• Study Resource
• Explore

Survey

* Your assessment is very important for improving the work of artificial intelligence, which forms the content of this project

Transcript
```Lesson 13.1 Skills Practice
Name_________________________________________________________ Date__________________________
Exponentially Speaking
Powers and Exponents
Vocabulary
Write the term that best completes each statement.
1. The exponent of a power is the number of times that the factor is repeatedly multiplied.
2. An expression used to represent a factor as repeated multiplication is called a power .
3. The base of a power is the repeated factor in a power.
Problem Set
State the base and the exponent of each power.
1. 94
The base is 9 and the exponent is 4.
3. 186
The base is 18 and the exponent is 6.
5. 2119
The base is 11 and the exponent is 9.
7. ​(215)​2​
The base is 215 and the exponent is 2.
9. ​​7​3​x3​ ​
The bases are 7 and x. The exponent is 3.
11. ​(222y)​4​
The base is 222y and the exponent is 4.
2. 102
The base is 10 and the exponent is 2.
4. 27
The base is 2 and the exponent is 7.
6. 24​8​3​
The base is 48 and the exponent is 3.
8.​(214)​6​
The base is 214 and the exponent is 6.
10.​(13x)​2​
The base is 13x and the exponent is 2.
12.​2(71y)​5​
The base is 71y and the exponent is 5.
Chapter 13 Skills Practice • 869
Lesson 13.1 Skills Practice
page 2
Write each power in expanded notation and evaluate to calculate the product.
13. 4
​ ​3​
14. 2
​ 4​ ​
The expanded notation The expanded notation is (4)(4)(4) 5 64.
is (2)(2)(2)(2) 5 16.
15. 5
​ 4​ ​
16. 7
​ 5​ ​
The expanded notation The expanded notation is (5)(5)(5)(5) 5 625.
is (7)(7)(7)(7)(7) 5 16,807.
17. 3
​ 8​ ​
18. ​63​ ​
The expanded notation The expanded notation is (3)(3)(3)(3)(3)(3)(3)(3) 5 6561.
is (6)(6)(6) 5 216.
Write each power in expanded notation and evaluate to calculate the product.
20. ​(24)​5​
The expanded notation is The expanded notation is (28)(28)(28) 5 2512.
(24)(24)(24)(24)(24) 5 21024.
21. ​(23)​4​
22. ​(26)​6​
The expanded notation is The expanded notation is (23)(23)(23)(23) 5 81.
(26)(26)(26)(26)(26)(26) 5 46,656.
23. ​2(2​5​)
24. ​2(5​6​)
The expanded notation is The expanded notation is 2(2)(2)(2)(2)(2) 5 232.
2(5)(5)(5)(5)(5)(5) 5 215,625.
25. ​2(7​2​)
26. ​2(8​3​)
The expanded notation is The expanded notation is 2(7)(7) 5 249.
2(8)(8)(8) 5 2512.
870 • Chapter 13 Skills Practice
19. ​(28)​3​
Lesson 13.1 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
Write each expression in expanded notation and evaluate to calculate the product.
27. ​(4)​2​​(25)​3​
28. ​(24)​4​​(7)​2​
The expanded notation is The expanded notation is (4)(4)(25)(25)(25) 5 22000.
(24)(24)(24)(24)(7)(7) 5 12,544.
29. ​(2)​5​​(26)​2​
30. ​(22)​5​​(8)​3​
The expanded notation is The expanded notation is (2)(2)(2)(2)(2)(26)(26) 5 1152.
(22)(22)(22)(22)(22)(8)(8)(8) 5 216,384.
31. ​2(3)​6​​(2)​4​
32. ​2(6)​3​​(4)​2​
The expanded notation is The expanded notation is 2(3)(3)(3)(3)(3)(3)(2)(2)(2)(2) 5 211,664.
2(6)(6)(6)(4)(4) 5 23456.
33. ​2(4)​3​​(8)​2​
34. ​2(5)​3​​(9)​2​
The expanded notation is The expanded notation is 2(4)(4)(4)(8)(8) 5 24096.
2(5)(5)(5)(9)(9) 5 210,125.
Write each expression in expanded notation.
35. 3
​ 2​ ​​x​3​
36. 2
​ ​5​​y​2​
The expanded notation is The expanded notation is (3)(3)(x)(x)(x).
(2)(2)(2)(2)(2)( y)( y).
37. 5
​ 2​ ​​y​4​
38. x
​ 5​ ​​z​2​
The expanded notation is The expanded notation is (5)(5)( y)( y)( y)( y).
(x)(x)(x)(x)(x)(z)(z).
Chapter 13 Skills Practice • 871
Lesson 13.1 Skills Practice
39. ​(5x)​4​
The expanded notation is page 4
40. ​(7x)​2​
The expanded notation is (7x)(7x).
(5x)(5x)(5x)(5x).
41. ​(6z)​3​
42. ​(2xy)​5​
The expanded notation is The expanded notation is (6z)(6z)(6z).
(2xy)(2xy)(2xy)(2xy)(2xy).
43. ​5x​3​
The expanded notation is 5(x)(x)(x).
44. ​4y​5​
The expanded notation is 4( y)( y)( y)( y)( y).
Write each expression in expanded notation.
46. ​2(6y)​3​
The expanded notation is The expanded notation is 2(7z)(7z)(7z)(7z)(7z).
2(6y)(6y)(6y).
47. ​(24a)​3​
48. ​(22b)​6​
The expanded notation is The expanded notation is (24a)(24a)(24a).
(22b)(22b)(22b)(22b)(22b)(22b).
49. 28z4
50. ​26m​7​
The expanded notation is The expanded notation is 28(z)(z)(z)(z).
26(m)(m)(m)(m)(m)(m)(m).
51. ​2(3x)​2​ ​y​4​
52. ​(28y)​3​ ​z​2​
The expanded notation is The expanded notation is 2(3x)(3x)( y)( y)( y)( y).
(28y)(28y)(28y)(z)(z).
872 • Chapter 13 Skills Practice
45. ​2(7z)​5​
Lesson 13.1 Skills Practice
page 5
Name_________________________________________________________ Date__________________________
Write each expression using exponents.
53. (7)(7)(7)(7)
54. 2  2  2  2  2  2  2  2
The power is ​7​ ​.
The power is ​2​8​.
4
55. 4  4  4  4  4  4
56. (5)(5)(5)(5)
The power is ​4​ ​.
The power is ​54​ ​.
6
57. (23)(23)(23)(23)(23)
The power is ​(23)​ ​.
5
59. 24  24  24  24  24
The power is ​(24)​5​.
61. 22  2  2  2  2  2  2  2
The power is ​22​8​.
58. 29  29  29  29  29  29  29
The power is ​(29)​7​.
60. 2(6)(6)(6)(6)
The power is ​26​4​.
62. 25  5  5
The power is ​25​3​.
Write each expression using exponents.
63. y  y  y  y  y
64. x  x  x  x  x  x  x  x  x
The power is ​y​ ​.
The power is ​x​9​.
5
65. (a)(a)(a)(a)(a)(a)(a)
66. b  b  b  b  b  b
The power is ​a​ ​.
The power is ​b6​ ​.
7
67. 2z  2z  2z  2z
68. (2m)(2m)(2m)(2m)(2m)(2m)
The power is ​(2z)​ ​.
4
69. 2b  2b  2b  2b  2b
The power is ​(2b)​5​.
71. 2(x)(x)(x)(x)
The power is ​2x​4​.
The power is ​(2m)​6​.
70. 2n  n  n  n  n
The power is ​2n​5​.
72. 2y  y  y  y  y  y  y  y
The power is ​2y​8​.
Chapter 13 Skills Practice • 873
Lesson 13.1 Skills Practice
page 6
Write each expression using exponents.
74. (2a)(2a)(2a)(2a)(c)(c)(c)(c)(c)
73. (2x)(2x)(2x)(2y)(2y)
The power is ​(2x)​3​(2y)​2​.
75. (2y)  (2y)  z  z  z  z  z
The power is ​(2y)​2​​z​5​.
The power is ​(2a)​4​​c​5​.
76. (2m)  (2m)  (2m)  (2m)  (2m)  n  n  n
The power is ​(2m)​5​​n​3​.
78. 26  6  6  b  b  h  h  h
77. (2a)(2a)(2a)(b)(b)(b)
The power is ​(2a)​3​​b​3​.
79. 24  4  4  4  4  m  m  m  n  n
The power is 263​b2​ ​​h​3.​
80. 23  3  3  3  3  y  y  y  y  z  z  z
The power is 245​m3​ ​​n​2​.
