elevation of -12,468 feet? 16 Chapter 1 Foundations for Algebra THINK AND DISCUSS 1. The difference of -7 and -5 is -2. Explain why the difference is greater than -7.
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Exercises KEYWORD: MA11 1-2 KEYWORD: MA7 Parent GUIDED PRACTICE 1. Vocabulary The sum of a number and its absolute value) SEE EXAMPLE 1 SEE EXAMPLE 2 p. 15 SEE EXAMPLE 3 SEE EXAMPLE 4 p. 16 3. -3.5 - 5 ( ) 4. 5.6 - 9.2 1 5. 3 - -6 _ 4 8. 15.6 + (-17.9) 5 1 +_ 9. - _ 16 8 Add. 6. 91 + (-11) p. 16 is always zero. (opposite or Add or subtract using a number line. 2. -4 + 7 p. 14 ? ( ) 3 + -3 _ 3 7. 4 _ 4 4 Subtract. 10. 23 - 36 11. 4.3 - 8.4 1 - 2_ 4 12. 1 _ 5 5 14. Economics The Dow Jones Industrial Average (DJIA) reports the average prices of stocks for 30 companies. Use the table to determine the total decrease in the DJIA for the two days. ( ) 7 - -_ 2 13. _ 5 10 DJIA 1987 Friday, Oct. 16 -108.35 Monday, Oct. 19 -507.99 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 15–18 19–22 23–26 27 1 2 3 4 Extra Practice Skills Practice p. S4 Application Practice p. S28 Add or subtract using a number line. 15. -2 + 6 16. 6 + (-2) 1 - 12 17. _ 4 2 +6 18. - _ 5 20. -2.3 + 3.5 21. -15 + 29 22. -4.8 + (-5.4) 25. 38 - 24.6 2 - -_ 1 26. _ 3 2 Add. 19. -18 + (-12) Subtract. 23. 12 - 22 ( ) 3 - -_ 1 24. - _ 4 4 ( ) 27. Meteorology A meteorologist reported that the day’s high temperature was 17 °F and the low temperature was -6 °F. What was the difference between the day’s high and low temperatures? 1- 2 Adding and Subtracting Real Numbers 17 Evaluate the expression n + (-5) for each value of n. 28. n = 312 7 30. n = - _ 12 29. n = 5.75 2 31. n = -7 _ 5 Add or subtract. 33. -9 + (-3) 34. 16 - (-16) 35. 100 - 63 7 3 2 2 -_ 36. 5.2 - 2.5 37. -4.7 - (-4.7) 38. _ - _ 39. _ 5 8 5 10 40. Business A restaurant manager lost $415 in business during the month of January. Business picked up in February, and he ended that month with a profit of $1580. a. What was the manager’s profit after January and February? b. What if…? The restaurant lost $245 in business during the month of March. What was the manager’s profit after January, February, and March? 32. -8 - 3 Compare. Write <, >, or =. 41. -4 - (-6) 44. -3 - 8 42. ⎪-51⎥ -7 - 3 ⎪0⎥ 45. ⎪-10 + 5⎥ -22 + 11 ⎪-15⎥ 43. 3 - (-3) 0 - (-3) 46. 9 + (-8) -12 + 13 47. Travel Death Valley National Park is located in California. Use the table to determine the difference in elevation between the highest and lowest locations. Death Valley National Park Location Elevation (ft) Badwater -282 Emigrant Pass 5,318 -210 Furnace Creek Airport Telescope Peak 11,049 Critical Thinking Tell whether each statement is sometimes, always, or never true. Explain. 48. The value of the expression -2 + n is less than the value of n. 49. When b is positive, the expression -b + (-b) is equal to 0. 50. When x is negative, the value of the expression x + 1 is negative. 51. /////ERROR ANALYSIS///// Which is incorrect? Explain the error. ! " ž
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 52. This problem will prepare you for the Multi-Step Test Prep on page 38. a. A plane flies at a height of 1800 feet directly over a 150-foot-tall building. How far above the building is the plane? Draw a diagram to explain your answer. b. The same plane then flies directly over a diver who is 80 feet below the surface of the water. How far is the plane above the diver? Draw a diagram to explain your answer. c. Subtract the diver’s altitude of -80 feet from the plane’s altitude of 1800 feet. Explain why this distance is greater than 1800 feet. 18 Chapter 1 Foundations for Algebra 53. Write About It Explain why addition and subtraction are called inverse operations. Use the following examples in your explanation: 8 + (-2) = 8 - 2 8 - (-2) = 8 + 2 54. Which expression is equivalent to ⎥ -3 + 5⎥ ? -3 - 5 -3 + 5 3-5 3+5 55. At midnight, the temperature was -12 °F. By noon, the temperature had risen 25 °F. During the afternoon, it fell 10 °F and fell another 3 °F by midnight. What was the temperature at midnight? 