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Chapter 1 Review

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- 1. WHY is it helpful to write numbers in different ways? The Number System Course 3, Lesson 1-1
- 2. To • write fractions as decimals • write decimals as fractions Course 3, Lesson 1-1 The Number System
- 3. • rational number • repeating decimal • terminating decimal Course 3, Lesson 1-1 The Number System
- 4. Course 3, Lesson 1-1 The Number System Words A rational number is a number that can be written as the ratio of two integers in which the denominator is not zero. Symbols , where a and b are integers and b ≠ 0 Model a b
- 5. 1 Need Another Example? 2 Step-by-Step Example 1. Write the fraction as a decimal. means 5 ÷ 8. 0.6 5.000 – 48 20 – 16 40 – 40 0 8 Divide 5 by 8. 25
- 6. Answer Need Another Example? Write as a decimal. 0.1875
- 7. 1 Need Another Example? 2 3 Step-by-Step Example 2. Write the mixed number as a decimal. can be rewritten as . Divide 5 by 3 and add a negative sign. The mixed number can be written as –1.6.
- 8. Answer Need Another Example? Write as a decimal. –3.18
- 9. 1 Need Another Example? 2 3 4 Step-by-Step Example 3. In a recent season, St. Louis Cardinals first baseman Albert Pujols had 175 hits in 530 at bats. To the nearest thousandth, find his batting average. To find his batting average, divide the number of hits, 175, by the number of at bats, 530. 175 Look at the digit to the right of the thousandths place. Since 1 < 5, round down. 530 0.3301886792 Albert Pujols’s batting average was 0.330.
- 10. Answer Need Another Example? When Juliana went strawberry picking, 28 of the 54 strawberries she picked weighed less than 2 ounces. To the nearest thousandth, find the fraction of the strawberries that weighed less than 2 ounces. 0.519
- 11. 1 Need Another Example? 2 Step-by-Step Example 4. Write 0.45 as a fraction. 0.45 = = 0.45 is 45 hundredths. Simplify.
- 12. Answer Need Another Example? Write 0.32 as a fraction in simplest form.
- 13. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 5. Write 0.5 as a fraction in simplest form. Assign a variable to the value 0.5. Let N = 0.555... . Then perform operations on N to determine its fractional value. N = 0.555... Simplify. Multiply each side by 10 because 1 digit repeats. 7 10(N) = 10(0.555...) Multiplying by 10 moves the decimal point 1 place to the right. 10N = 5.555... –N = 0.555... Subtract N = 0.555... to eliminate the repeating part. Divide each side by 9. 9N = 5 N = The decimal 0.5 can be written as .
- 14. Answer Need Another Example? Write 2.7 as a mixed number in simplest form.
- 15. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 6. Write 2.18 as a mixed number in simplest form. Assign a variable to the value 2.18. Let N = 2.181818…. Then perform operations on N to determine its fractional value. N = 2.181818... Simplify. Multiply each side by 100 because 2 digits repeat.100(N) = 100(2.181818...) Multiplying by 100 moves the decimal point 2 places to the right. 100N = 218.181818 –N = 2.181818… Subtract N = 2.181818… to eliminate the repeating part. Divide each side by 99. Simplify. 99N = 216 N = 7 The decimal 2.18 can be written as .
- 16. Answer Need Another Example? Write 5.45 as a mixed number in simplest form.
- 17. • To write and evaluate expressions using exponents Course3, Lesson 1-2 The Number System
- 18. • power • base • exponent Course3, Lesson 1-2 The Number System
- 19. 1 Need Another Example? 2 Step-by-Step Example 1. Write the expression using exponents. (–2) • (–2) • (–2) • 3 • 3 • 3 • 3 The base –2 is a factor 3 times, and the base 3 is a factor 4 times. (–2) • (–2) • (–2) • 3 • 3 • 3 • 3 = (–2)3 • 34
- 20. Answer Need Another Example? Write (–9) • (–9) • (–9) • (–9) • (–9) • 4 • 4 • 4 as an expression using exponents. (–9)5 • 43
- 21. 1 Need Another Example? 2 3 Step-by-Step Example 2. Write the expression using exponents. a • b • b • a • b Use the properties of operations to rewrite and group like bases together. The base a is a factor 2 times, and the base b is a factor 3 times. a • b • b • a • b = a • a • b • b • b = a2 • b3
- 22. Answer Need Another Example? Write x • y • x • y • x • x as an expression using exponents. x4 • y2
- 23. 1 Need Another Example? 2 Step-by-Step Example 3. Evaluate . = Write the power as a product. Multiply
- 24. Answer Need Another Example? Evaluate .