The power is ​23​5​​y​4​​z​3​.
81. 25  5  a  a  a  a  a  b  b
82 . 22  2  2  2  x  x  x  y  y  y  y
The power is ​25​ ​​a​ ​​b​ ​.
The power is ​22​4​​x​3​​y​4​.
2 5
2
Write each expression using exponents.
The power is ​(3x)​3​y​3​z5​ ​.
85. 5mn  5mn  5mn  5mn  l  l  l
The power is ​(5mn)​4​​l3​ ​.
84. 4a  4a  2b  2b  2b  c  c  c  c
The power is ​(4a)​2​​(2b)​3​​c​4​.
86. (22x)(22x)(22x)( y)( y)(z)(z)(z)
The power is ​(22x)​3​​y​2​​z​3​.
87. a  a  a  (24b)  (24b)  c  c
88. 26m  6m  6m  6m  n  n  n
The power is ​a​ ​​(24b)​ ​​c​ ​.
The power is ​2(6m)​4​​n​3​.
3
2 2
89. 23x  3x  3x  (22y)  (22y)  z  z  z  z
The power is 2(3x)3​(22y)​2​​z​4​.
874 • Chapter 13 Skills Practice
83. 3x  3x  3x  y  y  y  z  z  z  z  z
Lesson 13.2 Skills Practice
Name_________________________________________________________ Date__________________________
Digital Storage
Multiplying and Dividing Powers
Problem Set
Write each expression in expanded form.
1. 5
​ 2​ ​ ​54​ ​
2. 3
​ 4​ ​ ​35​ ​
555555
3. 7
​ 3​ ​ ​75​ ​
333333333
4. 2
​ 2​ ​ ​23​ ​
77777777
22222
Simplify each expression. Show your work.
5. ​(9)​2​​(9)​4​
6. ​(24)​3​​(24)​7​
​(9)​2​​(9)​4​ 5 ​92​ ​14
​
​(24)​3​​(24)​7​ 5 ​(24)​317
5 ​96​ ​
7. ​(22)​2​​(22)​5​
5 ​(24)​10​
8. ​(28)​4​​(28)​5​
​
​(22)​2​​(22)​5​ 5 ​(22)​215
​
​(28)​4​​(28)​5​ 5 ​(28)​415
5 ​(22)​7​
5 ​(28)​9​
Write each expression in expanded form.
9. x​ ​2​ ​x3​ ​
xxxxx
11. ​m5​ ​ ​m2​ ​
mmmmmmm
10. y  ​y2​ ​
yyy
12. n
​ 4​ ​ ​n5​ ​
nnnnnnnnn
Chapter 13 Skills Practice • 875
Lesson 13.2 Skills Practice
page 2
Simplify each expression. Show your work.
13. ​(2a)​3​​(2a)​4​
14. ​(2b)​2​​(2b)​4​
​(2a)​3​​(2a)​4​ 5 ​(2a)​3​14
​(2b)​2​​(2b)​4​ 5 ​(2b)​2​14
5 ​(2b)​6​
5 ​(2a)​7​
16. ​(2z)​6​​(2z)​2​
15. ​(2c)(2c)​3​
​
​(2c)(2c)​3​ 5 ​(2c)​113
​
​(2z)​6​​(2z)​2​ 5 ​(2z)​612
5 ​(2z)​8​
5 ​(2c)​4​
Simplify each expression. Show your work.
17. x​ ​2​​y​2​​x​3​​y​5​
​x​2​​y​2​​x​3​​y​5​ 5 ​(x​2​​x​3​)​( y​2​​y​5​)
​a3​ ​​b​2​​ab​4​ 5 ​(a​3​​a)(b​2​​b​4​)
5 ​x2​ 13
​ ​y2​ 15
​
5 ​a311
​ ​​b​214
​
5 ​x5​ ​​y​7​
5 ​a4​ ​​b​6​
19. ​7y​3​​z​4​ ​2yz​3​
20. ​3mn​3​ ​8m​6​​n​7​
​7y​3​​z​4​  ​2yz​3​ 5 (7  2)​( y​3​​y)(z​4​​z​3​)
​3mn​3​  ​8m​6​​n​7​ 5 (3  8)(m  ​m6​ ​)​(n​3​​n​7​)
5 ​14y​311​ ​z​4​13
5 ​24m​116​ ​n3​ ​17
5 ​14y​4​​z​7​
5 ​24m​7​​n​10
876 • Chapter 13 Skills Practice
18. a
​ 3​ ​​b​2​​ab​4​
Lesson 13.2 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
21. ​5y​3​ ​3yz​3​ ​z2​ ​
22. ​9b​2​ ​2a​5​ ​​a​2​b6​ ​
​5y​3​  ​3yz​3​  ​z2​ ​ 5 (5  3)​( y​3​y)​(z​3​​z​2​)
​9b​2​  ​2a​5​  ​​a2​ ​b6​ ​ 5 (9  ​2)(a​5​​a​2​)​(b​2​​b​6​)
5 ​15y​311
​ ​z3​ 12
​
5 ​18a​512
​ ​ b​216
​
5 ​15y​4​​z​5​
5 ​18a​7​​b​8​
23. ​(2y)​2​ ​(2y)​4​ z  z
24. ​(2m)​2​ ​n3​ ​ ​(2m)​4​ ​n2​ ​
​ ​
​(2y)​2​  ​(2y)​4​  z  z 5 ​(2y)​2​14 ​z111
​(2m)​2​  ​n3​ ​  ​(2m)​4​  ​n2​ ​ 5 ​[ ​(2m)​2​  ​(2m)​4​ ]​​(n​3​  ​n2​ ​)
5 ​(2m)​214
​ ​n312
​ ​
5 ​(2m)​6​​n​5​
5 ​(2y)​6​​z​2​
25. ​3x​3​ ​2x​2​ ​y3​ ​ y
26. (2b)  4a  ​(2b)​5​ ​2a​3​
(2b)  4a  ​(2b)5​​ ​2a3​​5 (4  2) (a  ​​a3​​)[(2b)(2b)5​​]
5 ​6x​312
​ ​y3​ 11
​
5 ​8a​113​ ​(2b)​1​15
5 ​6x​5​​y​4​
5 ​8a​4​​(2b)​6​
​3x​3​  ​2x​2​  ​y3​ ​  y 5 (3  ​2)(x​3​​​x​2​)( y​3​y)
Chapter 13 Skills Practice • 877
Lesson 13.2 Skills Practice
page 4
Simplify each expression. Show your work.
28. ​​(4​3​)​5​
27. ​(2​2​​)​3​
​(2​2​​)​3​ 5 ​22?3
​ ​
​ ​
​​(4​3​)​5​ 5 ​43?5
5 ​2​6​
5 ​4​15​
29. ​​(7​2​)4​ ​
30. ​​(3​3​)​2​
​
​​(7​2​)​4​ 5 ​72​ ?4
​ ​
​​(3​3​)2​ ​ 5 ​33?2
5 ​7​8​
5 ​3​6​
31. ​​2(2​2​)4​ ​
32. ​​2(6​5​)​2​
​ ​)
​​2(2​2​)4​ ​ 5 2(​22?4
​​2(6​5​)​2​ 5 ​2(6​5?2)
5 ​22​8​
5 ​26​10​
Simplify each expression. Show your work.
34. ​​( y​3​)2​ ​
​ ​
​​(x​5​)​2​ 5 ​x5?2
5 ​x10
​​
35. ​​(m​3​)​5​
​ ​
​​( y​3​)​2​ 5 ​y3?2
5 ​y​6​
36. ​​(n​4​)​3​
​ ​
​​(m​3​)​5​ 5 ​m3?5
​ ​
​​(n​4​)​3​ 5 ​n4?3
5 ​m15
​​
5 ​n​12​
37. ​​(2b​2​)​6​
38. ​​2(m​2​)​4​
​ ​)
​​(2b​2​)6​ ​ 5 ​(21)​6​(​b2?6
5 ​b​12​
878 • Chapter 13 Skills Practice
​ ​)
​​2(m​2​)​4​ 5 2(​m2?4
5 ​2m​8​
33. ​​(x​5​)​2​
Lesson 13.2 Skills Practice
page 5
Name_________________________________________________________ Date__________________________
Simplify each expression. Show your work.