0 °F 3 °F 12 °F 24 °F 56. The table shows the amounts Mr. Espinosa spent on lunch each day one week. What is the total amount Mr. Espinosa spent for lunch this week? Day Amount ($) Monday Tuesday Wednesday Thursday Friday 5.40 4.16 7.07 5.40 9.52 $21.83 $22.03 $31.55 $36.95 CHALLENGE AND EXTEND Simplify each expression. 1 + (-7.8) 1 + 2.1 57. -1 _ 58. - _ 5 5 ( ) 3 59. 9.75 + -7 _ 4 3 + 8.5 60. -2 _ 10 For each pattern shown below, describe a possible rule for finding the next term. Then use your rule to write the next 3 terms. 6, -_ 8, -_ 4, … 61. 14, 10, 6, 2, … 62. -2, - _ 5 5 5 63. Geography Sam visited two volcanoes, Cotapaxi and Sangay, and two caves, Sistema Huautla and Sistema Cheve. Cotapaxi, in Ecuador, has an elevation of 19,347 ft. Sangay, also in Ecuador, has an elevation of 17,159 ft. The main entrance of Sistema Huautla, in Mexico, has an elevation of 5051 ft. The main entrance of Sistema Cheve, also in Mexico, has an elevation of 9085 ft. What is the average elevation of these places? Mexico Central America Ecuador SPIRAL REVIEW Write each number as a terminating or repeating decimal. (Previous course) 3 15 2 4 64. _ 65. _ 66. _ 67. _ 16 9 12 11 Divide each polygon into triangles to find the sum of its angle measures. (Hint: Remember that the sum of the angle measures in a triangle is 180°.) (Previous course) 68. 69. 70. 1- 2 Adding and Subtracting Real Numbers 19 1-3 Multiplying and Dividing Real Numbers Objectives Multiply real numbers. Who uses this? Hot-air balloon pilots can determine how far away from liftoff they will land by using multiplication. (See Example 4.) Divide real numbers. Vocabulary reciprocal multiplicative inverse When you multiply or divide two numbers, the signs of the numbers determine whether the result is positive or negative. Numbers Product/Quotient Both positive Positive One negative Negative Both negative Positive Multiplying and Dividing Real Numbers WORDS NUMBERS Multiplying and Dividing Numbers with the Same Sign If two numbers have the same sign, their product or quotient is positive. 4 · 5 = 20 -15 ÷ (-3) = 5 Multiplying and Dividing Numbers with Different Signs If two numbers have different signs, their product or quotient is negative. EXAMPLE 1 6 (-3) = -18 -18 ÷ 2 = -9 (-7)2 = -14 10 ÷ (-5) = -2 Multiplying and Dividing Signed Numbers Find the value of each expression. A -12 · 5 -60 ( 4) 8 -_ 5 = -_ 40 = (_ 4 1 )( 4 ) 5 B 8 -_ = -10 The product of two numbers with different signs is negative. Multiply. The quotient of two numbers with different signs is negative. Find the value of each expression. 1a. 35 ÷ (-5) 1b. -11 (-4) 20 Chapter 1 Foundations for Algebra 1c. -6(7) Two numbers are reciprocals if their product is 1. A number and its reciprocal are called multiplicative inverses . Because multiplying by 1 does not change a number’s value, 1 is the multiplicative identity. Two numbers are multiplicative inverses if their product is the multiplicative identity. Inverse Property of Multiplication WORDS The product of a nonzero real number and its reciprocal is 1. NUMBERS ALGEBRA 1 =_ 1 ·4=1 4·_ 4 4 1 = -_ 1 · (-3) = 1 -3 · - _ 3 3 For any real number a (a ≠ 0), 1 _ 1 a ·_ a =a ·a=1 ( ) To divide by a number, you can multiply by its multiplicative inverse. EXAMPLE 2 Dividing with Fractions Divide. ( ) ( _) 8 4 ÷ -_ A -_ You can write the reciprocal of a number by switching the numerator and denominator. A number written without a denominator has a denominator of 1. 5 15 4 ÷ - 8 = -_ 4 - 15 -_ 5 5 15 8 ( _) 8 15 To divide by - ___ , multiply by - __ . 15 8 (-4 )(-15) = __ 5 (8) 60 3 =_=_ 40 2 Multiply the numerators and multiply the denominators. 8 - __45 and - ___ have the same sign, so 15 the quotient is positive. 1 B -4 ÷ 9 _ 4 37 1 = -_ 4 ÷_ -4 ÷ 9 _ 4 4 1 4 4 _ _ =- · 1 37 4 (4) 16 = -_ = -_ 37 1 (37) Write 4 as a fraction with a denominator of 1. Write 9 __14 as an improper fraction. 37 4 To divide by __ , multiply by __ . 4 37 - 4 and 9 __14 have different signs, so the quotient is negative. Divide. ( ) 3 ÷ (-9) 2a. - _ 4 6 3 ÷ -_ 2b. _ 5 10 5 ÷ 1_ 2 2c. - _ 6 3 The number 0 has special properties for multiplication and division. Properties of Zero WORDS NUMBERS ALGEBRA Multiplication by Zero The product of any number and 0 is 0. _1 · 0 = 0 3 0 (-17) = 0 a·0=0 0·a=0 Zero Divided by a Number The quotient of 0 and any nonzero number is 0. _0 = 0 6 _ 0÷ 2 =0 3 _0 = 0 a (a ≠ 0) Division by Zero Division by 0 is undefined. 