- 25. 1 Need Another Example? 2 3 4 Step-by-Step Example 4. The deck of a skateboard has an area of about 25 • 7 square inches. What is the area of the skateboard deck? 25 • 7 = 2 • 2 • 2 • 2 • 2 • 7 Write the power as a product. Associative Property= (2 • 2 • 2 • 2 • 2) • 7 = 32 • 7 or 224 The area of the skateboard deck is about 224 square inches. Multiply.
- 26. Answer Need Another Example? A racquetball court has an area of 2 • 42 • 52 square feet. What is the area of the racquetball court? 800 ft2
- 27. 1 Need Another Example? 2 3 Step-by-Step Example 5. Evaluate the expression if a = 3 and b = 5. a2 + b4 a2 + b4 = 32 + 54 Replace a with 3 and b with 5. Write the powers as products.= (3 • 3) + (5 • 5 • 5 • 5) = 9 + 625 or 634 Add.
- 28. Answer Need Another Example? Evaluate x3 + y5 if x = 4 and y = 2. 96
- 29. 1 Need Another Example? 2 3 Step-by-Step Example 6. Evaluate the expression if a = 3 and b = 5. (a – b)2 (a – b)2 = (3 – 5)2 Replace a with 3 and b with 5. Perform operations in the parentheses first. = (–2)2 = (–2) • (–2) or 4 Write the powers as products. Then simplify.
- 30. Answer Need Another Example? Evaluate (x + y)2 if x = 4 and y = 2. 36
- 31. • To multiply and divide powers Course 3, Lesson 1-3 The Number System
- 32. • monomial Course 3, Lesson 1-3 The Number System
- 33. Course 3, Lesson 1-3 The Number System Words To multiply powers with the same base, add their exponents. Numbers Algebra Examples or4 3 4 3 2 2 2 7 2 m n m n a a a
- 34. 1 Need Another Example? 2 3 4 Step-by-Step Example 1. Simplify using the Laws of Exponents. 52 • 5 52 • 5 = 52 • 51 5 = 51 The common base is 5.= 52 + 1 = 53 or 125 Add the exponents. Simplify. Check 52 • 5 = (5 • 5) • 5 = 5 • 5 • 5 = 53
- 35. Answer Need Another Example? Simplify 76 • 7 using the Laws of Exponents. 77
- 36. 1 Need Another Example? 2 Step-by-Step Example 2. Simplify using the Laws of Exponents. c3 • c5 c3 • c5 = c3 + 5 The common base is c. Add the exponents.= c8
- 37. Answer Need Another Example? Simplify r4 • r6 using the Laws of Exponents. r10
- 38. 1 Need Another Example? 2 3 Step-by-Step Example 3. Simplify using the Laws of Exponents. –3x2 • 4x5 –3x2 • 4x5 = (–3 • 4)( x2 • x5) Commutative and Associative Properties The common base is x.= (–12)(x2 + 5) = –12x7 Add the exponents.
- 39. Answer Need Another Example? Simplify −7x2 • 11x4 using the Laws of Exponents. −77x6
- 40. Course 3, Lesson 1-3 The Number System Words To divide powers with the same base, subtract their exponents. Numbers Algebra Examples or 7 7 3 3 3 3 3 4 3 , where 0 m m n n a a a a
- 41. 1 Need Another Example? 2 Step-by-Step Example 4. Simplify using the Laws of Exponents. = 48 – 2 The common base is 4. Simplify.= 46 or 4,096
- 42. Answer Need Another Example? Simplify using the Laws of Exponents. 610
- 43. 1 Need Another Example? 2 Step-by-Step Example 5. Simplify using the Laws of Exponents. = n9 – 4 The common base is n. Simplify.= n5
- 44. Answer Need Another Example? Simplify using the Laws of Exponents. a6
- 45. 1 Need Another Example? 2 3 4 Step-by-Step Example 6. Simplify using the Laws of Exponents. = Group by common base. Subtract the exponents.= 23 • 31 • 51 23 = 8= 8 • 3 • 5 Simplify.= 120
- 46. Answer Need Another Example? Simplify . 450
- 47. 1 Need Another Example? 2 3 Step-by-Step Example 7. Hawaii’s total shoreline is about 210 miles long. New Hampshire’s shoreline is about 27 miles long. About how many times longer is Hawaii’s shoreline than New Hampshire’s? To find how many times longer, divide 210 by 27. Quotient of Powers= 210 – 7 or 23 Hawaii’s shoreline is about 23 or 8 times longer.
