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7. 5.45 is a rational number because it is a repeatingdecimal. It is a real number because all rationalnumbers are real numbers.18 is an integer because it is equal to -3, which is8. -6a number with no fractional or decimal part. Everyinteger is also a rational number and a real number.21. The set of integers best describes the situation. Thescores are counting numbers, their opposites, and 0.22. Nathaniel is correct. A rational number is a number1 is a fraction.that can be written as a fraction, and11π23. A whole number. The diameter isπ mi, or 1 mi.9. True. Whole numbers are a subset of the set ofrational numbers and can be written as a fractionwith a denominator of 1.24. It can be a rational number that is not an integer, oran irrational number.25. The total number of gallons of milk is either a wholenumber or a mixed number in which the fractional1 . Therefore, the number is a rational number.part is210. True. Whole numbers are rational numbers.11. The set of integers best describes the situation.The change can be a whole dollar amount andcan be positive, negative, or 0.Focus on Higher Order Thinking26. The set of negative numbers also includesnon-integer rational numbers and irrational numbers.12. The set of rational numbers best describes the1 inch.situation. The ruler is marked every1613. Sample answer: Describe one set as being a subsetof another, or show their relationships in a Venndiagram.27. Sample answer: If the calculator shows a decimalthat terminates in fewer digits than what thecalculator screen allows, then you can tell that thenumber is rational. If not, you cannot tell from thecalculator display whether the number terminatesbecause you see a limited number of digits.It may be a repeating decimal (rational) or anon-terminating non-repeating decimal (irrational).1 1. Since28. It is a whole number. 3 · 0.3 3 ·33 · 0.3 is equal to 0.9, then 0.9 is equal to 1, whichis a whole number.Independent Practice14. - 9 is an integer because it is equal to -3. Everyinteger is also a rational number and a real number.15. 257 is a whole number because it is a positivenumber with no fractional or decimal part. Everywhole number is also an integer, a rational number,and a real number.16. 50 is an irrational number because 50 is awhole number that is not a perfect square. It is areal number because all irrational numbers are realnumbers.1 is a rational number because it can be17. 8217 . It is a real numberrepresented as the ratio2because all rational numbers are real numbers.29. Sample answer: In decimal form, irrational numbersnever terminate and never repeat. Therefore, nomatter how many decimal places you include, thenumber will never be precisely represented. Therewill always be more digits.LESSON 1.3Your Turn18. 16.6 is a rational number because it is a terminatingdecimal. It is a real number because all rationalnumbers are real numbers.19. 16 is a whole number because it is equal to 4,which is a positive number with no fractional ordecimal part. Every whole number is also an integer,a rational number, and a real number.16.6Integers182Irrational Numbers 505. 5 is between 2 and 3, but is closer to 2. So5 2.5. 3 is between 1 and 2, so 3 1.5. 3 , 5 , 2.5Whole Numbers 9 1612 is between 3 and 4, so 12 3.5.4. 12 6 3.5 6 9.5 6 is between 2 and 3, so 6 2.5.12 6 12 12.5 14.5Since 9.5 14.5, 12 6 12 6 .Real NumbersRational Numbers3. 2 is between 1 and 2, so 2 1.5. 2 4 1.5 4 5.5 4 22 4 2 2 4Since 5.5 4, 2 4 2 4 .257 3020. The set of real numbers best describes the situation.The height can be any number greater than 0.Copyright by Houghton Mifflin Harcourt.All rights reserved.50.511.5 522.533.54