12 ÷ 0 ✗ -5 ✗ _ 0 a÷0✗ a ✗ _ 0 1- 3 Multiplying and Dividing Real Numbers 21 EXAMPLE 3 Multiplying and Dividing with Zero Multiply or divide if possible. A 0 ÷ 16.568 Zero is divided by a nonzero number. The quotient of zero and any nonzero number is 0. 0 7 B 63 _ ÷0 A number is divided by zero. 8 undefined Division by zero is undefined. C 1·0 A number is multiplied by zero. The product of any number and 0 is 0. 0 Multiply or divide if possible. ( ) 1 3a. 0 ÷ - 8 _ 6 EXAMPLE 4 3c. (-12,350) (0) 3b. 2.04 ÷ 0 Recreation Application A hot-air balloon is taken for a 2.5-hour trip. The wind speed (and the speed of the balloon) is 4.75 mi/h. The balloon travels in a straight line parallel to the ground. How many miles away from the liftoff site will the balloon land? Find the distance traveled at a rate of 4.75 mi/h for 2.5 hours. To find distance, multiply rate by time. rate times time 4.75 · 2.5 4.75 · 2.5 11.875 The hot-air balloon will land 11.875 miles from the liftoff site. 4. What if…? On another hot-air balloon trip, the wind speed is 5.25 mi/h. The trip is planned for 1.5 hours. The balloon travels in a straight line parallel to the ground. How many miles away from the liftoff site will the balloon land? THINK AND DISCUSS 1. Explain how to use mental math to find the missing value: __45 · ? = 1. 2. GET ORGANIZED Copy and complete the graphic organizer. In each blank, write “pos” or “neg” to indicate positive or negative. 22 Chapter 1 Foundations for Algebra Տ̈«Þˆ˜}Ê>˜`Ê ˆÛˆ`ˆ˜}Ê Õ“LiÀà Տ̈«ˆV>̈œ˜ ˆÛˆÃˆœ˜ Ê«œÃ Ê«œÃʵÊÊÊÊ Ê«œÃ Ê«œÃʕÊÊÊÊÊ ʘi} Ê«œÃʵÊÊÊÊ ʘi} ˜i}ʕÊÊÊÊ ʘi} ˜i}ʵÊÊÊÊ ʘi} ˜i}ʕÊÊÊÊ Ê«œÃ ˜i}ʵÊÊÊÊ Ê«œÃ Ê«œÃʕÊÊÊÊ 1-3 Exercises KEYWORD: MA11 1-3 KEYWORD: MA7 Parent GUIDED PRACTICE SEE EXAMPLE 1 2. -72 ÷ (-9) p. 20 SEE EXAMPLE 2 p. 21 SEE EXAMPLE 1. Vocabulary How do you find the reciprocal of __12 ? Find the value of each expression. p. 22 SEE EXAMPLE 4 p. 22 4. -7.2 ÷ 3.6 Divide. ( ) 7 4 ÷ -_ 6. _ 5 5 5 5. 5 ÷ _ 7 3 3. 11(-11) ( ) 2 ÷ -_ 1 7. -_ 3 3 ( ) 16 ÷ -_ 4 8. -_ 5 25 Multiply or divide if possible. 10. 0 (-27) 9. 3.8 ÷ 0 7 ÷0 12. _ 8 2 11. 0 ÷ _ 3 13. Entertainment It is estimated that 7 million people saw off-Broadway shows in 2002. Assume that the average price of a ticket was $30. How much money was spent on tickets for off-Broadway shows in 2002? PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 14–16 17–20 21–24 25 1 2 3 4 Extra Practice Skills Practice p. S4 Application Practice p. S28 Find the value of each expression. 14. -30 ÷ (-6) 16. -25 (-12) 15. 8 (-4) Divide. ( ) 3 ÷ -_ 1 17. -_ 20 6 3 ÷_ 15 18. _ 14 28 ( ) 1 ÷ 1_ 1 19. - 4 _ 2 2 3 ÷ -1 _ 1 20. 2 _ 4 2 23. 0 ÷ 1 0 ÷3 24. _ 1 Multiply or divide if possible. 21. 0 · 15 22. -0.25 ÷ 0 25. Weather A cold front changes the temperature by -3 °F each day. If the temperature started at 0 °F, what will the temperature be after 5 days? Multiply or divide. 26. 21 ÷ (-3) 27. -100 ÷ 25 28. -6 ÷ (-14) 29. -6.2 (10) 1 ÷_ 1 30. _ 2 2 31. -3.75 (-5) 1 (-3) 32. -12 _ 2 1 33. 17 _ 17 ( ) 34. Critical Thinking What positive number is the same as its reciprocal? _ Evaluate each expression for a = 4, b = -3, and c = - 1 . 2 35. ab 36. a ÷ c 37. bc 38. c ÷ a Let p represent a positive number, n represent a negative number, and z represent zero. Tell whether each expression is positive, negative, zero, or undefined. n 39. pn 40. pnz 41. _ 42. -pz p p 43. - _ n 44. -(pn) pn 45. _ z z 46. _ n 1- 3 Multiplying and Dividing Real Numbers 23 Evaluate the expression y ÷ _3 for each value of y. 4 9 48. y = - _ 16 3 47. y = _ 4 3 49. y = _ 8 1 50. y = -2 _ 4 _ Evaluate the expression 1 ÷ m for each value of m. 2 5 7 4 _ 51. m = 52. m = _ 53. m = _ 2 8 9 Diving 54. m = -5 55. Education Benjamin must have 120 credit hours of instruction to receive his college degree. Benjamin wants to graduate in 8 semesters without attending summer sessions. How many credit hours must Benjamin take on average each semester to graduate in 8 semesters? 