- 48. Answer Need Another Example? One centimeter is equal to 10 millimeters, and one kilometer is equal to 106 millimeters. How many centimeters are in one kilometer? 105 cm
- 49. To find the • power of a power, • power of a product Course 3, Lesson 1-4 The Number System
- 50. Course 3, Lesson 1-4 The Number System Words To find the power of a power, multiply the exponents. Numbers Algebra Examples 3 2 2 • 3 65 = 5 or 5 •= n m m na a
- 51. 1 Need Another Example? 2 Step-by-Step Example 1. Simplify using the Laws of Exponents. (84)3 (84)3 = 84 • 3 Simplify.= 812 Power of a Power
- 52. Answer Need Another Example? Simplify (52)8 . 516
- 53. 1 Need Another Example? 2 Step-by-Step Example 2. Simplify using the Laws of Exponents. (k7)5 (k7)5 = k7 • 5 Simplify.= k35 Power of a Power
- 54. Answer Need Another Example? Simplify (a3)7 . a21
- 55. Course 3, Lesson 1-4 The Number System Words To find the power of a product, find the power of each factor and multiply. Numbers Algebra Examples 33 32 2 66 = 6 or 216x x x = m m mab a b
- 56. 1 Need Another Example? 2 Step-by-Step Example 3. Simplify using the Laws of Exponents. (4p3)4 (4p3)4 = 44 • p3 • 4 Simplify.= 256p12 Power of a Power
- 57. Answer Need Another Example? Simplify (3c4)3. 27c12
- 58. 1 Need Another Example? 2 Step-by-Step Example 4. Simplify using the Laws of Exponents. (–2m7n6)5 (–2m7n6)5 = (–2)5m7 • 5n6 • 5 Simplify.= –32m35n30 Power of a Product
- 59. Answer Need Another Example? Simplify (−4p5q)2. 16p10q2
- 60. 1 Need Another Example? 2 3 4 5 Step-by-Step Example 5. A magazine offers a special service to its subscribers. If they scan the square logo shown on a smartphone, they can receive special offers from the magazine. Find the area of the logo. A = s2 Replace s with 7a4b.A = (7a4b)2 Area of a square Power of a ProductA = 72(a4)2(b1)2 Simplify.A = 49a8b2 The area of the logo is 49a8b2 square units.
- 61. Answer Need Another Example? Find the volume of a cube with side lengths of 6mn7 . Express as a monomial. 216m3n21 cubic units
- 62. To • use negative exponents, • multiply and divide powers using negative exponents Course 3, Lesson 1-5 The Number System
- 63. Course 3, Lesson 1-5 The Number System Words Any nonzero number to the zero power is 1. Any nonzero number to the negative n power is the multiplicative inverse of its nth power. Numbers Algebra Examples 05 1 3 or -3 1 1 1 17 7 7 7 7 0 1, 0x x , - 0 1 n n xx x
- 64. 1 Need Another Example? Step-by-Step Example 1. Write the expression using a positive exponent. 6–3 6–3 = Definition of negative exponent.
- 65. Answer Need Another Example? Write 4−4 using a positive exponent.
- 66. 1 Need Another Example? Step-by-Step Example 2. Write the expression using a positive exponent. a–5 a–5 = Definition of negative exponent.
- 67. Answer Need Another Example? Write c−7 using a positive exponent.
- 68. 1 Need Another Example? Step-by-Step Example 3. Write the fraction as an expression using a negative exponent other than –1. = 5–2 Definition of negative exponent.
- 69. Answer Need Another Example? Write as an expression using a negative exponent. f–5
- 70. 1 Need Another Example? 2 Step-by-Step Example 4. Write the fraction as an expression using a negative exponent other than –1. = 6–2 Definition of exponent Definition of negative exponent
- 71. Answer Need Another Example? Write as an expression using a negative exponent. 3−2
- 72. 1 Need Another Example? 2 3 4 Step-by-Step Example 5. One human hair is about 0.001 inch in diameter. Write the decimal as a power of 10. = 10–3 1,000 = 103 Definition of negative exponent 0.001 = Write the decimal as a fraction. = A human hair is 10–3 inch thick.
