chapter 7.pdf - Exponents and Polynomials 7A Exponents 7-1 Integer Exponents 7-2 Powers of 10 and Scientific Notation Lab Explore Properties of

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Unformatted text preview: Exponents and Polynomials 7A Exponents 7-1 Integer Exponents 7-2 Powers of 10 and Scientific Notation Lab Explore Properties of Exponents 7-3 Multiplication Properties of Exponents 7-4 Division Properties of Exponents 7-5 Rational Exponents 7B Polynomials 7-6 Polynomials Lab Model Polynomial Addition and Subtraction 7-7 Adding and Subtracting Polynomials Lab Model Polynomial Multiplication 7-8 Multiplying Polynomials 7-9 Special Products of Binomials • Use exponents and scientific notation to describe numbers. • Use laws of exponents to simplify monomials. • Perform operations with polynomials. Every Second Counts How many seconds until you graduate? The concepts in this chapter will help you find and use large numbers such as this one. KEYWORD: MA7 ChProj 456 Chapter 7 Vocabulary Match each term on the left with a definition on the right. A. a number that is raised to a power 1. Associative Property 2. coefficient B. a number that is multiplied by a variable 3. Commutative Property C. a property of addition and multiplication that states you can add or multiply numbers in any order 4. exponent D. the number of times a base is used as a factor 5. like terms E. terms that contain the same variables raised to the same powers F. a property of addition and multiplication that states you can group the numbers in any order Exponents Write each expression using a base and an exponent. 6. 4 · 4 · 4 · 4 · 4 · 4 · 4 7. 5 · 5 9. x · x · x 8. (-10)(-10)(-10)(-10) 10. k · k · k · k · k 11. 9 Evaluate each expression. 12. 3 4 13. -12 2 14. 5 3 15. 2 5 16. 4 3 17. (-1)6 19. 25.25 × 100 20. 2.4 × 6.5 Evaluate Powers Multiply Decimals Multiply. 18. 0.006 × 10 Combine Like Terms Simplify each expression. 21. 6 + 3p + 14 + 9p 22. 8y - 4x + 2y + 7x - x 23. (12 + 3w - 5) + 6w - 3 - 5w 24. 6n - 14 + 5n Squares and Square Roots Tell whether each number is a perfect square. If so, identify its positive square root. 25. 42 26. 81 27. 36 28. 50 29. 100 30. 4 31. 1 32. 12 Exponents and Polynomials 457 Key Vocabulary/Vocabulario Previously, you • wrote and evaluated • exponential expressions. simplified algebraic expressions by combining like terms. You will study • properties of exponents. • powers of 10 and scientific • notation. how to add, subtract, and multiply polynomials by using properties of exponents and combining like terms. binomial binomio degree of a monomial grado de un monomio degree of a polynomial grado de un polinomio leading coefficient coeficiente principal monomial monomio perfect-square trinomial trinomio cuadrado perfecto polynomial polinomio scientific notation notación científica standard form of a polynomial forma estándar de un polinomio trinomial trinomio 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. You can use the skills in this chapter • to model area, perimeter, and • • volume in geometry. to express very small or very large quantities in science classes such as Chemistry, Physics, and Biology. in the real world to model business profits and population growth or decline. 