56. Diving An underwater exploration team is swimming at a depth of -20 feet. Then they dive to an underwater cave that is at 7 times this depth. What is the depth of the underwater cave? Multiply or divide. Then compare using <, >, or =. Florida is home to more than 300 freshwater springs, some of which are explored by cave divers. ( ) 1 57. 10 - _ 2 20 ÷ 4 60. 20 ÷ 4 3 ÷ -_ 1 _ 4 2 ( ) 58. 16 ÷ (-2) -2 (-4) 2 ÷3 59. -2 _ 3 61. 2.1 (-3.4) -3.4(2.1) 3 62. 0 - _ 5 ( ) 5(-2.4) 1 ÷_ 1 _ 2 2 63. Critical Thinking There is a relationship between the number of negative factors and the sign of the product. a. What is the sign of the product of an even number of negative factors? b. What is the sign of the product of an odd number of negative factors? c. Explain why the number of negative factors affects the sign of the product. d. Does the number of positive factors affect the sign of the product? Explain. Write each division expression as a multiplication expression. 80 64. 12 ÷ (-3) 65. 75 ÷ 15 66. _ -8 -121 67. _ 11 Determine whether each statement is sometimes, always, or never true. Explain. t 68. When t is negative, the expression __ is negative. 10 69. When n is positive, the expression -6n is positive. 70. The value of the expression 4c is greater than the value of c. 71. Write About It The product of two factors is positive. One of the factors is negative. Explain how you can determine the sign of the second factor. 72. This problem will prepare you for the Multi-Step Test Prep on page 38. d a. You swam 20 feet in 5 seconds. Use the formula r = __ to determine how fast t you were swimming. b. A diver descended at a rate of 15 feet per minute. Make a table to show the diver’s depth after 1, 2, and 5 minutes. c. Show two ways to find how far the diver descended in 5 minutes. Remember that multiplication is repeated addition. 24 Chapter 1 Foundations for Algebra 73. A recipe for lemonade calls for 1__12 cups of lemon juice per batch. Berto estimates that he can get about __14 cup of lemon juice from each lemon that he squeezes. Lemons cost $0.45 each. What is the approximate amount Berto will need to spend on lemons to make a batch of lemonade? $3.70 $2.70 $1.70 $0.70 74. Robyn is buying carpet for her bedroom floor, which is a 15-foot-by-12-foot rectangle. If carpeting costs $1.25 per square foot, how much will it cost Robyn to carpet her bedroom? $225 $68 $144 $180 75. Short Response In music notation, a half note is played __12 the length of a whole note. A quarter note is played __14 the length of a whole note. In a piece of music, the clarinets play 8 half notes. In the same length of time, the flutes play x quarter notes. Determine how many quarter notes the flutes play. Explain your method. CHALLENGE AND EXTEND Find the value of each expression. ( )( ) 5 2 · _ 81. ⎪- _ 5⎥ ⎪2⎥ 5 ·_ 5 77. _ 7 7 76. (-2)(-2)(-2) ⎪ ⎥ 3 4 -_ 78. 5 - _ 5 4 1 · ⎪20⎥ 79. - _ 4 80. 5 · 4 · 3 · 2 · 1 3 ·_ 1 ·_ 2 ·_ 4 82. _ 2 3 4 5 83. (- _34 ) (- _34 ) (- _34 ) 84. ⎪(-4)(-4)(-4)⎥ For each pattern shown below, describe a possible rule for finding the next term. Then use your rule to write the next 3 terms. 3, … 1 , -_ 1 ,_ 1, -_ 85. -1, 2, -4, 8, … 86. _ 7 63 21 7 87. -5, 10, -15, 20, -25, … 88. 0.5, 0.25, 0.125, 0.0625, … 89. A cleaning service charges $49.00 to clean a one-bedroom apartment. If the work takes longer than 2 hours, the service charges $18.00 for each additional hour. What would be the total cost for a job that took 4 hours to complete? SPIRAL REVIEW Find the surface area of each rectangular prism. (Previous course) 90. 91. 4 cm 21 in. 3 cm 12 cm 5 in. 25 in. 92. A prepaid phone card has a credit of 200 minutes. Write an expression for the number of minutes left on the card after t minutes have been used. (Lesson 1-1) Compare. Write <, >, or =. (Lesson 1-2) 93. -12 + 7 95. ⎪- 7 + 11⎥ 10 + (-5) ⎪-4⎥ 94. ⎪-14⎥ -2 96. -20 + (-35) -35 -20 1- 3 Multiplying and Dividing Real Numbers 25 1-4 Powers and Exponents Objective Simplify expressions containing exponents. Vocabulary power base exponent Who uses this? Biologists use exponents to model the growth patterns of living organisms. When bacteria divide, their number increases exponentially. This means that the number of bacteria is multiplied by the same factor each time the bacteria divide. Instead of writing repeated multiplication to express a product, you can use a power. A power is an expression written with an exponent and a base or the value of such an expression. 