39. ​(3x​2​​y)​4​
40. ​(2a​2​​​b​3​)5​ ​
​(3x​2​​y)​4​ 5 ​34​ ​​​(x​2​)4​ ​​y​4​
​(2a​2​​​b​3​)5​ ​ 5 ​25​ ​​​(a​2​)5​ ​​​(b​3​)5​ ​
5 ​81(x)​24​​y​4​
5 ​25​ ​​(a)​25​​(b)​35​
5 ​81x​8​​y​4​
5 ​32a​10​​b​15​
41. ​​(24mn​3​)3​ ​
42. ​(25y​3​​​z​2​)5​ ​
​​(24mn​3​)3​ ​ 5 ​(24)​3​​m​3​​​(n​3​)3​ ​
​(25y​3​​​z​2​)5​ ​ 5 ​(25)​5​​​( y​3​)5​ ​​​(z​2​)5​ ​
5 ​264m​3​​(n)​33​
5 2​3125( y)​35​​(z)​25​
5 ​264m​3​​n​9​
52​3125 y​15​​z​10​
43. ​​(22cd​2​)4​ ​
44. ​(22x​3​y4​​ )​2​
​(22x​3​​​y​4​)​2​ 5 ​(22)​2​​​(x​3​)​2​​​( y​4​)​2​
5 ​16c​4​​(d)​24​
5 ​4(x)​3?2​​( y)​4?2​
5 ​16c​4​​d​8​
5 ​4x​6​​y​8​
​​(22cd​2​)4​ ​ 5 ​(22)​4​​c​4​​​(d​2​)4​ ​
Chapter 13 Skills Practice • 879
Lesson 13.2 Skills Practice
page 6
Write each numerator and denominator in expanded form.
4
45. __
​ ​72​ ​ ​​
​7​
8
46. __
​ ​45​  ​
​4​
? 7
__________
​ 7 ? 7 ? 7 ​

7?7
​9​6​
47. __
​  2 ​
​9​ ​
? 4 ? 4 ?
4?4
______________________
​ ​4 ? 4 ? 4 ? 4
​
4?4?4?4?4
9
48. __
​ 5​3 ​
5​
?
9 ? 9
________________
​ 9 ? 9 ? 9 ? 9 ​
9?9
5 ? 5 ​
? 5 ?
5?5
_________________________
​ 5 ? 5 ? 5 ? 5 ?
5?5?5
Simplify each expression. Show your work.
68 ​
50. ​ __
65
5
49. __
​ ​32​ ​ ​​
3​
__
3​522​
​ ​32 ​ 5 ​
__
​ 6  ​ 5 ​6825
​ ​
5 ​33​​
5 ​63​ ​
8
​5​
65
3​
9
52. ____
​  3  2 ​
23
( 12​ )
( 3 )
12​
___
_____
​ 212​
​ 5

2​ ​  4 ​​  ​
4
3
__
____
​  3  2 ​ 5
2​ ​  2 ​  ​
5 2(​12​824​)
5 2(​3922
​ ​)
5212​4​
5 ​23​7​
8
12​
8
9
23
9
Write each numerator and denominator in expanded form.
5
y4
53. __
​ x3 ​
54. __
​  3 ​
x
y
x ? x ? x ? x
​ _____________
? x
​
x?x?x
​z​7​
55. __
​  2 ​
​z​ ​
z ? z ? z ? z ?  z
? z
? z​
​__________________
z?z
880 • Chapter 13 Skills Practice
y?y?y?y
__________
​

​
y?y?y
6
56. __
​ a3  ​
a
a ? a ? a ? a ?
a?a
​________________
a ? a ? a    ​
​212​8​
51. _____
​  4 ​

​  12​ ​
Lesson 13.2 Skills Practice
page 7
Name_________________________________________________________ Date__________________________
Simplify each expression. Show your work.
​m​9​
58. ___
​  6 ​
​m​ ​
​n5​ ​
57. __
​  4 ​
​n​ ​
5
__
​ ​n​ ​  ​5 ​n​524​
9
​ m​926​
___

​ ​m6​ ​ 5 ​
​n​ ​
​m​ ​
5 n
5 ​m3​ ​
4
​  b​6​
59. ____
​  4 ​

​2b​ ​
​2h​14​
60. _____
​  6 ​

​h​ ​
(  )
6
6
​
____
2​__
​ ​b​ ​  ​​
​ ​   b​ ​ 5
​ b​4​
2
​b4​ ​
14
​ 6​ ​
h
6
5 ​2b​624​
5 2(​h1426
​ ​)
5 ​2b​2​
5 2h​8​
​212x​7​

61. ______
​  2 ​
​3x​ ​
(  3 )(​x  ​​)
​24y​8​

62. ____
​  4 ​
​4y​ ​
( 4 )(​y  ​​)
7
7
​  _____
______

​ 212
​​__
​ ​x​ ​  ​​
​212x​
​ 5 ​
​
​y​ ​
​24y​ ​ ___
_____

​ 24 ​   ​​__
​  4 ​  ​
​  4 ​ 5 ​
5 24(​x722
​ ​)
5 6( y​824​)
5 ​24x​5​
5 ​6y​4​
​3x​2​
(  h )
14
​  _____
_____
​ ​2h​ ​ 5 ​

​ 2h  ​
​
45​a​7​​b​3​

63. ______
​  4  ​
25​a​ ​b
2
)(​a  ​​)( b )
(25
7 3
7
3
​
_______
​ 45  ​  ​​__
​ ​a4​ ​  ​​​__
​ ​b​ ​  ​​
5 ​___
​ 45​a​4​b ​ ​
25​a​ ​b
8
8
​4y​ ​
10​m​8​​n​3​
64. _______
​

​
2​m7​ ​
(  2 )( m )
8
_______
​ 10m n ​ 5 ​

___
​ 10 ​  ​​ ___
​ m  ​  ​(n3)
8 3
2m7
7
5 29(​a​724​)(​b321
​ ​)
5 5(​​m​827​)(n​3​)
5 ​29a​3​​b​2​
5 ​5mn​3​
Chapter 13 Skills Practice • 881
Lesson 13.2 Skills Practice
page 8
Simplify each expression. Show your work.
65. ​3a​2​ ​4a​3​
66. ​25m​8​ ​7m​2​
​3a​2​  ​4a​3​ 5 (3  ​4)(a​2​  ​a3​ ​)
​25m​8​  ​7m​2​ 5 (25  ​7)(m​8​  ​m2​ ​)
5 ​(12)(a​213
​ )
5 (235)(​m812
​ ​)
5 ​12a​5​
5 ​235m​10​
67. ​(22b)​3​ ​3b​4​
68. ​​(2n​2​)4​ ​
​(22b)​3​  ​3b​4​ 5 ​(22)​3​​(b​3​)  ​3b​4​
​​(2n​2​)​4​ 5 ​(2​4​)(​n2​  4​)
5 ​(22)​3​​(3)(b​3​  ​b4​ ​)
5 ​16n​8​
5 (28)(3)(​b314
​ ​)
52​24b​7​
69. ​​(25mn​3​)2​ ​
70. ​​(4y​3​)2​ ​ ​(23y)​2​
​​(25mn​3​)​2​ 5 ​(25)​2​​(m​2​)(​n3​  2​)
​​(4y​3​)2​ ​  ​(23y)​2​ 5 ​(4​2​)(​​ y3​  2​)(23)​2​​( y​2​)
5 ​25m​2​​n​6​
5 ​​(16)( y​6​)(9)( y​2​)
5 ​144y​8​
​12g​5​
71. _____
​  3 ​

​2g​ ​
5
72. _____
​ 14h4 ​
2h
( 2 )
(  2 )
​12g​ ​ ___
_____
​  3 ​ 5 ​

​ 12 ​   ​​g​523​
5
​  ___
_____
​ ​14h​ ​​ 5 ​
​ 14 ​  ​​​h​524​
5 ​6g​2​
5 7h
5
​2g​ ​
882 • Chapter 13 Skills Practice
2h​4
5 ​144( y​612
​ )
Lesson 13.2 Skills Practice
page 9
Name_________________________________________________________ Date__________________________
​232x​8​​y​5​
________
73. ​

​
​4x​6​y
40a​5​​b​7
74. ​ _______
​​
2​8a​2​​b​6
(  4 )( x​ )( y )
(  8 )( a )( b​ )
8 ​y​ ​
232​x​ ​​y​ ​ _____
________

​5 ​ ​ ​232
​

​​​ __
​ ​x​ ​ ​  ​​​​ __
​   ​ ​
​
________
​  40​a2​ ​b​6
​ 5 ​
_____
​ 240
​​ __
​ ​a​​2 ​  ​​ __
​ ​b6​  ​  ​
​
5 ​(28)(x​82​ 6​ )( y​521
​ )
5 ​(25)(a​52​ 2​ )(b​726
​ )
5 ​28x​2​​y​4​
5 ​25a​3​b
8 5
5
5
4​x6​ ​y
28​a​ b​
6
(  26 )(
)( n )
​4
​
m ? ​m
_____________
​ ​n​ ​ 5 ​

_____
​ 218 ​  ​​ ​ _______
​

​​ __
​ n  ​​ ​ ​
​ ​218​m​ ​ ? ​m
2 4
26​m ​n
​3
5
7
​56xy​3​ ​(x​3​​y)​2​ y
76. ______________
​
​
8​x6​ ​y
2​18m​5​ ​m2​ ​​n​4​
_____________
75. ​

​
26​m3​ ​n
5
7
5​
m​3
2
56xy3​ ? ​​(x​3​)​2​​y​2​ ? y
56xy​3​ ? ​(x​3​​y)​2​y _______________

​5 ​​
​​
​ _____________
6
8​x​ ​y
8​x6​ ​y
(  )(
)(
)
5 (3)(​m​51223​)(​n421
​ ​)
3?2 ​y3​
​ ? ​y​2​ ? y
​​
5 ​ ___
​ 56 ​  ​​ ______
​ x ? ​x​6 ​
​​​ ​ ________
​
​​
y
8
x
5 ​3m​4​​n​3​
5 ​(7)(x​116–6 ​)(​ y​3121121​)
5 ​7xy​5
Chapter 13 Skills Practice • 883
884 • Chapter 13 Skills Practice
Lesson 13.3 Skills Practice
Name_________________________________________________________ Date__________________________
Extending the Rules
Zero and Negative Exponents
Problem Set
Simplify each expression using the properties of powers. Express your solution in expanded notation,
and then calculate the value.