1. Very large and very small numbers are often encountered in the sciences. If notation means a method of writing something, what might scientific notation mean? 2. A polynomial written in standard form may have more than one algebraic term. What do you think the leading coefficient of a polynomial is? 3. A simple definition of monomial is “an expression with exactly one term.” If the prefix mono- means “one” and the prefix bi- means “two,” define the word binomial . 4. What words do you know that begin with the prefix tri-? What do they all have in common? Define the word trinomial based on the prefix tri- and the information given in Problem 3. 458 Chapter 7 Reading Strategy: Read and Understand the Problem Follow this strategy when solving word problems. • Read the problem through once. • Identify exactly what the problem asks you to do. • Read the problem again, slowly and carefully, to break it into parts. • Highlight or underline the key information. • Make a plan to solve the problem. From Lesson 6-6 29. Multi-Step Linda works at a pharmacy for $15 an hour. She also baby-sits for $10 an hour. Linda needs to earn at least $90 per week, but she does not want to work more than 20 hours per week. Show and describe the number of hours Linda could work at each job to meet her goals. List two possible solutions. Step 1 Identify exactly what the problem asks you to do. • Show and describe the number of hours Linda can work at each job and earn at least $90 per week, without working more than 20 hours per week. • List two possible solutions of the system. Step 2 Step 3 Break the problem into parts. Highlight or underline the key information. Make a plan to solve the problem. • Linda has two jobs. She makes $15 per hour at one job and $10 per hour at the other job. • She wants to earn at least $90 per week. • She does not want to work more than 20 hours per week. • Write a system of inequalities. • Solve the system. • Identify two possible solutions of the system. Try This For the problem below, a. identify exactly what the problem asks you to do. b. break the problem into parts. Highlight or underline the key information. c. make a plan to solve the problem. 1. The difference between the length and the width of a rectangle is 14 units. The area is 120 square units. Write and solve a system of equations to determine the length and the width of the rectangle. (Hint: The formula for the area of a rectangle is A = w.) Exponents and Polynomials 459 7-1 Integer Exponents Objectives Evaluate expressions containing zero and integer exponents. Who uses this? Manufacturers can use negative exponents to express very small measurements. Simplify expressions containing zero and integer exponents. In 1930, the Model A Ford was one of the first cars to boast precise craftsmanship in mass production. The car’s pistons had a diameter of 3 __78 inches; this measurement could vary by at most 10 -3 inch. You have seen positive exponents. Recall that to simplify 3 2, use 3 as a factor 2 times: 3 2 = 3 · 3 = 9. But what does it mean for an exponent to be negative or 0? You can use a table and look for a pattern to figure it out. Base x4 Exponent Power 55 54 53 52 51 Value 3125 625 125 25 5 ÷5 ÷5 ÷5 50 5 -1 5 -2 ÷5 When the exponent decreases by one, the value of the power is divided by 5. Continue the pattern of dividing by 5: 5 =1 50 = _ 5 1 1 ÷5=_ 1 =_ 5 -2 = _ 5 25 5 2 1 =_ 1 5 -1 = _ 5 51 Integer Exponents WORDS Zero exponent—Any nonzero number raised to the zero power is 1. 