32 is an example of a power. The base, 3, is the number that is used as a factor. The exponent, 2, tells how many times the base, 3, is used as a factor. When a number is raised to the second power, we usually say it is “squared.” The area of a square is s · s = s 2, where s is the side length. à à When a number is raised to the third power, we usually say it is “cubed.” The volume of a cube is s · s · s = s 3, where s is the side length. à à EXAMPLE 1 Writing Powers for Geometric Models Write the power represented by each geometric model. There are 3 rows of 3 dots. 3 × 3 A The factor 3 is used 2 times. 32 The figure is 4 cubes long, 4 cubes wide, and 4 cubes tall. 4 × 4 × 4 B The factor 4 is used 3 times. 43 Write the power represented by each geometric model. 1a. 1b. x x 26 Chapter 1 Foundations for Algebra x à There are no easy geometric models for numbers raised to exponents greater than 3, but you can still write them using repeated multiplication or a base and exponent. Reading Exponents Words Multiplication 3 3 3 2 9 3·3·3 3 3 27 3·3·3·3 34 81 3 3·3 3 to the second power, or 3 squared 3 to the third power, or 3 cubed 3 to the fourth power 3·3·3·3·3 3 to the fifth power 2 Value 1 3 to the first power EXAMPLE Power 3 5 243 Evaluating Powers Simplify each expression. A (-2)3 (-2)(-2)(-2) Use -2 as a factor 3 times. -8 B -5 2 In the expression -5 2, 5 is the base because the negative sign is not in parentheses. In the expression (-2)3, -2 is the base because of the parentheses. -1 · 5 · 5 -1 · 25 -25 C Think of a negative sign in front of a power as multiplying by -1. Find the product of -1 and two 5’s. (_2 ) 2 3 2 2 _·_ 3 3 2 ·_ 2 =_ 4 _ 3 3 9 Use __23 as a factor 2 times. Simplify each expression. 2a. (-5)3 EXAMPLE 3 () 3 2c. _ 4 2b. -6 2 3 Writing Powers Write each number as a power of the given base. A 8; base 2 2·2·2 23 B -125; base -5 (-5)(-5)(-5) (-5)3 The product of three 2’s is 8. The product of three -5’s is -125. Write each number as a power of the given base. 3a. 64; base 8 3b. -27; base -3 1- 4 Powers and Exponents 27 EXAMPLE 4 Problem-Solving Application A certain bacterium splits into 2 bacteria every hour. There is 1 bacterium on a slide. If each bacterium on the slide splits once per hour, how many bacteria will be on the slide after 6 hours? 1 Understand the Problem The answer will be the number of bacteria on the slide after 6 hours. List the important information: • There is 1 bacterium on a slide that divides into 2 bacteria. • Each bacterium then divides into 2 more bacteria. 2 Make a Plan Draw a diagram to show the number of bacteria after each hour. First bacterium After 1 hour After 2 hours After 3 hours 3 Solve Notice that after each hour, the number of bacteria is a power of 2. After 1 hour: 1 · 2 = 2 or 2 1 bacteria on the slide After 2 hours: 2 · 2 = 4 or 2 2 bacteria on the slide After 3 hours: 4 · 2 = 8 or 2 3 bacteria on the slide So, after the 6th hour, there will be 2 6 bacteria. 2 6 = 2 · 2 · 2 · 2 · 2 · 2 = 64 Multiply six 2’s. After 6 hours, there will be 64 bacteria on the slide. 4 Look Back The numbers quickly become too large for a diagram, but a diagram helps you recognize a pattern. Then you can write the numbers as powers of 2. 4. What if…? How many bacteria will be on the slide after 8 hours? THINK AND DISCUSS 1. Express 8 3 in words two ways. 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, give an example and tell whether the expression is positive or negative. 28 Chapter 1 Foundations for Algebra Ûi˜ Ý«œ˜i˜Ì *œÃˆÌˆÛiÊ >Ãi i}>̈ÛiÊ >Ãi "`` Ý«œ˜i˜Ì 1-4 Exercises KEYWORD: MA7 1-4 KEYWORD: MA7 Parent GUIDED PRACTICE 1. Vocabulary What does the exponent in the expression 5 6 tell you? SEE EXAMPLE 1 p. 26 Write the power represented by each geometric model. 2. 3. 4. ™ ™ SEE EXAMPLE 2 SEE EXAMPLE Simplify each expression. 6. (-2)4 5. 7 2 p. 27 3 p. 27 p. 28 4 Write each number as a power of the given base. 9. 81; base 9 12. 10; base 10 SEE EXAMPLE 4 () 1 8. - _ 2 7. (-2)5 10. 100,000; base 10 11. -64; base -4 13. 81; base 3 14. 