3
1. __
​ 53 ​
5
__
​ 53 ​ 5
5323
4
__
2424
​
​ 2  ​ 5 ​
5 ​50​ ​
5 ​20​ ​
5 1
5 1
3
5
3. __
​ 7  ​
72
2
24
6
4. __
​ 36 ​
3
2
__
​
​ 7  ​  5 ​7​222
__
36​ 26
​
​ 3  ​ 5 ​
5 ​70​ ​
5 ​30​ ​
5 1
5 1
7
2
4
2. __
​ 24 ​
2​
5. __
​ 4  ​
49
9
6
36
6. __
​ 6  ​
62
2
9
__
4929
​ 49 ​ 5
__
​ 6  ​ 5 ​
6222
​ ​
5 40
5 ​60​ ​
5 1
5 1
4
2
62
Chapter 13 Skills Practice • 885
Lesson 13.3 Skills Practice
page 2
Simplify each expression using the Quotient Rule of Powers.
7. __
​ 2  ​
24
0
8. __
​ 9  ​
95
0
__
2024
​ ​
​ 24 ​ 5 ​
2
9
5 ​224
​ ​
5​ 9​25​
0
9. __
​ 8  ​
83
0
0
__
​ 95 ​  5 ​9​025​
10. __
​ 4  ​
43
0
0
__
​ 43 ​  5 ​4​023​
0
__
​ 83 ​  5 ​8​023​
8
4
5 ​8​23​
5 ​4​23​
50
11. __
​  8 ​
5
12. __
​ 3  ​
32
0
0
__
​ 32 ​  5 ​3​022​
0
__
​ 58 ​  5 ​5​028​
5
3
5 ​5​28​
5 ​3​22​
Rewrite each expression so that the exponent is positive.
__
​ 1  ​
32
15. 6
​ 23
​ ​
__
​ 1  ​
63
​17. 4
​ 28
​ ​
__
​ 1  ​
48
886 • Chapter 13 Skills Practice
14. 5
​ 29
​ ​
__
​ 1  ​
59
16. 2
​ 25
​ ​
__
​ 1
25
18. 7
​ 23
​ ​
__
​ 1  ​
73
13. 3
​ 22
​ ​
Lesson 13.3 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
Rewrite each expression so that the exponent is positive.
19. x​ ​23​
20. ​y​28​
__
​ 1 ​
x​ 3​ ​
__
​ 1  ​
y8
21. a
​ 28
​ ​​b​22​
22. ​m24
​ ​​n​25​
____
​  1  ​
a8​b2​ ​
23. y​ 24
​ ​​z​23​
_____
​  1   ​
m4​n5​ ​
24. x​ 22
​ ​​y​26​
____
​  1  ​
y4​z3​ ​
____
​  1  ​
x2​y6​ ​
Rewrite each expression so that the exponent is positive.
25. ___
​  1  ​
225
​
26. ___
​  1
623
​ 5​ ​
2
27. ___
​  1  ​
924
​ 3​ ​
6
28. ___
​  1  ​
422
​ 2​ ​
4
​ 4​ ​
9
29. ___
​  1  ​
723
​
30. ___
​  1
524
​ 4​ ​
5
​73​ ​
Rewrite each expression so that the exponent is positive.
31. ___
​  1  ​
y24
y​ ​4​
32. ___
​  1  ​
x27
x​ ​7​
33. ______
​  1   ​
x26​y22
​ ​
34. ______
​  1   ​
a23​b28
​ ​
x​ 6​ ​​y​2​
​ 3​ ​​b​8​
a
35. _______
​  1   ​
m25 ​n​29​
36. _______
​  1   ​
g27 ​h24
​ ​
​m​5​​n​9​
​g7​ ​​h​4​
Chapter 13 Skills Practice • 887
Lesson 13.3 Skills Practice
page 4
Simplify each expression. Show your work.
y22
38. ___
​  22 ​​
y
4
37. __
​ ​x4 ​​
x
4
__
x424
​ x4 ​ 5
x
y22
y22 2 (22​)
​ 5
​ ___
y22
5 ​x​0​
5 ​y0​ ​
5 1
5 1
40. ​​(2​0​)(2​5​)
39. ​​(6​0​)(6​3​)
​​(6​0​)(6​3​) 5 ​(1)(6​3​)
​​(2​0​)(2​5​) 5 ​(1)(2​5​)
5 ​6​3​
5 ​25​ ​
5 216
5 32
12x2y3
41. ______
​  2  ​​
4x y
15a5b4 ​​
42. ​ ______
3a2​b4​
(  4 )
(  3 )
12x y
______
​ 5 ​
​  2
___
​ 12 ​  ​ x​222​y321​
​
______
​ 15a2 b4 ​ 5 ​
___
​ 15 ​  ​a522​
​ ​b424​
​
5 ​3x​0​​y​2​
5 ​5a​3​b0​ ​
5 ​3y​2​
5 ​5a​3​
2 3
3a b
4
4x y
5
888 • Chapter 13 Skills Practice
Lesson 13.3 Skills Practice
page 5
Name_________________________________________________________ Date__________________________
Simplify each expression. Then, calculate the value. Do not leave your answer as a power.
2
43. __
​ 34  ​
3
__
3224
​ 34 ​ 5
5
__
​
4528​
​ 48 ​ 5 ​
5 322​
5 ​423
​ ​
1  ​​
5 ​ __
32
1  ​​
5 ​ __
43
1 ​​
5 ​ __
9
1  ​​
5 ​ ___
64
2
3
45. ​​(6​5​)(6​27​)
5
44. __
​​ 48  ​​
4
4​
46. ​​(2​4​)(2​28​)
​
​​(6​5​)(6​27​) 5 ​651(27)​
​​​(2​4​)(2​28​) 5 2​41(28)​
5 ​622​
​
5 ​224
​ ​
1  ​​
5 ​ __
62
1  ​
5 ​ __
24
1  ​
5 ​ ___
36
1  ​
5 ​ ___
16
523 ​
47. ​ ___
526
25
48. ___
​ 322 ​
3
523 ​ 5
5232(26)
​ ___
526
325 ​ 5
3​252(22)​
​ ___
322
5 5​3​
5 ​323
​ ​
5 125
1  ​
5 ​ __
33
1  ​
5 ​ ___
27
Chapter 13 Skills Practice • 889
Lesson 13.3 Skills Practice
page 6
Simplify each expression so that the exponent is positive. Show your work.