- 73. Answer Need Another Example? A grain of salt has a mass of about 0.0001 gram. Write the decimal as a power of 10. 10–4
- 74. 1 Need Another Example? 2 3 Step-by-Step Example 6. Simplify the expression. 53 • 5–5 = or Simplify. Write using positive exponents. Simplify. 53 • 5–5 = 53 + (–5) Product of Powers = 5–2
- 75. Answer Need Another Example? Simplify 4−5 · 4−3.
- 76. 1 Need Another Example? 2 Step-by-Step Example 7. Simplify the expression. Subtract the exponents. = w–1 – (–4) Quotient of Powers = w(–1) + 4 or w3
- 77. Answer Need Another Example? Simplify . c2
- 78. To • write numbers in standard form, • write numbers in scientific notation Course 3, Lesson 1-6 The Number System
- 79. • scientific notation Course 3, Lesson 1-6 The Number System
- 80. Course 3, Lesson 1-6 The Number System Words Scientific notation is when a number is written as the product of a factor and an integer power of 10. When the number is positive the factor must be greater than or equal to 1 and less than 10. Symbols , where 1 ≤ a < 10 and n is an integer Example 425,000,000 = ×10na 84.25×10
- 81. 1 Need Another Example? Step-by-Step Example 1. Write the number in standard form. 5.34 × 104 5.34 × 104 = 53,400.
- 82. Answer Need Another Example? Write 9.62 × 105 in standard form. 962,000
- 83. 1 Need Another Example? Step-by-Step Example 2. Write the number in standard form. 3.27 × 10–3 3.27 × 10−3 = 0.00327
- 84. Answer Need Another Example? Write 2.85 × 10−5 in standard form. 0.0000285
- 85. 1 Need Another Example? 2 Step-by-Step Example 3. Write the number in scientific notation. 3,725,000 3,725,000 = 3.725 × 1,000,000 The decimal point moves 6 places. = 3.725 × 106 Since 3,725,000 > 1, the exponent is positive.
- 86. Answer Need Another Example? Write 931,500,000 in scientific notation. 9.315 × 108
- 87. 1 Need Another Example? 2 Step-by-Step Example 4. Write the number in scientific notation. 0.000316 0.000316 = 3.16 × 0.0001 The decimal point moves 4 places. = 3.16 × 10–4 Since 0 < 0.000316 < 1, the exponent is negative.
- 88. Answer Need Another Example? Write 0.0044 in scientific notation. 4.4 × 10−3
- 89. 1 Need Another Example? 2 Step-by-Step Example 5. Refer to the table at the right. Order the countries according to the amount of money visitors spent in the United States from greatest to least. 1.06 × 107 Canada and United Kingdom ↓ 1.06 > 1.03 Group the numbers by their power of 10. Mexico and India ↓ 1.03 × 107 7.15 × 106 1.83 × 106 > 7.15 > 1.83 Compare the decimals.↑ United Kingdom ↑ Canada ↑ Mexico ↑ India
- 90. Answer Need Another Example? The following table lists the maximum frequency for the colors of the visible light spectrum. List the colors from greatest to least frequency. violet, blue, green, orange, red
- 91. 1 Need Another Example? Step-by-Step Example 6. If you could walk at a rate of 2 meters per second, it would take you 1.92 × 108 seconds to walk to the moon. Is it more appropriate to report this time as 1.92 × 108 seconds or 6.09 years? Explain your reasoning. The measure 6.09 years is more appropriate. The number 1.92 × 108 seconds is very large so choosing a larger unit of measure is more meaningful.
- 92. Answer Need Another Example? One light year is about 9.46 × 1012 kilometers or 9.46 × 1018 millimeters. Is it more appropriate to report this length as 9.46 × 1012 kilometers or 9.46 × 1018 millimeters? Explain your reasoning. 9.46 × 1012 kilometers; the number is very large so choosing a larger unit of measure is more meaningful.
- 93. To • add, subtract, multiply and divide with numbers written in scientific notation Course 3, Lesson 1-7 The Number System
- 94. 1 Need Another Example? 2 3 4 5 Step-by-Step Example 1. Evaluate (7.2 × 103)(1.6 × 104). Express the result in scientific notation. (7.2 × 103)(1.6 × 104) = (7.2 × 1.6)(103 × 104) = (11.52)(103 × 104) = 11.52 × 103 + 4 = 11.52 × 107 = 1.152 × 108 Commutative and Associative Properties Multiply 7.2 by 1.6. Product of Powers Add the exponents. Write in scientific notation.