36; base -6 15. Technology Jan wants to predict the number of hits she will get on her Web page. Her Web page received 3 hits during the first week it was posted. If the number of hits triples every week, how many hits will the Web page receive during the 5th week? PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 16–18 19–22 23–28 29 1 2 3 4 Extra Practice Write the power represented by each geometric model. 16. 17. x 18. Î x Î Î Simplify each expression. 20. (-4)2 19. 3 3 21. -4 2 22. Skills Practice p. S4 Application Practice p. S28 (- _35 ) 2 Write each number as a power of the given base. 23. 49; base 7 24. 1000; base 10 25. -8; base -2 26. 1,000,000; base 10 27. 64; base 4 28. 343; base 7 29. Biology Protozoa are single-celled organisms. Paramecium aurelia is one type of protozoan. The number of Paramecium aurelia protozoa doubles every 1.25 days. There was one protozoan on a slide 5 days ago. How many protozoa are on the slide now? 30. Write About It A classmate says that any number raised to an even power is positive. Give examples to explain whether your classmate is correct. Compare. Write <, >, or =. 31. 3 2 35. -2 3 33 (-2)3 32. 5 2 36. -3 2 25 (-3)2 33. 4 2 37. 10 2 24 26 34. 1 9 14 38. 2 2 41 1- 4 Powers and Exponents 29 Write each expression as repeated multiplication. Then simplify the expression. 39. 2 3 40. 1 7 41. (-4)3 43. (-1)3 44. (-1)4 45. 42. -4 3 (_13 ) 3 46. -2.2 2 47. Geometry The diagram shows an ornamental tile design. a. What is the area of the whole tile? b. What is the area of the white square? c. What is the area of the two shaded regions? Èʈ˜° Îʈ˜° Èʈ˜° Write each expression using a base and an exponent. 48. 3 · 3 · 3 · 3 49. 6 · 6 50. 8 · 8 · 8 · 8 · 8 51. (-1)(-1)(-1)(-1) 52. (-7)(-7)(-7) 53. (_19 )(_19 )(_19 ) 5454. Art A painting is made of 3 concentric squares. The side length of the largest square is 24 cm. What is the area of the painting? 55. Estimation A box is shaped like a cube with edges 22.7 centimeters long. What is the approximate volume of the box? Write the exponent that makes each equation true. 56. 2 = 4 57. 4 = 16 58. (-2) = 16 59. 5 = 625 60. -2 = -8 61. 10 = 100 62. 5 = 125 63. 3 = 81 64. Entertainment Mark and Becky play a coin toss game. Both start with one point. Every time the coin comes up heads, Mark doubles his score. Every time the coin comes up tails, Becky triples her score. The results of their game so far are shown in the table. a. What is Mark’s score? b. What is Becky’s score? c. What if…? If they toss the coin 50 more times, who do you think will win? Why? Coin Toss Results Heads Tails ✓ ✓ ✓ ✓ ✓ ✓ ✓ ✓ 65. Critical Thinking The number of zeros in powers of 10 follow a pattern. a. Simplify each of the following: 10 2, 10 3, 10 4. b. Explain what relationship you see between the exponent of a power of 10 and the number of zeros in the answer. 66. This problem will prepare you for the Multi-Step Test Prep on page 38. F The formula p = _ shows that pressure p is the amount of force F exerted over an A area A in square units. a. A 50-pound bag of flour sits on a block and exerts a force over an area of 100 in 2. What is the pressure exerted on the block by the bag of flour? b. A weight exerts 64 pounds on each square foot of a diver’s body. What force is exerted on each square inch of the diver’s body? (Hint: Determine how many square inches are in one square foot.) 30 Chapter 1 Foundations for Algebra 67. Which of the following is equal to 92? 27 9·2 -9 2 34 68. Which expression represents the same value as the product (-16)(-16)(-16)(-16)? (-16)4 (-16)4 -(16 · 4) -16 4 69. A number raised to the third power is negative. What is true about the number? The number is even. The number is positive. The number is odd. The number is negative. 70. A pattern exists as a result (-1)n (-1)1 of raising -1 to consecutive Value -1 whole numbers. Which is the best representation of the value of -1 raised to the 100th power? -1 100 -1 (-1)2 (-1)3 (-1)4 (-1)5 (-1)6 1 -1 1 -1 1 0 1 CHALLENGE AND EXTEND Simplify each expression. 