2 -4 is read “2 to the negative fourth power.” Negative exponent—A nonzero number raised to a negative exponent is equal to 1 divided by that number raised to the opposite (positive) exponent. NUMBERS 3 0 = 1 123 0 = 1 (-16) 0 = 1 ALGEBRA If x ≠ 0, then x 0 = 1. (_37 ) = 1 0 1 1 =_ 3 -2 = _ 9 32 1 1 =_ 2 -4 = _ 16 24 If x ≠ 0 and n is an integer, 1. then x -n = _ xn Notice the phrase “nonzero number” in the table above. This is because 0 0 and 0 raised to a negative power are both undefined. For example, if you use the pattern given above the table with a base of 0 instead of 5, you would get 0 0 = __00 . 1 Also, 0 -6 would be __ = __10 . Since division by 0 is undefined, neither value exists. 6 0 460 Chapter 7 Exponents and Polynomials EXAMPLE 1 Manufacturing Application The diameter for the Model A Ford piston could vary by at most 10 -3 inch. Simplify this expression. 1 1 =_ 1 10 -3 = _ =_ 10 3 10 · 10 · 10 1000 1 inch, or 0.001 inch. 10 -3 inch is equal to ____ 1000 1. EXAMPLE 2 A sand fly may have a wingspan up to 5 -3 m. Simplify this expression. Zero and Negative Exponents Simplify. A 2 -3 B 50 1 1 =_ 1 2 -3 = _ =_ 23 2 · 2 · 2 8 to the zero power is 1. C (-3)-4 In (-3) , the base is negative because the negative sign is inside the parentheses. In -3 -4 the base (3) is positive. -4 1 = __ 1 1 (-3)-4 = _ =_ (-3) 4 (-3)(-3)(-3)(-3) 81 D -3 -4 1 1 = -_ 1 -3 -4 = - _ = -_ 3·3·3·3 81 34 Simplify. 2a. 10 -4 EXAMPLE 5 0 = 1 Any nonzero number raised 3 2b. (-2)-4 2c. (-2)-5 2d. -2 -5 Evaluating Expressions with Zero and Negative Exponents Evaluate each expression for the given value(s) of the variable(s). A x -1 for x = 2 2 -1 2 -1 Substitute 2 for x. 1 1 =_ =_ 21 2 1 . Use the definition x -n = _ xn B a 0b -3 for a = 8 and b = -2 8 0 · (-2)-3 Substitute 8 for a and -2 for b. 1· 1· 1 _ (-2)3 1 __ (-2)(-2)(-2) _ 1· 1 -8 1 -_ 8 Simplify expressions with exponents. Write the power in the denominator as a product. Simplify the power in the denominator. Simplify. Evaluate each expression for the given value(s) of the variable(s). 3a. p -3 for p = 4 3b. 8a -2b 0 for a = -2 and b = 6 7-1 Integer Exponents 461 What if you have an expression with a negative exponent in a denominator, 1 such as ___ ? -8 x 1 , or _ 1 = x -n x -n = _ xn xn 1 = x -(-8) _ x -8 Definition of negative exponent Substitute -8 for n. = x8 Simplify the exponent on the right side. So if a base with a negative exponent is in a denominator, it is equivalent to the same base with the opposite (positive) exponent in the numerator. An expression that contains negative or zero exponents is not considered to be simplified. Expressions should be rewritten with only positive exponents. EXAMPLE 4 Simplifying Expressions with Zero and Negative Exponents Simplify. -4 B _ -4 A 3y -2 3y -2 =3·y k -4 = -4 · 1 _ k -4 k -4 _ -2 _ = 3 · 12 y 3 = _2 y = -4 · k 4 = -4k 4 x -3 C _ 0 5 a y x -3 = _ 1 _ 0 5 3 a y x · 1 · y5 1 =_ x 3y 5 1. a 0 = 1 and x -3 = _ x3 Simplify. r -3 4b. _ 7 4a. 2r 0m -3 g4 4c. _ h -6 THINK AND DISCUSS -3 s =_ 1 , ? -2 = _ 2,_ 1 1. Complete each equation: 2b ? = _ b2 k ? s3 t2 2. GET ORGANIZED Copy and complete the graphic organizer. In each box, describe how to simplify, and give an example. -ˆ“«ˆvވ˜}Ê Ý«ÀiÃȜ˜ÃÊÜˆÌ…Ê i}>̈ÛiÊ Ý«œ˜i˜Ìà œÀÊ>ʘi}>̈ÛiÊiÝ«œ˜i˜ÌÊ ˆ˜Ê̅iʘՓiÀ>̜ÀÊ°Ê°Ê° 462 Chapter 7 Exponents and Polynomials œÀÊ>ʘi}>̈ÛiÊiÝ«œ˜i˜ÌÊ ˆ˜Ê̅iÊ`i˜œ“ˆ˜>̜ÀÊ°Ê°Ê° 7-1 Exercises KEYWORD: MA7 7-1 KEYWORD: MA7 Parent GUIDED PRACTICE SEE EXAMPLE 1 1. Medicine A typical virus is about 10 -7 m in size. Simplify this expression. p. 461 SEE EXAMPLE 2 p. 461 SEE EXAMPLE 3 p. 461 SEE EXAMPLE 4 p. 462 Simplify. 