6. An approximate value of π is 3.14. So,π2 9.8956. 75 is between 8 and 9, so 75 8.5.2 75 , π , 10 7588.5 3π2909.5 10 10.5 11 11.5 121. 3 is between 1 and 2, so 3 1.5. 3 2 1.5 2 3.5 3 3 1.5 3 4.5Since 3.5 4.5, 3 2 3 3.8 is between 2 and 3 but very close to 3. Use 2.8.2. 8 17 2.8 17 19.8 11 is between 3 and 4, so 11 3.5. 11 15 3.5 15 18.5Since 19.8 18.5, 8 17 11 15.7 2.259 , -3.75,8, 342216. a. A 3.5 12.25 m2b. C π · 4 4π m4π 4 · 3.14 12.6 m2c. The circle would give her more space to plantbecause it has a greater area.17. a. 60 is between 7.7 and 7.8, so 60 7.75.58 7.2587.3 7.333 7.60757.75 7.25 7.33 7.60 29.93 7.482544The average is 7.4825 km.is between1 and 2, so 1.5.12 - 2 12 - 1.5 10.5 8 is between 2 and 3, so 8 2.514 - 8 14 - 2.5 11.5Since 10.5 11.5, 12 - 2 14 - 8 .7 is between 2 and 3, so 7 2.5.7. 7 2 2.5 2 4.5 10 is between 3 and 4, so 10 3.5. 10 - 1 3.5 - 1 2.5Since 4.5 2.5, 7 2 10 - 1.b. 56 7.4833, which is slightly greater than, butvery close to, Winnie’s estimate.17 is between 4 and 5, but very close to 4. Use 4.1.8. 17 3 10 - 2.5 7.5 11 is between 3 and 4, so 11 3.5.3 11 3 3.5 6.5Since 7.5 6.5, 17 3 3 11 .Copyright by Houghton Mifflin Harcourt.All rights reserved.645917 is between 4 and 5, but very close to 4. Use 4.1.5. 17 - 3 4.1 - 3 1.1 5 is between 2 and 3, but very close to 2. Use 2.2.-2 5 -2 2.2 0.2Since 1.1 0.2, 17 - 3 -2 5 .4 8 8 is between 2 and 3, so 8 2.5 and 1.75.2 8, 2, 72is between 3.1 and 3.2, so 10 3.15. π 3.14.13. 10π, 10 , 3.5220 is between 14 and 15, so 220 14.5.14. 100 10-10, 100 , 11.5, 22015. 8 is between 2 and 3, so 8 2.5.9 34. 9 3 3 3 6 3 is between 1 and 2, so 3 1.5.9 3 9 1.5 10.5Since 6 10.5, 9 3 9 3 .6.312. 7 is between 2 and 3, so 7 2.5. 22Independent Practice6 is between 2 and 3, so 6 2.5.3. 6 5 2.5 5 7.5 5 is also between 2 and 3, but will be a bit lessthan 2.5.Use 2.3.6 5 6 2.3 8.3Since 7.5 8.3, 6 5 6 5 . 212π10. 17 is between 4and 5 but very close to 4, so 17 4.1, and 17 - 2 2.1.π 2.57.π 1.57, and 1 π 3.14, so2212 2.45π km, 2.5 km,12 km, 17 - 2 km1 5211. Sample answer: Convert each number to a decimalequivalent, using estimation to find equivalents forirrational numbers. Graph each number on a numberline. Read the numbers from left to right to order thenumbers from least to greatest. Read the numbersfrom right to left to order the numbers from greatestto least.10 3.1 3.57.3; 332 10 is between 3 and 4, but is very close to 3.So 10 3.3.10 mi, 1 mi, 3.45 mi,10 mi323Guided Practice9. 3 is between 1.7 and 1.8, so 3 1.75.π 3.14,so 2π 6.281.5, 3 , 2π18. Sample answer: 3.7.19. Sample answer: 31 .20.6115 is between 10.7 and 10.8, so 115 10.75.115 10.4511Neither student is correct. The answer should be:115 , 10.5624, 115 .11