- 95. Answer Need Another Example? Evaluate (1.1 × 10–3)(2.5 × 109). Express the result in scientific notation. 2.75 × 106
- 96. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 2. In 2010, the world population was about 6,860,000,000. The population of the United States was about 3 × 108. About how many times larger is the world population than the population of the United States? Estimate the population of the world and write in scientific notation. 6,860,000,000 ≈ 7,000,000,000 or 7 × 109 Find . = ≈ 2.3 × Associative Property Divide 7 by 3. Round to the nearest tenth. Quotient of Powers 7 ≈ 2.3 × 101 So, the population of the world is about 23 times larger than the population of the United States. ≈ 2.3 × 109 – 8 Subtract the exponents.
- 97. Answer Need Another Example? The largest planet in our solar system is Jupiter with a diameter of about 143,000 kilometers. The smallest planet in our solar system is Mercury with a diameter of about 5 × 103 kilometers. About how many times greater is the diameter of Jupiter than the diameter of Mercury? Sample answer: 3 × 101 or 30 times greater
- 98. 1 Need Another Example? 2 3 4 Step-by-Step Example 3. Evaluate the expression. Express the result in scientific notation. (6.89 × 104) + (9.24 × 105) (6.89 × 104) + (9.24 × 105) = (6.89 + 92.4) × 104 Write 9.24 × 105 as 92.4 × 104. Distributive Property Rewrite in scientific notation.= 9.929 × 105 = (6.89 × 104) + (92.4 × 104) = 99.29 × 104 Add 6.89 and 92.4.
- 99. Answer Need Another Example? Evaluate (2.85 × 107) + (1.61 × 109). Express the result in scientific notation. 1.6385 × 109
- 100. 1 Need Another Example? 2 3 4 5 Step-by-Step Example 4. Evaluate the expression. Express the result in scientific notation. (7.83 × 108) – 11,610,000 (7.83 × 108) – (1.161 × 107) = (78.3 × 107) – (1.161 × 107) Rewrite 11,610,000 in scientific notation. Write 7.83 × 108 as 78.3 × 107. Subtract 1.161 from 78.3.= 77.139 × 107 = (78.3 – 1.161) × 107 Distributive Property (7.83 × 108) – (1.161 × 107) Rewrite in scientific notation.= 7.7139 × 108
- 101. Answer Need Another Example? Evaluate (8.23 × 106) – 391,000. Express the result in scientific notation. 7.839 × 106
- 102. 1 Need Another Example? 2 3 4 Step-by-Step Example 5. 593,000 + (7.89 × 106) = (0.593 × 106) + (7.89 × 106) = (5.93 × 105) + (7.89 × 106) Rewrite 593,000 in scientific notation. Write 5.93 × 105 as 0.593 × 106 Add 0.593 and 7.89.= 8.483 × 106 = (0.593 + 7.89) × 106 Distributive Property 593,000 + (7.89 × 106)
- 103. Answer Need Another Example? Evaluate 6,450,000,000 – (8.27 × 107). Express the result in scientific notation. 6.3673 × 109
- 104. To • find the square roots of perfect squares, • solve equations with square and cube roots Course 3, Lesson 1-8 The Number System
- 105. • square root • perfect square • radical sign • cube root • perfect cube Course 3, Lesson 1-8 The Number System
- 106. Need Another Example? Step-by-Step Example 1 1. Find the square root. √64 Find the positive square root of 64; 82 = 64. √64 = 8
- 107. Answer Need Another Example? 15 Find √225.
- 108. Need Another Example? Step-by-Step Example 1 2. Find both square roots of 1.21; 1.12 = 1.21. ± √1.21 = ± 1.1 Find the square root. ± √1.21
- 109. Answer Need Another Example? ±1.2 Find ±√1.44.
- 110. 1 Need Another Example? Step-by-Step Example 3. Find the square root. Find the negative square root of
- 111. Answer Need Another Example?
- 112. 1 Need Another Example? Step-by-Step Example 4. There is no real square root because no number times itself is equal to –16. Find the square root. √ –16
- 113. Answer Need Another Example? no real square root Find √ –81.