71. (2 2)(2 2)(2 2) 72. (2 3)(2 3)(2 3) 74. Design The diagram shows the layout of a pool and the surrounding path. The path is 2.5 feet wide. a. What is the total area of the pool and path? b. What is the area of the pool? c. What is the area of the path? d. One bag of pebbles covers 10 square feet. How many bags of pebbles are needed to cover the path? 73. (-4 2)(-4 2)(-4 2)(-4 2) 30 ft 30 ft 75. Exponents and powers have special properties. 2 3 a. Write both 4 and 4 as a product of 4’s. b. Write the product of the two expressions from part a. Write this product as a power of 4. c. Write About It Add the exponents in the expressions 4 2 and 4 3. Describe any relationship you see between your answer to part b and the sum of the exponents. SPIRAL REVIEW Find the mean of each data set by dividing the sum of the data by the number of items in the data set. (Previous course) 76. 7, 7, 8, 8 77. 1, 3, 5, 7, 9 78. 10, 9, 9, 12, 12 Give two ways to write each algebraic expression in words. (Lesson 1-1) 79. 5 - x 80. 6n 81. c ÷ d 82. a + b 85. -20(-14) 1 -_ 4 86. _ 5 2 Multiply or divide if possible. (Lesson 1-3) 8 4 ÷_ 83. _ 5 25 6 84. 0 ÷ _ 7 ( ) 1- 4 Powers and Exponents 31 1-5 Roots and Real Numbers Objectives Simplify expressions containing roots. Why learn this? Square roots can be used to find the side length of a square garden when you know its area. (See Example 3.) Classify numbers within the real number system. Vocabulary square root principal square root perfect square cube root natural numbers whole numbers integers rational numbers terminating decimal repeating decimal irrational numbers A number that is multiplied by itself to form a product is a square root of that product. The radical symbol √ is used to represent square roots. For nonnegative numbers, the operations of squaring and finding a square root are inverse operations. In other words, for x ≥ 0, √ x · √ x = x. Positive real numbers have two square roots. The principal square root of a number is the positive square root and is represented by √. A negative square root is represented by - √. The symbol ± √ is used to represent both square roots. Positive square root of 16 Negative square root of 16 A perfect square is a number whose positive square root is a whole number. Some examples of perfect squares are shown in the table. 0 1 4 9 16 25 36 49 64 81 100 02 12 22 32 42 52 62 72 82 92 10 2 A number that is raised to the third power to form a product is a cube root 3 3 of that product. The symbol √ indicates a cube root. Since 2 3 = 8, √ 8 = 2. 4 4 4 Similarly, the symbol √ indicates a fourth root: 2 = 16, so √ 16 = 2. EXAMPLE 1 Finding Roots Find each root. A √ 49 The small number to the left of the root is the index. In a square root, the index is understood to be 2. In other words, √ is the 2 same as √. √ 49 = √ 72 =7 Think: What number squared equals 49? B - √ 36 - √ 36 = - √ 62 = -6 C Think: What number squared equals 36? √ -125 3 3 (-5 3) Think: What number cubed equals -125? √  -125 = √ (-5)(-5)(-5) = 25(-5) = -125 = -5 3 Find each root. 1a. √ 4 32 Chapter 1 Foundations for Algebra 1b. - √ 25 4 1c. √ 81 EXAMPLE 2 Finding Roots of Fractions √_14 .  1 =  _ √ 4 √(_12 ) 1 =_ 1 _ √ 4 2 Find 2 _ Think: What number squared equals 1 ? 4 Find each root.  4 2a. _ 9 √ 2b. √_18 √  4 2c. - _ 49 3 Square roots of numbers that are not perfect squares, such as 15, are not whole numbers. A calculator can approximate the value of √ 15 as 3.872983346… Without a calculator, you can use the square roots of perfect squares to help estimate the square roots of other numbers. EXAMPLE 3 Gardening Application Nancy wants to plant a square garden of wildflowers. She has enough wildflower seeds to cover 19 ft 2. Estimate to the nearest tenth the side length of a square with an area of 19 ft 2.  ft. Since the area of the square is 19 ft 2, then each side of the square is √19 19 is not a perfect square, so find the two consecutive perfect squares that 19 is between: 16 and 25. √ 19 is between √ 16 and √ 25 , or 4 and 5. Refine the estimate. The symbol ≈ means “is approximately equal to.” 4.3: 4.3 2 = 18.49 too low √ 19 is greater than 4.3. 4.4: 4.4 = 19.36 too high √ 19 is less than 4.4. 4.35: 4.35 2 = 18.9225 too low √ 19 is greater than 4.35. 