​y3
49. __
​  5 ​​
y
50. __
​ x  ​
x9
4
y
__
​   ​ 5 ​
y325​
​
y
x
5 ​y22
​ ​
5 ​x​25​
1  ​​
5 ​ __
y2
5 __
​ 1 ​
x5
3
5
a25 ​​
51. ​​ ___
a2
4
__
​ x9 ​  5 ​x​429​
8
52. ___
​ c24  ​​
c
a25 ​ 5 ​
1   ​
____
​ ___
a2
a2a5​
c8  ​ 5
c82(24)​
​ ___
c24
1   ​
5 ​ ____
a215​
5 ​c12
​​
1
5 ​ __7  ​
a​
x4y23
53. _____
​  2 ​

xy
3 2
54. _____
​ m5n6
​
mn​
xy
_____
​   ​​ 5 ​

x421
​ ​y2322​
​
3 2
_____
​ ​m5n6 ​ 5
​m325
​ ​​n​226​
5 x3​y25
​ ​
5 ​m22
​ ​​n​24​
x3 ​
5 ​ __
y5
1   ​
5 ​ _____
m2​n​4​
4 23
xy
mn
2
890 • Chapter 13 Skills Practice
Lesson 13.3 Skills Practice
page 7
Name_________________________________________________________ Date__________________________
(x2y25) ? (xy)
55. ___________
​
​​

x4y3
18a2b2
56. ​ __________
​​
6ab3 ? ab24
x211 y2511
(x2y25) ? (xy) _________
​
​ 5 ​

​ ___________
x4y3
x4y3​
18a2b2
18a2b2
_________
​ 5 ​
​
​ __________

3
24
6ab ? ​ab​ ​ 6a111 b324​
x3y24
5 ​ _____
​​
x4y3
3a2b2 ​​
5 ​ _____
a2b21
5 x324​y2423
​ ​
5 3a​222​​b​22(21)​
5 ​x21
​ ​​y​27​
5 ​3a​0​​b​3​
1  ​​
5 ​ ___
xy7
5 ​3b​3​
4 6
n

​
57. __________
​  36m
2 3
4m n  mn8​
n
36m n
___________

​ 5 ​ _________
​​
​  36m
2 3
8
211 318
5c215d311 ​​
5c2d3 ? c5d ​ 5 ​
_________
​ __________
4
7
40c ? cd
40c411d7
9m4n6 ​
5 ​ ______
m3n11​
cd
5 ​ _____
​​
8c5d7
5 9m423​n​6211​
1 ​ c725d427
5 ​ __
8
5 ​9m​1​ ​n25​
​
1 ​ c
5 ​ __
​2​​d​23​
8
9m ​
5 ​ ____
n5​
c   ​
5 ​ ____
8d3
4 6
4m n ? mn
2 3
5
​
58. __________
​ 5c d4 ? c d
40c ? cd7
4 6
4m
n
7
4
2
Chapter 13 Skills Practice • 891
892 • Chapter 13 Skills Practice
Lesson 13.4 Skills Practice
Name_________________________________________________________ Date__________________________
Let’s Get Scientific!
Scientific Notation
Vocabulary
Match each definition to its corresponding term.
1. a way to express a very large or very a. scientific notation
small number as the product of a number
between 1 and 10 and a power of 10
a. scientific notation
2. the decimal part of a number written in scientific notation; the number a in a 3 10
b. order of magnitude
n
c. mantissa
3. an estimate of size expressed as a c. mantissa
power of 10
b. order of magnitude
4. the exponent of 10 for a number written d. characteristic
in scientific notation; the number
n in a 3 10n
d. characteristic
Chapter 13 Skills Practice • 893
Lesson 13.4 Skills Practice
page 2
Problem Set
Write each number in scientific notation. Show your work.
2. 3700
1. 45,000,000
45,000,000 5 (4.5)(10,000,000)
3700 5 (3.7)(1000)
5 (4.5)(107)
5 (3.7)(103)
The number written in scientific notation is
The number written in scientific notation is
4.5 3 10 .
3.7 3 103.
7
4. 92,400,000,000
3. 562,000
562,000 5 (5.62)(100,000)
92,400,000,000 5 (9.24)(10,000,000,000)
5 (5.62)(105)
5 (9.24)(1010)
The number written in scientific notation The number written in scientific notation is 5.62 3 10 .
is 9.24 3 1010.
5
5. There are over 29,000 grains of long-grain rice in a one-pound bag.
29,000 5 (2.9)(10,000)
5 (2.9)(104)
The number written in scientific notation is 2.9 3 104.
894 • Chapter 13 Skills Practice
Lesson 13.4 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
6. There are about 100,000,000,000 stars in our galaxy.
100,000,000,000 5 (1) (100,000,000,000)
5 (1)(1011)
The number written in scientific notation is 1 3 1011.
7. The distance from the Earth to the Moon is about 385,000,000 meters.
385,000,000 5 (3.85)(100,000,000)
5 (3.85)(108)
The number written in scientific notation is 3.85 3 108.
8. The average distance between the Earth and the Sun is about 93,000,000 miles.
93,000,000 5 (9.3)(10,000,000)
5 (9.3)(107)
The number written in scientific notation is 9.3 3 107.
Write each number in standard notation. Show your work.
9. 5.2 3 107
5.2 3 107 5 (5.2)(107)
5 (5.2)(10,000,000)
The number written in standard notation is 52,000,000.
Chapter 13 Skills Practice • 895
Lesson 13.4 Skills Practice
page 4
10. 3.7 3 108
3.7 3 108 5 (3.7)(108)
5 (3.7)(100,000,000)
The number written in standard notation is 370,000,000.
11. 2.871 3 1011
2.871 3 1011 5 (2.871)(1011)
5 (2.871)(100,000,000,000)
The number written in standard notation is 287,100,000,000.
12. 4.26 3 109
4.26 3 109 5 (4.26)(109)
5 (4.26)(1,000,000,000)
13. There are about 1 3 ​10​5​strands of hair on the human head.
1 3 105 5 (1)(105)
5 (1)(100,000)
The number written in standard notation is 100,000.
896 • Chapter 13 Skills Practice
The number written in standard notation is 4,260,000,000.
Lesson 13.4 Skills Practice
page 5
Name_________________________________________________________ Date__________________________
14. The population of Texas is about 2.4 3 107 people.
2.4 3 107 5 (2.4)(107)
5 (2.4)(10,000,000)
The number written in standard notation is 24,000,000.
15. The population of California is about 3.7 3 ​10​7​people.
3.7 3 ​10​7​ 5 ​(3.7)(10​7​)
5 (3.7)(10,000,000)
The number written in standard notation is 37,000,000.
16. The circumference of the Earth at the equator is about 4.008 3 ​10​7​meters.
4.008 3 ​10​7​ 5 ​(4.008)(10​7​)
5 (4.008)(10,000,000)
The number written in standard notation is 40,080,000.
Chapter 13 Skills Practice • 897
Lesson 13.4 Skills Practice
page 6
Write each number in scientific notation. Show your work.
18. 0.000831
17. 0.000067
0.000067 5 (6.7)(0.00001)
0.000831 5 (8.31)(0.0001)
1   ​
0.00001 5 ​ ________
100,000
1   ​
0.0001 5 ​ _______
10,000
1  ​
5 ​ ___
105
1  ​
5 ​ ___
104
5 ​10​25​
5 ​10​24​
The number written in scientific notation is
The number written in scientific notation is
6.7 3 ​10​ ​.
8.31 3 ​10​24​.
25
20. 0.00000092
0.00000000253 5 (2.53)(0.000000001)
0.00000092 5 (9.2)(0.0000001)
1
​
0.000000001 5 ​ _____________
1,000,000,000
1
​
0.0000001 5 ​ ___________
10,000,000
5 ___
​  1  ​
109
5 ___
​  1  ​
107
​5 10​29​
​5 10​27​
The number written in scientific notation is
The number written in scientific notation is
2.53 3 ​10​ ​.
9.2 3 1027.
29
898 • Chapter 13 Skills Practice
19. 0.00000000253
Lesson 13.4 Skills Practice
page 7
Name_________________________________________________________ Date__________________________
21. A grain of rice weighs about 0.025 grams.
0.025 5 (2.5)(0.01)
1  ​
0.01 5 ​ ____
100
1  ​
5 ​ ___
102
5 ​10​22​
The number written in scientific notation is 2.5 3 ​10​22​.
22. The diameter of a red blood cell is about 0.00004 inches.
0.00004 5 (4)(0.00001)
1   ​
0.00001 5 ​ ________
100,000
1  ​
5 ​ ___
105
5 ​10​25​
The number written in scientific notation is 4 3 ​10​25​.
Chapter 13 Skills Practice • 899
Lesson 13.4 Skills Practice
page 8
23. A grain of salt weighs about 0.0000585 grams.
0.0000585 5 (5.85)(0.00001)
0.00001 5 ________
​  1   ​
100,000
5 ___
​  1  ​
105
​5 10​25​
The number written in scientific notation is 5.85 3 ​10​25​.