7 2.65 and 8 2.83, е is between21. a. Since 7 and 8 . 9 3 and 10 3.16, π is betweenb. Since 9 and 10 .5.Focus on Higher Order Thinking22 3.142922. a.73.143.1403.1413.142x 76.103.1431-107. Each side measures 200 ft.14.12 198.81; 14.22 201.64 so,200 is between 14.1 and 14.2. 200 14.1Each side is approximately 14.1 feet long. 3.1429. It is closer to π on the number line.7x 3.1416c.113x 113 · 3.1416x 355.0008355121is a whole number because8. 121121 11, and 11 is a positive number with121 11121no fractional or decimal part. Every whole number isalso an integer, a rational number, and a realnumber.23. 2; Rational numbers can have the same location,and irrational numbers can have the same location,but they cannot share a location.24. She did not consider that 12.6 12.66 .MODULE 1π is an irrational number because π is an irrational9.2number and dividing π by 2 gives another irrationalnumber. It is a real number because all irrationalnumbers are real numbers.Ready to Go On?0.351. 20 7.00-60100-10000.3510. True; Integers can be written as the quotient of twointegers.11. 8 is between 2 and 3, so 8 2.5. 8 3 2.5 3 5.5 3 is between 1 and 2, so 3 1.5.8 3 8 1.5 9.5Since 5.5 9.5, 8 3 8 3 .x 1.27( 100 )x ( 100 )1.27100x 127.27Becausex 1.27, subtract x from one sideand 1.27 fromthe other.100x 127.27-x-1.2799x 12612699x 999914 115 is between 2 and 3, so 5 2.5.12. 5 11 2.5 11 13.5 11 is between 3 and 4, so 11 3.5.5 11 5 3.5 8.5Since 13.5 8.5, 5 11 5 11 .213. π 9.87, 9.8 9.88, and 99 9.95.Therefore, the orderfrom least to greatestshould be π2, 9.8, 99 .157 3. 1881.8758 15.000-870-64605640-4001.875Copyright by Houghton Mifflin Harcourt.All rights reserved.11 1 ; -1 - 10010100101 and22b.2.x3 3433 33 x 343227π4. 81 9; - 81 -99 and -914.1 1 0.20 5251 0.250.2 0.22 11 , 0.2,2544 15. Sample answer: Real numbers, such as the rational1 , can describe amounts used in cooking.number47

[ ( 6 - 1 )2 ]210.( 3 2 )3[ ( 5 )2 ]2 ( 5 )3( 25 )2 ( 5 )3625 125 52 3-511. (2 ) - (10 - 6 )3 · 42 )33-5( 2 -4 ·4 26 - 43 · 4-5() 26 - 43 -56-2 2 -41 64 -1615 6316MODULE 2 Exponents and Scientific NotationAre You Ready?1. 10210 1010032. 1010 10 10100053. 1010 10 10 10 10100,00074. 1010 10 10 10 10 10 1010,000,0005. 45.3 1045.3 100045,3003Guided Practice6. 7.08 107.08 1000.07081. As the exponent decreases by 1, the value of thepower is divided by 8.80 1 118 827. 0.00235 100.00235 1,000,0002,35062. As the exponent decreases by 1, the value of thepower is divided by 6.60 1-116 616-2 368. 3,600 103,600 1,0000.3649. 0.5 100.5 1005023. Any number raised to the power 0 equals 1.2560 110. 67.7 1067.7 100,0000.00067754. As the exponent increases by 1, the value of thepower is multiplied by 10.100 1101 10102 10011. 0.0057 100.0057 1,0005745. As the exponent increases by 1, the value of thepower is multiplied by 5.50 151 552 2553 12554 62512. 195 10195 1,000,0000.0001956LESSON 2.1Your Turn6. As the exponent decreases by 1, the value of thepower is divided by 2.20 1-112 212-2 412-3 8-42 116-52 1327. ( 2 · 11 )2 222 4842 )32·3(8. 2 2 26 643-4-13-4-19. 5 · 5 · 5 5-2 51 521 25Copyright by Houghton Mifflin Harcourt.All rights reserved.8

90-109 0-1018. 6 · 6 · 6 67. As the exponent decreases by 1, the value of thepower is divided by 4.40 1-114 4-24 11614-3 6414-4 25614-5 1024 6-11 611 62710 · 10 102 7-519.510 104 10,000220. (10 - 6 )3 · 4 ( 10 2)232 4 · 4 122 43 2 122 45 122 1,024 144 1,1688. Any number raised to the power 0 equals 1.890 19. As the exponent decreases by 1, the value of thepower is divided by 11.110 1-1111 11111-2 121111-3 1331( 12 - 5 )721.[ (3 4)2 ]277 (72)277 72·277 7410. 4 · 4 · 4 41 · 41 · 41 41 1 1 4311. 77-4 73 343···2· (21 · 21) · (21 · 21 · 21) (21 1) · (21 1 1) 22 · 23 25(22)(22)22. Sample answer: When multiplying powers with thesame base, you add the exponents. When dividingpowers with the same base, you subtract theexponents. When raising a product to a power, youraise each factor to that power. When raising apower to a power, you multiply the exponents.7612.656·6·6·6·6·6·6 6·6·6·6·61·61 · 61 · 61 · 61 · 61 · 616 61 · 61 · 61 · 61 · 611 1 1 1 1 1 16 61 1 1 1 176 6523. 5-7 · 512 · 5-2 5-7 12-2 53 125()7 -2() 812 · 87 · -224. 8 · 812 812 · 8-14 67-5 812-14 62 8-21 821 6481213.89 812-925. 5 · (3 · 5 )2 5 · 152 83 5 · 2251014. 5 · 5 · 5 510 · 51 · 51 510 1 1 5122926.5915. 7 · 7 78 5 7132·482 4 616. ( 6 ) 685 9 9-31 931 7293317. ( 8 · 12)3 8 · 12Copyright by Houghton Mifflin Harcourt.All rights reserved. 1,1252-59