2. 6 -2 3. 3 0 4. -5 -2 7. -8 -3 8. 10 -2 9. (4.2)0 5. 3 -3 6. 1 -8 10. (-3)-3 11. 4-2 Evaluate each expression for the given value(s) of the variable(s). 12. b -2 for b = -3 13. (2t)-4 for t = 2 14. (m - 4)-5 for m = 6 15. 2x 0y -3 for x = 7 and y = -4 Simplify. 16. 4m 0 17. 3k -4 7 18. _ r -7 x 10 19. _ d -3 20. 2x 0y -4 f -4 21. _ g -6 c4 22. _ d -3 23. p 7q -1 PRACTICE AND PROBLEM SOLVING Independent Practice For See Exercises Example 24 25–36 37–42 43–57 1 2 3 4 Extra Practice Skills Practice p. S16 Application Practice p. S34 24. Biology One of the smallest bats is the northern blossom bat, which is found from Southeast Asia to Australia. This bat weighs about 2 -1 ounce. Simplify this expression. Simplify. 25. 8 0 26. 5 -4 27. 3 -4 29. -6 -2 30. 7 -2 31. 33. (-3)-1 34. (-4)2 (_25 ) 1 35. (_ 2) 0 -2 28. -9 -2 32. 13 -2 36. -7 -1 Evaluate each expression for the given value(s) of the variable(s). (_23 v) -3 37. x -4 for x = 4 38. 0 39. (10 - d) for d = 11 40. 10m -1n -5 for m = 10 and n = -2 -2 1 and b = 8 41. (3ab) for a = _ 2 42. 4w vx v for w = 3, v = 0, and x = -5 for v = 9 Simplify. 43. k -4 44. 2z -8 1 45. _ 2b -3 46. c -2d 47. -5x -3 48. 4x -6y -2 2f 0 49. _ 7g -10 r -5 50. _ s -1 s5 51. _ -12 t 3w -5 52. _ x -6 53. b 0c 0 2 m -1n 5 54. _ 3 q -2r 0 55. _ s0 a -7b 2 56. _ c 3d -4 3 -1 k _ 57. h 6m 2 7-1 Integer Exponents 463 Evaluate each expression for x = 3, y = -1, and z = 2. 58. z -5 62. 66. 59. (x + y)-4 61. (xyz)-1 60. (yz) 0 63. x -y 64. (yz) -x 65. xy -4 (xy - 3)-2 /////ERROR ANALYSIS///// Look at the two equations below. Which is incorrect? Explain the error.  .q , *  XXXXXXX .q , . .q , XXXXX q, Simplify. Biology (a 2b -7) 0 67. a 3b -2 68. c -4d 3 69. v 0w 2y -1 70. 2a -5 72. _ b -6 2a 3 73. _ b -1 m2 74. _ n -3 x -8 75. _ 3y 12 77. Biology Human blood contains red blood cells, white blood cells, and platelets. The table shows the sizes of these components. Simplify each expression. Tell whether each statement is sometimes, always, or never true. When bleeding occurs, platelets (which appear green in the image above) help to form a clot to reduce blood loss. Calcium and vitamin K are also necessary for clot formation. 71. -5y -6 20p -1 76. - _ 5q -3 Blood Components Part Size (m) 125,000 -1 Red blood cell White blood cell 3(500)-2 Platelet 3(1000)-2 1 . 78. If n is a positive integer, then x -n = __ xn 79. If x is positive, then x -n is negative. 80. If n is zero, then x -n is 1. 81. If n is a negative integer, then x -n = 1. 82. If x is zero, then x -n is 1. 83. If n is an integer, then x -n > 1. 84. Critical Thinking Find the value of 2 3 · 2 -3. Then find the value of 3 2 · 3 -2. Make a conjecture about the value of a n · a -n. 1 85. Write About It Explain in your own words why 2 -3 is the same as __ . 3 2 Find the missing value. 1 =2 86. _ 4 1 87. 9 -2 = _ 1 90. 7 -2 = _ 91. 10 1 =_ 1000 1 = 88. _ 64 -2 3 92. 3 · 4 -2 = _ 89. _ = 3 -1 3 1 =2·5 93. 