24. A credit card is about 0.0299 inches thick.
0.0299 5 (2.99)(0.01)
​  1  ​
0.01 5 ____
100
​  1  ​
5 ___
102
5 ​10​22​
The number written in scientific notation is 2.99 3 ​10​22​.
900 • Chapter 13 Skills Practice
Lesson 13.4 Skills Practice
page 9
Name_________________________________________________________ Date__________________________
Write each number in standard notation. Show your work.
25. 8.3 3 ​10​25​
26. 6.22 3 ​10​29​
8.3 3 ​10​25​ 5 ​(8.3)(10​25​)
6.22 3 ​10​29​ 5 ​(6.22)(10​29​)
(  )
5 (8.3)​___
​  1  ​  ​
105
5 (6.22)​___
​  1  ​  ​
109
5 (8.3)​________
​  1   ​
​
100,000
)
1    ​  ​
5 (6.22)​ ​ _____________
1,000,000,000​
5 (8.3)(0.00001)
5 (6.22)(0.000000001)
(
(
)
The number written in standard notation is
The number written in standard notation is
0.000083.
0.00000000622.
27. 4.7 3 ​10​23​
28. 3.85 3 ​10​212​
4.7 3 ​10​23​ 5 ​(4.7)(10​23​)
(  )
(  )
5 (4.7)​( ​  1   ​ )​
1000
5 (4.7)​___
​  1  ​  ​
103
_____
5 (4.7)(0.001)
3.85 3 ​10​212​ 5 ​(3.85)(10​212​)
(  )
1
5 (3.85)​​( ​
​​  ​
1,000,000,000,000 )
5 (3.85)​____
​  1  ​
​
1012
_________________

5 (3.85)(0.000000000001)
The number written in standard notation is The number written in standard notation is
0.0047.
0.00000000000385.
Chapter 13 Skills Practice • 901
Lesson 13.4 Skills Practice
page 10
29. An oxygen atom has a radius of about 4.8 3 ​10​211​meters.
4.8 3 ​10​11​ 5 ​(4.8)(10​211​)
2
(  )
1
5 (4.8)​​( ​
​  ​​
100,000,000,000 )
5 (4.8)​____
​  1  ​
​
1011
________________

5 (4.8)(0.00000000001)
The number written in standard notation is 0.000000000048.
30. The diameter of a white blood cell is about 1 3 ​10​25​meters.
1 3 ​10​25​ 5 ​(1)(10​25​)
(  )
5 (1)​___
​  1  ​  ​
105
5 (1)​________
​  1   ​
​
100,000
5 (1)(0.00001)
(
)
The number written in standard notation is 0.00001.
902 • Chapter 13 Skills Practice
Lesson 13.4 Skills Practice
page 11
Name_________________________________________________________ Date__________________________
31. A nitrogen atom has a radius of about 5.6 3 ​10​211​meters.
5.6 3 ​10​211​ 5 ​(5.6)(10​211​)
(  )
1
5 (5.6)​​( ​
​  ​​
100,000,000,000 )
5 (5.6)​____
​  1  ​
​
1011
________________

5 (5.6)(0.00000000001)
The number written in standard notation is 0.000000000056.
32. The width of a human hair is about 1.8 3 ​10​23​centimeters.
1.8 3 ​10​23​ 5 ​(1.8)(10​23​)
(  )
5 (1.8)​(_____
​  1   ​  ​
1000 )
5 (1.8)​___
​  1  ​  ​
103
5 (1.8)(0.001)
The number written in standard notation is 0.0018.
Chapter 13 Skills Practice • 903
904 • Chapter 13 Skills Practice
Lesson 13.5 Skills Practice
Name_________________________________________________________ Date__________________________
Are We There Yet? What Is the Distance?
Operations with Scientific Notation
Problem Set
Calculate each product. Express the product in scientific notation.
2. (3 3 ​10​7​)(2 3 ​10​5​)
1. (4 3 ​10​3​)(2 3 ​10​6​)
(4 3 ​10​3​)(2 3 ​10​6​) 5 ​​(4)(2)(10​3​)(10​6​)
(3 3 ​10​7​)(2 3 ​10​5​) 5 ​​(3)(2)(10​7​)(10​5​)
5 ​(8)(10​316
​ )
5 ​(6)(10​715
​ )
5 8 3 ​10​9​
5 6 3 ​10​12​
3. (8 3 ​10​6​)(1 3 ​10​23​)
4. (2 3 ​10​23​)(3 3 ​10​8​)
(8 3 106)(1 3 ​10​23​) 5 ​​(8)(1)(10​6​)(10​23​)
(2 3 ​10​23​)(3 3 ​10​8​) 5 ​​(2)(3)(10​23​)(10​8​)
5 ​(8)(10​6​1 (23))
5 ​(6)(10​2318
​ )
5 8 3 ​10​3​
5 6 3 ​10​5​
6. (7.9 3 ​10​3​)(2.6 3 ​10​5​)
5. (3.7 3 ​10​4​)(5.2 3 ​10​9​)
(3.7 3 ​10​4​)(5.2 3 ​10​9​) 5 ​​(3.7)(5.2)(10​4​)(10​9​)
(7.9 3 ​10​3​)(2.6 3 ​10​5​) 5 ​​(7.9)(2.6)(10​3​)(10​5​)
5 ​(19.24)(10​419
​ )
5 ​(20.54)(10​315
​ )
5 19.24 3 ​10​13​
5 20.54 3 ​10​8​ 5 1.924 3 ​10​14​
5 2.054 3 ​10​9​
Chapter 13 Skills Practice • 905
Lesson 13.5 Skills Practice
page 2
8. (4.2 3 ​10​24​)(9.1 3 ​10​26​)
7. (2.8 3 ​10​28​)(6.31 3 ​10​4​)
(2.8 3 ​10​28​)(6.31 3 ​10​4​) 5 ​​(2.8)(6.31)(10​28​)(10​4​)
(4.2 3 ​10​24​)(9.1 3 ​10​26​) 5 ​​(4.2)(9.1)(10​24​)(10​26​)
5 ​(17.668)(10​2814
​ )
5 ​(38.22)(10​241(26)​)
5 17.668 3 ​10​24​ 5 38.22 3 ​10​210​
5 1.7668 3 ​10​23​
5 3.822 3 ​10​29​
9. Seismosaurus is the longest known dinosaur. It measured 1800 inches. How far would
3000 Seismosaurus dinosaurs span if they were placed head to tail? Use scientific notation
to calculate the length.
1800 5 1.8 3 ​10​3​
3000 5 3 3 ​10​3​
(1.8 3 ​10​3​)(3 3 ​10​3​) 5 ​​(1.8)(3)(10​3​)(10​3​)
5 ​(5.4)(10​313
​ )
5 5.4 3 ​10​6​
The 3000 Seismosaurus dinosaurs would span 5.4 3 ​10​6​ inches.
906 • Chapter 13 Skills Practice
Lesson 13.5 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
10. According to many scientists, Argentinosaurus is the heaviest known dinosaur. It weighed
220,000 pounds. How much would 1500 Argentinosaurus dinosaurs weigh? Use scientific
notation to calculate the weight.
220,000 5 2.2 3 ​10​5​
1500 5 1.5 3 ​10​3​
(2.2 3 ​10​5​)(1.5 3 ​10​3​) 5 ​​(2.2)(1.5)(10​5​)(10​3​)
5 ​(3.3)(10​513
​ )
5 3.3 3 ​10​8​
The 1500 Argentinosaurus dinosaurs would weigh 3.3 3 ​10​8​ pounds.
11. The bee hummingbird is the world’s smallest bird. It is about 0.05842 meters long. How far would
2,000,000 bee hummingbirds span if they were placed head to tail? Use scientific notation to
calculate the length.
0.05842 5 5.842 3 ​10​22​
2,000,000 5 2 3 ​10​6​
(5.842 3 ​10​22​)(2 3 ​10​6​) 5 ​​(5.842)(2)(10​22​)(10​6​)
5 ​(11.684)(10​2216​)
5 11.684 3 ​10​4​ 5 1.1684 3 ​10​5​
The 2,000,000 bee hummingbirds would span 1.1684 3 ​10​5​ meters.
Chapter 13 Skills Practice • 907
Lesson 13.5 Skills Practice
page 4
12. The pygmy shrew is the world’s smallest mammal by weight. It weighs just 0.0013 kilograms.
How much would 3,000,000 pygmy shrews weigh? Use scientific notation to calculate the weight.