- 114. 1 Need Another Example? 2 3 Step-by-Step Example 5. Solve t2 = 169. Check your solution(s). t2 = 169 Write the equation. t = 13 and –13 Definition of square root Check 13 · 13 = 169 and (–13)(–13) = 169 t = ±√169
- 115. Answer Need Another Example? x2 = 225 ±15
- 116. Course 3, Lesson 1-8 The Number System Words A square root of a number is one of its two equal factors. Symbols Example 2If = ,then is a square root of .x y x y 25 =25 so 5 is a square root of 25.
- 117. 1 Need Another Example? Step-by-Step Example 6. Find the cube root. √125 53 = 5 · 5 · 5 or 125 3 √125 = 5 3
- 118. Answer Need Another Example? 3 Find √27. 3
- 119. 1 Need Another Example? Step-by-Step Example 7. (–3)3 = (–3) · (–3) · (–3) or –27 Find the cube root. √ –27 3 √–27 = –3 3
- 120. Answer Need Another Example? –10 Find √ –1,000. 3
- 121. 1 Need Another Example? 2 3 4 5 Step-by-Step Example 8. Dylan has a planter in the shape of a cube that holds 8 cubic feet of potting soil. Solve the equation 8 = s3 to find the side length s of the container. Write the equation.8 = s3 Take the cube root of each side. So, each side of the container is 2 feet. Definition of cube root2 = s Check (2)3 = 8 √8 = s 3
- 122. Answer Need Another Example? A desktop organizer that is shaped like a cube has a volume of 125 cubic inches. What is the length of one side of the organizer? 5 in.
- 123. Course 3, Lesson 1-8 The Number System Words A cube root of a number is one of its three equal factors. Symbols 3If = ,then is the cube root of .x y x y
- 124. To • estimate square roots and cube roots to the nearest integer Course 3, Lesson 1-9 The Number System
- 125. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 1. Estimate √83 to the nearest integer. 7 • The largest perfect square less than 83 is 81. • The smallest perfect square greater than 83 is 100. √81 = 9 √100 = 10 Plot each square root on a number line. Then estimate √83. 81 < 83 < 100 Write an inequality. 92 < 83 < 102 81 = 92 and 100 = 102 √92 < √83 < √102 Find the square root of each number. 9 < √83 < 10 Simplify. So, √83 is between 9 and 10. Since √83 is closer to √81 than √100, the best integer estimate for √83 is 9.
- 126. Answer Need Another Example? Estimate √54 to the nearest integer. 7
- 127. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 2. Estimate √320 to the nearest integer. • The largest perfect cube less than 320 is 216. • The smallest perfect cube greater than 320 is 343. 216 < 320 < 343 Find the cube root of each number. Simplify. So, √320 is between 6 and 7. Since 320 is closer to 343 than 216, the best integer estimate for √320 is 7. Write an inequality. 63 < 320 < 73 216 = 63 and 343 = 73 3 √63 < √320 < √73 6 < √320 < 7 √216 = 6. √343 = 7. 3 3 3 3 3 3 3 3
- 128. Answer Need Another Example? Estimate √100 to the nearest integer. 5 3
- 129. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 3. Wyatt wants to fence in a square portion of the yard to make a play area for his new puppy. The area covered is 2 square meters. How much fencing should Wyatt buy? √2 is between 1.41 and 1.42. Wyatt will need 4 • √2 meters of fencing. The square root of 2 is between 1 and 2 so 4 • √2 is between 4 and 8. Is this the best approximation? You can truncate the decimal expansion of √2 to find better approximations. Truncate, or drop, the digits after the first decimal place. √2 is between 1.4 and 1.5. √2 ≈ 1.414213562 Use a calculator. 5.6 < 4√2 < 6.0 To find a closer approximation, expand √2 then truncate the decimal expansion after the first two decimal places. 7 Estimate √2 by truncating, or dropping, the digits after the first decimal place, then after the second decimal place, and so on until an appropriate approximation is reached. 4 • 1.4 = 5.6 and 4 • 1.5 = 6.0 4 • 1.41 = 5.64 and 4 • 1.42 = 5.685.64 < 4√2 < 5.68 The approximations indicate that Wyatt should buy 6 yards of fencing. √2 ≈ 1.414213562 √2 ≈ 1.414213562
- 130. Answer Need Another Example? A square flower garden has an area of 250 square feet. A stone path runs along the outermost edge of the flower garden. Find three sets of approximations for the length of the path. Then determine the length of the path. Sample answer: 60 ft and 64 ft; 63.2 ft and 63.6 ft; 63.24 ft and 63.28 ft; 63.2 ft
- 131. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 4. The golden rectangle is found frequently in the nautilus shell. The length of the longer side divided by the length of the shorter side is equal to . Estimate this value. 4 = 22 and 9 = 32 4 < 5 < 9 4 and 9 are the closest perfect squares. √22 < √5 < √32 First estimate the value of √5. Find the square root of each number. Since 5 is closer to 4 than 9, the best integer estimate for √5 is 2. Use this value to evaluate the expression. 7 22 < 5 < 32 2 < √5 < 3 Simplify.