2 Since 4.35 is too low and 4.4 is too high, √ 19 is between 4.35 and 4.4. Rounded to the nearest tenth, √ 19 ≈ 4.4. The side length of the plot is √ 19 ≈ 4.4 ft. 3. Estimate to the nearest tenth the side length of a cube with a volume of 26 ft3 . Real numbers can be classified according to their characteristics. Natural numbers are the counting numbers: 1, 2, 3, … Whole numbers are the natural numbers and zero: 0, 1, 2, 3, … Integers are the whole numbers and their opposites: …, -3, -2, -1, 0, 1, 2, 3, … To show that one or more digits repeat continuously, write a bar over those digits. − 1.333333333… = 1.3 −− 2.14141414… = 2.14 Rational numbers are numbers that can be expressed in the form __ab , where a and b are both integers and b ≠ 0. When expressed as a decimal, a rational number is either a terminating decimal or a repeating decimal. • A terminating decimal has a finite number of digits after the decimal point (for example, 1.25, 2.75, and 4.0). • A repeating decimal has a block of one or more digits after the decimal point that repeat continuously (where all digits are not zeros). 1- 5 Roots and Real Numbers 33 Irrational numbers are all real numbers that are not rational. They cannot be expressed in the form __ab where a and b are both integers and b ≠ 0. 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Ê Ó 7 œiÊ Õ“LiÀÃÊ­ϓ® Note the symbols for the sets of numbers. : real numbers : rational numbers : integers : whole numbers : natural numbers ÊÊÊ ÊÚÚÚ Ê£ä ££ û Classifying Real Numbers Write all classifications that apply to each real number. A 8 9 _ _8 is in the form _a , where a and b are integers and b ≠ 0. 9 b 8 ÷ 9 = 0.8888… − = 0.8 rational, repeating decimal _8 can be written as a repeating decimal. 9 B 18 18 a 18 = _ 18 can be written in the form _. 1 b 18 = 18.0 18 can be written as a terminating decimal. rational, terminating decimal, integer, whole, natural C √ 20 irrational 20 is not a perfect square, so √ 20 is irrational. Write all classifications that apply to each real number. 4 4a. 7 _ 4b. -12 9 4c. √ 10 4d. √ 100 34 Chapter 1 Foundations for Algebra THINK AND DISCUSS 1. Write __23 and __35 as decimals. Identify what number classifications the two numbers share and how their classifications are different. 2. GET ORGANIZED Copy the graphic organizer and use the flowchart to classify each of the given numbers. Write each number in the box with the most specific classification that applies. 4, √ 25 , 0, __13 , -15, -2.25, __14 , 2 4 ( √ 21 , 2 , -1) ,>̈œ˜>Ê˜Õ“LiÀ Yes ÀÀ>̈œ˜>Ê˜Õ“LiÀ No Integer Yes No No 1-5 Exercises Whole number Natural number Yes No KEYWORD: MA11 1-5 KEYWORD: MA7 Parent GUIDED PRACTICE 1. Vocabulary Give an example of a square root that is not a rational number. SEE EXAMPLE 1 p. 32 SEE EXAMPLE Find each root.  2. √64 6. √ 81 2 10. p. 33 14. SEE EXAMPLE 3 p. 33 SEE EXAMPLE 4 p. 34 1 _ √ 16  1 √_ 36 3. - √ 225 3 4. √ -64 4 5. √ 625 7. - √ 27 8. - √ -27 9. - √ 16 3 11. 15. 8 _ √ 27  1 √_ 64 3 3 3 √ √  1 12. - _ 9  4 16. - _ 81 13. 17. 9 _ √ 64 1 -_ √ 125 3 18. A contractor is told that a potential client’s kitchen floor is in the shape of a square. The area of the floor is 45 ft 2. Estimate to the nearest tenth the side length of the floor. Write all classifications that apply to each real number. 1 19. -27 20. _ 21. √ 33 6 22. -6.8 PRACTICE AND PROBLEM SOLVING Find each root. 23. √ 121 27.  1 √_ 25 3 24. √ -1000 28.  1 √_ 16 4 25. - √ 100 29. 1 -_ √ 8 3 4 26. √ 256 30. -  25 √_ 36 31. An artist makes glass paperweights in the shape of a cube. He uses 68 cm3 of glass to make each paperweight. Estimate to the nearest tenth the side length of a paperweight. 1- 5 Roots and Real Numbers 35 Independent Practice For See Exercises Example 23–26 27–30 31 32–35 1 2 3 4 Extra Practice Skills Practice p. S5 Application Practice p. S28 Write all classifications that apply to each real number. 5 32. _ 49 34. -3 33. √ 12 36. Geometry The cube root of the volume of a cube gives the length of one side of the cube. a. Find the side length of the cube shown. b. Find the area of each face of the cube. Compare. Write <, >, or =. √63  37. 8 38. √ 88 35. √ 18 6œÕ“iÊÊÎ{ÎÊV“Î 39. 6 9 Travel During a cross-country road trip, Madeline recorded the distance between several major cities and the