2 · _ 5 94. This problem will prepare you for the Multi-Step Test Prep on page 494. a. The product of the frequency f and the wavelength w of light in air is a constant v. Write an equation for this relationship. b. Solve this equation for wavelength. Then write this equation as an equation with f raised to a negative exponent. c. The units for frequency are hertz (Hz). One hertz is one cycle per second, which is often written as __1s . Rewrite this expression using a negative exponent. 464 Chapter 7 Exponents and Polynomials 95. Which is NOT equivalent to the other three? 1 _ 25 5 -2 0.04 -25 (-6)(-6) 1 -_ 6·6 1 _ 6·6 a 3b 2 _ -c a3 _ -b 2c c _ 3 2 ab 96. Which is equal to 6 -2? 6 (-2) a 3b -2 . 97. Simplify _ c -1 3 ac _ 2 b 98. Gridded Response Simplify ⎡⎣2 -2 + (6 + 2)0⎤⎦. 99. Short Response If a and b are real numbers and n is a positive integer, write a simplified expression for the product a -n · b 0 that contains only positive exponents. Explain your answer. CHALLENGE AND EXTEND 100. Multi-Step Copy and complete the table of values below. Then graph the ordered pairs and describe the shape of the graph. -4 x y=2 -3 -2 -1 0 1 2 3 4 x 101. Multi-Step Copy and complete the table. Then write a rule for the values of 1 n and (-1)n when n is any negative integer. n -1 -2 -3 -4 -5 1n (-1)n SPIRAL REVIEW Solve each equation. (Lesson 2-3) 102. 6x - 4 = 8 103. -9 = 3 (p - 1) y 104. _ - 8 = -12 5 105. 1.5h - 5 = 1 106. 2w + 6 - 3w = -10 1n+2-n 107. -12 = _ 2 Identify the independent and dependent variables. Write a rule in function notation for each situation. (Lesson 4-3) 108. Pink roses cost $1.50 per stem. 109. For dog-sitting, Beth charges a $30 flat fee plus $10 a day. Write the equation that describes each line in slope-intercept form. (Lesson 5-7) 110. slope = 3, y-intercept = -4 111. slope = __13 , y-intercept = 5 112. slope = 0, y-intercept = __23 113. slope = -4, the point (1, 5) is on the line. 7-1 Integer Exponents 465 7-2 Powers of 10 and Scientific Notation Objectives Evaluate and multiply by powers of 10. Why learn this? Powers of 10 can be used to read and write very large and very small numbers, such as the masses of atomic particles. (See Exercise 44.) Convert between standard notation and scientific notation. Vocabulary scientific notation The table shows relationships between several powers of 10. ÷ 10 ÷ 10 2 Power 10 3 10 Value 1000 100 ÷ 10 Nucleus of a silicon atom ÷ 10 ÷ 10 ÷ 10 10 1 10 0 10 -1 10 -2 10 -3 10 1 1 = 0.1 _ 10 1 = 0.01 _ 100 1 = 0.001 _ 1000 × 10 × 10 × 10 × 10 × 10 × 10 • Each time you divide by 10, the exponent decreases by 1 and the decimal point moves one place to the left. • Each time you multiply by 10, the exponent increases by 1 and the decimal point moves one place to the right. Powers of 10 WORDS NUMBERS Positive Integer Exponent 10 4 = 1 0, 0 0 0 ⎧  ⎨  ⎩ If n is a positive integer, find the value of 10 n by starting with 1 and moving the decimal point n places to the right. 4 places Negative Integer Exponent EXAMPLE 1 1 = 0.0 0 0 0 0 1 10 -6 = _ 10 6 ⎧   ⎨   ⎩ If n is a positive integer, find the value of 10 -n by starting with 1 and moving the decimal point n places to the left. 6 places Evaluating Powers of 10 Find the value of each power of 10. A 10 -3 You may need to add zeros to the right or left of a number in order to move the decimal point in that direction. 