- 132. Answer Need Another Example? To estimate the time in seconds it will take an object to fall h feet, you can use the expression . About how long will it take an object to fall from a height of 38 feet? 1.5 s
- 133. To • classify numbers, • compare and order real numbers Course 3, Lesson 1-10 The Number System
- 134. • irrational number • real number Course 3, Lesson 1-10 The Number System
- 135. Course 3, Lesson 1-10 The Number System Words Examples Rational Number A rational number is a number that can be expressed as the ratio , where a and b are integers and b ≠ 0. Irrational Number An irrational number is a number that cannot be expressed as the ratio, where a and b are integers and b ≠ 0. 7 , -12 8 76-2,5,3. 1.414213562....2
- 136. 1 Need Another Example? Step-by-Step Example 1. Name all sets of numbers to which the real number belongs. 0.2525... The decimal ends in a repeating pattern. It is a rational number because it is equivalent to .
- 137. Answer Need Another Example? Name all sets of numbers to which 0.090909… belongs. rational
- 138. 1 Need Another Example? Step-by-Step Example 1. Name all sets of numbers to which the real number belongs. √36 Since √36 = 6, it is a natural number, a whole number, an integer, and a rational number.
- 139. Answer Need Another Example? Name all sets of numbers to which √25 belongs. natural, whole, integer, rational
- 140. 1 Need Another Example? Step-by-Step Example 1. Name all sets of numbers to which the real number belongs. –√7 –√7 ≈ –2.645751311… The decimal does not terminate nor repeat, so it is an irrational number.
- 141. Answer Need Another Example? Name all sets of numbers to which −√12 belongs. irrational
- 142. 1 Need Another Example? 2 3 4 Step-by-Step Example 4. Fill in the with <, >, or = to make a true statement. √7 ≈ 2.645751311… 2 = 2.666666666… Since 2.645751311… is less than 2.66666666…, √7 < 2 .
- 143. Answer Need Another Example? Replace the in √15 3 with <, >, or = to make a true statement. <
- 144. 1 Need Another Example? 2 3 4 Step-by-Step Example 5. Fill in the with <, >, or = to make a true statement. 15.7% √0.02 15.7% = 0.157 √0.02 ≈ 0.141 Since 0.157 is greater than 0.141, 15.7% > √0.02.
- 145. Answer Need Another Example? Replace the in 12.3% √0.01 with <, >, or = to make a true statement. >
- 146. 1 Need Another Example? 2 3 4 5 6 Step-by-Step Example 6. Order the set √30 , 6, 5 , 5.36 from least to greatest. Verify your answer by graphing on a number line. Write each number as a decimal. Then order the decimals. √30 ≈ 5.48 5.36 ≈ 5.37 7 6 = 6.00 5 = 5.80 From least to greatest, the order is 5.36, √30, 5 , and 6.
- 147. Answer Need Another Example? 3, √15, 4.21, 4 Order the set √15, 3, 4 , 4.21 from least to greatest. Verify your answer by graphing on a number line.
- 148. 1 Need Another Example? 2 3 4 Step-by-Step Example 7. On a clear day, the number of miles a person can see to the horizon is about 1.23 times the square root of his or her distance from the ground in feet. Suppose Frida is at the Empire Building observation deck at 1,250 feet and Kia is at the Freedom Tower observation deck at 1,362 feet. How much farther can Kia see than Frida? Use a calculator to approximate the distance each person can see. Frida: 1.23 • √1,250 ≈ 43.49 Kia can see 45.39 – 43.49 or 1.90 miles farther than Frida. Kia: 1.23 • √1,362 ≈ 45.39
- 149. Answer Need Another Example? The time in seconds that it takes an object to fall d feet is . About how many seconds would it take for a volleyball thrown 32 feet up in the air to fall from its highest point to the sand? 1.4 s