0.0013 5 1.3 3 ​10​23​
3,000,000 5 3 3 ​10​6​
(1.3 3 ​10​23​)(3 3 ​10​6​) 5 ​​(1.3)(3)(10​23​)(10​6​)
5 ​(3.9)(10​2316
​ )
5 3.9 3 ​10​3​
The 3,000,000 pygmy shrews would weigh 3.9 3 ​10​3​ kilograms.
Simplify each expression. Express the quotient in scientific notation.
(  )(  )
5
(8 3 ​10​5​)
_________
​ 5 ​

__
​ 8 ​  ​​___
​ 10  ​  ​
​
2 102
(2 3 ​10​2​)
(12 3 108)
14. __________
​

​
(6 3 103)
)
( 6 )(10
(12 3 10 ) ___
__________

​ 12 ​   ​​___
​ 103 ​  ​
​
3  ​ 5 ​
8
(6 3 10 )
8
5 ​(4)(10​522​)
5 ​(2)(10​823​)
5 ​(4)(10​3​)
5 ​(2)(10​5​)
5 4 3 ​10​3​
5 2 3 ​10​5​
908 • Chapter 13 Skills Practice
(8 3 105)
13. ________
​

​
(2 3 102)
Lesson 13.5 Skills Practice
page 5
Name_________________________________________________________ Date__________________________
(18 3 1022)
16. ___________
​

​
(3 3 1026)
(15 3 1027)
15. ___________
​

​
(5 3 104)
( 5 )( 10 )
27
(15 3 10 ) ___
___________

​ 15 ​   ​​____
​ 10 4 ​

​
​
4  ​ 5 ​
27
(5 3 10 )
22
(3 3 10 )
5 ​(3)(10​2724​)
5 ​(6)(10​222(26)​)
5 ​(3)(10​211​)
5 ​(6)(10​4​)
5 3 3 ​10​211​
5 6 3 ​10​4​
(1.508 3 107)
17. ____________
​

​
(2.6 3 103)
(5.92 3 106)
18. ___________
​

​
(3.7 3 104)
)
(  2.6 )(10
(  )(  )
7
(1.508 3 10 )
1.508
____________
______
​

​​___
​ 103 ​  ​
​
3  ​ 5 ​​
7
(2.6 3 10 )
)
( 3 )(10
22
(18 3 10 ) ___
___________
​ 5 ​

​ 18 ​   ​​____
​ 1026 ​
​
​
26
(5.92 3 106)
5.92 106

​   ​  ​​ ___
​   ​  ​
​ ___________
​ 5 ​ _____
4
3.7 104
(3.7 3 10 )
5 ​(0.58)(10​723​)
5 ​(1.6)(10​624​)
5 ​(0.58)(10​4​)
5 ​(1.6)(10​2​)
5 5.8 3 ​10​3​
5 1.6 3 ​10​2​
(8.82 3 1027)
20. ____________
​

​
(2.52 3 1022)
(3.68 3 1028)

19. ____________
​
​
(3.2 3 1024)
)
( 3.2 )(10
28
(3.68 3 10 ) _____
3.68 ​
____________

​​____
​ 1024 ​
​
​
24  ​ 5 ​​
28
(3.2 3 10 )
)(10
)
(2.52
27
(8.82 3 10 ) _____
8.82 ​
____________

​​____
​ 1022 ​
​
​
22  ​ 5 ​​
27
(2.52 3 10 )
5 ​(1.15)(10​282(24)​)
5 ​(3.5)(10​272(22)​)
5 ​(1.15)(10​24​)
5 ​(3.5)(10​25​)
5 1.15 3 ​10​24​
5 3.5 3 ​10​25​
Chapter 13 Skills Practice • 909
Lesson 13.5 Skills Practice
page 6
21. (5.9 3 ​10​5​)(4.8 3 ​10​3​)
22. (7.83 3 ​10​3​)(2.9 3 ​10​8​)
(5.9 3 ​10​5​)(4.8 3 ​10​3​) 5 ​​(5.9)(4.8)(10​5​)(10​3​)
(7.83 3 ​10​3​)(2.9 3 ​10​8​) 5 ​​(7.83)(2.9)(10​3​)(10​8​)
5 ​(28.32)(10​513​)
5 ​(22.707)(10​318​)
5 28.32 3 ​10​8​
5 22.707 3 ​10​11​
5 2.8 3 ​10​9​
5 2.3 3 ​10​12​
24. (5.31 3 ​10​5​)(6 3 ​10​28​)
(2.90 3 ​10​27​)(4.860 3 ​10​22​)
(5.31 3 ​10​5​)(6 3 ​10​28​) 5 ​​(5.31)(6)(10​5​)(10​28​)
5 ​​(2.90)(4.860)(10​27​)(10​22​)
5 ​(31.86)(10​51(28)​)
5 ​(14.094)(10​271(22)​)
5 31.86 3 ​10​23​
5 14.094 3 ​10​29​
5 3 3 ​10​22​
5 1.41 3 ​10​28​
(4.5 3 ​10​5​)

​
25. __________
​
(3.8 3 ​10​2​)
)(10
)
(3.8
5
(4.5 3 ​10​ ​) ___
__________
​ 5 ​

​ 4.5 ​  ​​___
​ 102 ​  ​
​
2
5
(3.8 3 ​10​ ​)
(5.12 3 ​10​6​)
26. ___________
​

​
(8 3 ​10​3​)
(  8 )(10  )
6
(5.12 3 ​10​ ​) _____
5.12
___________
​ 5 ​

​   ​
​​___
​ 10  ​  ​
​
6
(8 3 ​10​ ​)
3
3
5 ​(1.184210 . . . )(10​522​)
5 ​(0.64)(10​623​)
5 ​(1.184210 . . . )(10​3​)
5 0.64 3 ​10​3​ 5 1.2 3 ​10​3​
5 6 3 ​10​2​
910 • Chapter 13 Skills Practice
23. (2.90 3 ​10​27​)(4.860 3 ​10​22​)
Lesson 13.5 Skills Practice
page 7
Name_________________________________________________________ Date__________________________
27. A football is about 280 millimeters long. A football field is about 100,000 millimeters long. How
many footballs placed end to end would be needed to span the field? Use scientific notation to
calculate the number of footballs.
280 5 2.8 3 ​10​2​
100,000 5 1 3 ​10​5​
)(10
)
(2.8
5 ​( ___
​  1  ​  ​​( 10
2.8 )
5
0​5​)
___________
​  (1 3 ​1
​  1  ​  ​​___
​ 102 ​  ​
​ 5 ​___
(2.8 3 ​10​2​)
522 )
​
5 ​(0.357142 . . . )(10​3​)
5 4 3 ​10​2​ 5 400
About 400 footballs would span the football field.
Chapter 13 Skills Practice • 911
Lesson 13.5 Skills Practice
page 8
28. A large Siberian tiger can weigh up to 305,000 grams. A small Sand cat can weigh as little as
1500 grams. How many times greater is the weight of a Siberian tiger than the weight of a
Sand cat? Use scientific notation to calculate the difference in weight.
305,000 5 3.05 3 ​10​5​
1500 5 1.5 3 ​10​3​
)
( 1.5 )(10
5
(3.05 3 ​10​ ​) _____
___________
​ 5 ​​ 3.05 ​
​

​​___
​ 103 ​  ​
3
5
(1.5 3 ​10​ ​)
__
5 (2.0​​3 ​)(10​
523​)
5 (2.0​​3 ​)(10​
2​)
5 2.0 3 ​10​2​ 5 200
__
The weight of a Siberian tiger is about 200 times greater than the weight of a Sand cat.
912 • Chapter 13 Skills Practice
Lesson 13.6 Skills Practice
Name_________________________________________________________ Date__________________________
Identifying the Properties of Powers
Problem Set
Justify each step to simplify the expression. Choose the properties from the box.