466 B 10 2 C 10 0 Start with 1 and move the decimal point three places to the left. Start with 1 and move the decimal point two places to the right. Start with 1 and move the decimal point zero places. 0. 0 0 1 1 0 0 1 0.001 100 Chapter 7 Exponents and Polynomials Find the value of each power of 10. 1a. 10 -2 1b. 10 5 EXAMPLE 2 1c. 10 10 Writing Powers of 10 Write each number as a power of 10. A 10,000,000 If you do not see a decimal point in a number, it is understood to be at the end of the number. B 0.001 C 10 The decimal point is seven places to the right of 1, so the exponent is 7. The decimal point is three places to the left of 1, so the exponent is -3. The decimal point is one place to the right of 1, so the exponent is 1. 10 7 10 -3 10 1 Write each number as a power of 10. 2a. 100,000,000 2b. 0.0001 2c. 0.1 You can also move the decimal point to find the product of any number and a power of 10. You start with the number instead of starting with 1. Multiplying by Powers of 10 125 × 10 5 = 12,5 0 0, 0 0 0 ⎧   ⎨   ⎩ If the exponent is a positive integer, move the decimal point to the right. 5 places 36.2 × 10 = 0.0 3 6 2 ⎧ ⎨ ⎩ If the exponent is a negative integer, move the decimal point to the left. -3 3 places EXAMPLE 3 Multiplying by Powers of 10 Find the value of each expression. A 97.86 × 10 6 97.8 6 0 0 0 0 Move the decimal point 6 places to the right. 97,860,000 B 19.5 × 10 -4 0 0 1 9.5 Move the decimal point 4 places to the left. 0.00195 Find the value of each expression. 3a. 853.4 × 10 5 3b. 0.163 × 10 -2 Scientific notation is a method of writing numbers that are very large or very small. A number written in scientific notation has two parts that are multiplied. The first part is a number that is greater than or equal to 1 and less than 10. The second part is a power of 10. 7- 2 Powers of 10 and Scientific Notation 467 EXAMPLE 4 Astronomy Application Jupiter has a diameter of about 143,000 km. Its shortest distance from Earth is about 5.91 × 10 8 km, and its average distance from the Sun is about 778,400,000 km. Jupiter’s orbital speed is approximately 1.3 × 10 4 m/s. £{Î]äääʎ“ A Write Jupiter’s shortest distance from Earth in standard form. 5.91 × 10 8 5.9 1 0 0 0 0 0 0 Move the decimal point 8 places to the right. Standard form refers to the usual way that numbers are written. 591,000,000 km B Write Jupiter’s average distance from the Sun in scientific notation. ⎩   ⎨   ⎧ 778,400,000 7 7 8, 4 0 0, 0 0 0 8 places 7.784 × 10 8 km Count the number of places you need to move the decimal point to get a number between 1 and 10. Use that number as the exponent of 10. 4a. Use the information above to write Jupiter’s diameter in scientific notation. 4b. Use the information above to write Jupiter’s orbital speed in standard form. EXAMPLE 5 Comparing and Ordering Numbers in Scientific Notation Order the list of numbers from least to greatest. 1.2 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5, 1 × 10 -1, 9.9 × 10 -4 Step 1 List the numbers in order by powers of 10. 9.9 × 10 -4, 1.2 × 10 -1, 1 × 10 -1, 8.2 × 10 4, 6.2 × 10 5, 2.4 × 10 5 Step 2 Order the numbers that have the same power of 10. 9.9 × 10 -4, 1 × 10 -1, 1.2 × 10 -1, 8.2 × 10 4, 2.4 × 10 5, 6.2 × 10 5 5. Order the list of numbers from least to greatest. 5.2 × 10 -3, 3 × 10 14, 4 × 10 -3, 2 × 10 -12, 4.5 × 10 30, 4...
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