2·5(62)527. 6 8866106 8635. 4-5 610-8· (4 · 9)5 · 9-3 4-5 · (45 · 95 ) · 9-3 Power of aProductProperty-5553 (4 · 4 ) · (9 · 9 ) AssociativeProperty 62 361110 1110-(3 5)28.113 · 115 1110-8 112 1212 307-52·3 20 27-529. ( 2 ) 2 2 · 2 2 26 20 22 69-2· ( 7 - 4)2 (7 4 )2 3-2 (3 )2 112 3-2 2 112 1 · 92Zero ExponentProperty 92Identity Property ofMultiplication 81Definition of anexponentIndependent Practice 30 11237. The exponents cannot be added because the basesare not the same. 1 121 122538. To express 3 as a product of powers, the basesshould be 3 and the powers should add up to 5;Sample answer: 35 · 30; 34 · 31; 33 · 324431. 10 · [ ( 8 2)2 ]-3 10 · [ ( 10 )2 ]-3 104 · 102 · (-3)7439. 22 22 ;722224 227-4 104 · 10-6 104-6 10-21 1021 100545( )7 ( 2 5 )4 7( 7)732.7(8 - 1)7 75 4-7 223 10,648The distance from Earth to Neptune is the greaterdistance. It is about 10,648 times greater than thedistance from Earth to the moon.40. The student is not correct because()1 1 , which is less83 · 8-5 83 -5 8-2 82 64than 1.2 n-641. (b ) b2n -6-62n 22n -3(b2)-3 b-6 72 49[ (4 2 ][ ( 6 )3 ]5 33.12(9 - 3)( 6 )1263 · 5 612615 612 615-12)3 542. 63 216(4 · 643 · 63 34.3(5 1)63m69x ·x xm 6 9m 6-6 9-6m 3x3 · x6 x925)3y y643.yn25 - n 625 - n - 25 6 - 25-n -19-n · (-1) -19 · (-1)n 1925y y6y19 43 · 63-3 43 · 60 64 · 1 64Copyright by Houghton Mifflin Harcourt.All rights reserved.Product of PowersProperty36. Sample answer: The product of two fractions is theproduct of the numerators over the product of thedenominators. Writing this as a product of fractionslets you simplify them separately, using the Quotientof Powers Property to simplify the second fraction. 64 1 430. 3 40 · 9210

52. -32 -9-33 -27-34 -81-35 -243;( -3 )2 9( -3 )3 -27( -3 )4 81( -3 )5 -243For -an, you get -9, -27, -81, and -243.For (-a)n, you get 9, -27, 81, and -243. No, it doesnot appear that -an (-a)n. When n is even, thetwo expressions are opposites. When n is odd, thetwo expressions are equal.44. Sample answer: Dividing is the same as multiplyingby the reciprocal. So when dividing powers with thesame base, you add the opposite of the exponent inthe denominator. This is the same as subtracting theexponents.2 103045.2 10273010 1027 1030-27 103 1,000103 kg, or 1,000 kg53. Let the number equal x.x12 125x912-9x 125x3 125The cube root of 125 is the number 5.46. 2 · 2 210 30 240240 bytes10307-247. x · x7 ( -2) x x5;x7 x2 x7-2 x5LESSON 2.2Your Turn3. Move the decimal in 6,400 to the left to get 6.4.6,400 6.4 1,000 1036,400 6.4 103757-2x ; SampleBoth expressions equal x , so x · x x2answer: When multiplying powers with the samebase, you add exponents: 7 ( -2 ) 5. Whendividing powers with the same base, you subtractexponents: 7 - 2 5. In cases like this,n-mxn .x · x xm48. The number of cubes in each row is 3 raised to therow number.4. Move the decimal in 570,000,000,000 to the left toget 5.7.570,000,000,000 5.7 100,000,000,000 1011570,000,000,000 5.7 10115. Move the decimal in 9,461,000,000,000 to the left toget 9.461.9,461,000,000,000 9.461 1,000,000,000,000 10129,461,000,000,000 9.461 1012 km98. To write 7.034 10 in standard notation, move thedecimal 9 places to the right.7.034 109 7,034,000,00049. Since the number of cubes in each row is 3 raised tothe row number, the number of cubes in Row 6 willbe 3 raised to the power 6, and the number of cubesin Row 3 will be 3 raised to the power 3.36 729;3633 36-3 33 27The number of cubes in Row 6 will be 36, or 729.There will be 33, or 27, times the number of cubes inRow 6 as there are in Row 3.59. To write 2.36 10 in standard notation, move thedecimal 5 places to the right.2.36 105 236,000610. To write 5 10 in standard notation, move thedecimal 6 places to the right.5 106 5,000,000 gGuided Practice1. Move the decimal in 58,927 to the left to get 5.8927.58,927 5.8927 10,000 10458,927 5.8927 10450. 3 3 3 3 3 3 3 9 27 81 243 729 1,092The total number of cubes in the triangle is 1,092;Sample answer: I evaluated 31, 32, 33, 34, 35, and 36and added these numbers together.1234562. Move the decimal in 1,304,000,000 to the left toget 1.304.1,304,000,000 1.304 1,000,000,000 1091,304,000,000 1.304 1093. Move the decimal in 6,730,000 to the left to get 6.73.66,730,000 6.73 1,000,000 1066,730,000 6.73 10Focus on Higher Order Thinking51. Sample answer: No, I do not agree, because26 · 6 6·66 1 1. 362 36 · 36 6 · 6 · 6 · 6 6 · 6 36Copyright by Houghton Mifflin Harcourt.All rights reserved.4. Move the decimal in 13,300 to the left to get 1.33.413,300 1.33 10,000 10413,300 1.33 1011

22. 1,000 10.5 10,50010,500 1.05 10,000 10410,500 1.05 104 mosquitoes5. Move the decimal in97,700,000,000,000,000,000,000 to the left toget 9.77.97,700,000,000,000,000,000,000 9.77 102297,700,000,000,000,000,000,000 9.77 10222,400 words40 words 60 23.1 minute 601 hour2.6 105 260,000260,000 2,400 108.31 hours, or 108 hours and 20 minutes10836. Move the decimal in 384,000 to the left to get 3.84.5384,000 3.84 100,000 105384,000 3.84 1057. To write 4 10 in standard notation, move thedecimal 5 places to the right.4 105 400,00024. a. Write 1.182 in standard notation, 1,182, and thenmultiply by your weight.98. To write 1.8499 10 in standard notation, move thedecimal 9 places to the right.1.8499 109 1,849,900,000b. Sample answer: 94,560 lb; 9.456 10425. 230 20 4,6004,600 4.6 1,000 1034,600 4.6 103 lb9. To write 6.41 10 in standard notation, move thedecimal 3 places to the right.6.41 103 6,41034126. 9.999 10 and 2 10 ; numbers in scientificnotation are written as the product of a numbergreater than or equal to 1 and less than 10, and apower of 10, so 0.641 103 and 4.38 510 are notwritten in scientific notation.710. To write 8.456 10 in standard notation, move thedecimal 7 places to the right.8.456 107 84,560,000511. To write 8 10 in standard notation, move thedecimal 5 places to the right.8 105 800,00027. a. None of the girls have the correct answer.b. Polly and Samantha have the decimal in thewrong places, causing their exponents to beincorrect. Esther has the decimal in the correctplace but miscounted the number of places thedecimal moved.1012. To write 9 10 in standard notation, move thedecimal 10 places to the right.9 1010 90,000,000,000413. To write 5.4 10 in standard notation, move thedecimal 4 places to the right.5.4 104 54,000 s28. Sample answer: Scientific notation is a quicker wayto write large numbers. Also, it’s easier to read, it’sused b

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