Product Rule
Power to a
Negative
Quotient Rule
of Powers
Power Rule
Exponent Rule
of Powers
Zero Power Simplify Powers
Identity
Commutative
Rule
Property of
Property of
Multiplication
Multiplication
1. 4​x5​ ​ 6​x2​ ​​y​6​ xy
4​x5​ ​  6​x2​ ​​y6​ ​  xy 5 (4  6)(​x5​ ​​x2​ ​x)( y6​ ​y) Commutative Property of Multiplication
5 (24)(​x51211
​
​)( y611
​ ​) Product Rule of Powers
5 24​x​8​​y​7​ Simplify Powers
2. 3​a2​ ​​b​3​ 7a​b5​ ​ ​b2​ ​
3​a2​ ​​b3​ ​  7a​b5​ ​  ​b2​ ​ 5 (3  7)(​a2​ ​a)(​b3​ ​​b5​ ​​b2​ ​) Commutative Property of Multiplication
5 (21)(​a211
​ ​)(​b31512
​
​)
Product Rule of Powers
5 21​a​3​​b​10​
Simplify Powers
Chapter 13 Skills Practice • 913
Lesson 13.6 Skills Practice
page 2
3. ​(4​m2​ ​​n​5​)3​ ​
​(4​m2​ ​​n5​ ​)3​ ​ 5 ​43​ ​​(m)​23​​(n)​53​ Power to a Power Rule
5 64​m​6​​n​15​
Simplify Powers
4. ​(23​x7​ ​​y​3​)5​ ​
​(23​x7​ ​​y3​ ​)5​ ​ 5 ​(23)​5​​(x)​75​​( y)​35​ Power to a Power Rule
5 2243​x​35​​y​15​ Simplify Powers
27​y8​ ​​z​5​
5. _______
​  4 2 ​

23​y​ ​​z​ ​
(  )(  ) (  )
​​5 29( y824
​ ​)(z522
​ ​)
Quotient Rule of Powers
​5 29y​4​​z​3​
Simplify Powers
296​m9​ ​​n​2​
6. _________
​

​
8​m2​ ​​n​6​
(  )(  ) (  )
9
2
2
296m9n ​ 5 ​
​ _________

_____
​ 296
​​ ___
​ m2 ​  ​​ __
​ n6 ​  ​ Commutative Property of Multiplication
​
2 6
8
m n
8m n
5 ​​212(m​922​)(n​226​)
Quotient Rule of Powers
5 ​212m7​ ​​n24
​ ​
Simplify Powers
m​
​
5 _______
​ 212​
​
​ ​4​
n
Negative Exponent Rule
7
914 • Chapter 13 Skills Practice
y8 z5
27​y8​ ​​z​5​ 27 __
_______

​    ​  ​ ​  4 ​ ​​___
​   ​​​  ​ Commutative Property of Multiplication
​  4 2 ​
5 ​___
23​y​ ​​z​ ​ 23 y z2
Lesson 13.6 Skills Practice
page 3
Name_________________________________________________________ Date__________________________
7. ​22x​5​​y​3​ ​8x​2​​y​25​ ​x29
​ ​​y​2​
​22x​5​​y​3​  ​8x​2​​y​25​  ​x​29​​y​2​ 5 (22  ​8)(x​5​​x​2​​​x​29​)( y​3​​y​25​​y​2​) Commutative Property of Multiplication
5 ​​(216)(x5121(29)
​
​)( y31(25)12
​
​) Product Rule of Powers
5 ​216x22
​ ​​y0​ ​ Simplify Powers
5 ​216x​22​(1) Zero Power Rule
216
5 ​ _____

​ Negative
Exponent Rule
x​ 2​ ​
​m
​ ​​n​
​

8. _____________
​ ​42m​ ​​n​6 ​5 ​
6​m​ ​​n​ ​
5 3
4 2
Commutative Property of Multiplication
(  6 )(  m ) (  n )
5 (7)​( ​ m  ​
​​ ​ n  ​  ​ Product Rule of Powers
m )( n )
5 3
4 2
5
4
3 2
n  ___
_____________

​ 5 ​
​ 42 ​  ​​ _____
​ m m
​
​​ ____
​ n n ​
​ ​ 42m n 6 m
5
6
6m n
5
514
312
_____
____

5
6
5 ​​(7)(m51426
​
​)(n31225
​
​)
Quotient Rule of Powers
5 ​7m3​ ​​n0​ ​
Simplify Powers
5 ​7m3​ ​(1)
Zero Power Rule
5 ​7m3​ ​ Identity Property of Multiplication
Chapter 13 Skills Practice • 915
Lesson 13.6 Skills Practice
page 4
Choose which solution is correct.
c.
b.
9. a.
​3x​ ​ ​4x​ ​5 (3  ​4)(x​ ​​x​ ​)
2
3
2 3
​3x​ ​ ​4x​ ​5 (3  ​4)(x​ ​​x​ ​)
2
3
2 3
​3x​2​ ​4x​3​5 (3  ​4)(x​2​​x​3​)
5 ​(12)(x​213​)
5 ​(12)(x​223​)
5 (12)(​x23
​ ​)
5 ​(12)(x​5​)
5 ​(12)(x​21​)
5 ​(12)(x​6​)
5 ​12x​5​
12 ​ ​
5 ​ ​ ___
x
5 ​12x​6​
The correct solution is a.
10. a.
b.
​25a​ ​ ​2a​
6
3
c.
​25a​ ​ ​2a​ ​
6
​25a​6​ ​2a​3​
3
5 (25  ​2)(a​6​​a3​ ​)
5 (25  ​2)(a​6​​a3​ ​)
5 (25  ​2)(a​6​​a3​ ​)
5 ​(210)(a​623​)
5 ​(210)(a​63​)
5 ​(210)(a​613​)
5 ​(210)(a​3​)
5 ​(210)(a​18​)
5 ​(210)(a​9​)
5 ​210a​3​
5 ​210a​18​
5 ​210a​9​
The correct solution is c.
( 4 )( ​m​​)
c.
( 4 )( ​m​​)
36 ​m​ ​
36​m​ ​ ___
______
​
5 ​​   ​   ​​____

​   ​
​
​
36​m​ ​ ___
36 ​m​ ​
______
​
​
5 ​​   ​   ​​____

​   ​
​
36 ​m​ ​
36​m​ ​ ___
______
​
5 ​​   ​   ​​____

​   ​
​
​
5 ​(9)(m​1244​)
5 ​(9)(m​1224​)
5 ​(9)(m​1214​)
5 ​(9)(m​3​)
5 ​(9)(m​8​)
5 ​(9)(m​16​)
5 ​9m​3​
5 ​9m​8​
5 ​9m​16​
12
4​m​ ​
4
( 4 )( ​m​​)
b.
12
4
12
4​m​ ​
4
The correct solution is b.
916 • Chapter 13 Skills Practice
12
4
12
4​m​ ​
4
12
4
11. a.
Lesson 13.6 Skills Practice
page 5
Name_________________________________________________________ Date__________________________
12. a.
b.
( 5 )( ​x​​)
60​x2​ ​ ___
60 ​x​2​
____
5 ​​   ​   ​​__
​   ​

​   ​  ​
5​x8​ ​
8
c.
60​x​ ​ ___
60 ​x​ ​
____
5 ​​   ​   ​​__
​   ​

​   ​  ​
x​ ​
60 ​x​ ​
​
​5 ​ ​   ​  ​​ ​   ​  ​
(​ ​ 60​
5​x​ ​ ) ( 5 )(​x​ ​)
( 5 )( ​x​​)
2
2
2
5​x8​ ​
____
8
8
2
___  __
8
5 ​(12)(x​822​)
5 ​(12)(x​218​)
5 ​(12)(x​228​)
5 ​(12)(x​6​)
5 ​(12)(x​10​)
5 ​(12)(x​26​)
5 ​12x​6​
5 ​12x​10​
12
5 ___
​  6 ​
​x​ ​
The correct solution is c.
13. a.
b.
​​(5x​ ​)​ ​5 ​(5)​ ​​​(x​ ​)​ ​
2 3
3
2 3
c.
​​(5x​ ​)​ ​5 ​(5)​ ​​​(x​ ​)​ ​
2 3
3
2 3
​​(5x​2​)3​ ​5 ​(5)​3​​​(x​2​)3​ ​
3
5 ​(125)(x​23​)
5 ​(125)(x​213​)
5 (125)(x​2​ ​)
5 ​(125)(x​6​)
5 ​(125)(x​5​)
5 ​(125)(x​8​)
5 ​125x​6​
5 ​125x​5​
5 ​125x​8​
The correct solution is a.
14. a.
​​(4y​ ​)​ ​5 ​(4)​ ​​​( y​ ​)​ ​
c.
b.
23 3
3
23 3
​​(4y​ ​)​ ​5 ​(4)​ ​​​( y​ ​)​ ​
23 3
3
23 3
​​(4y​23​)3​ ​5 ​(4)​3​​​( y​23​)3​ ​
5 ​(64)( y​2313​)
5 ​(64)( y​233​)
5 ​(64)( y​2323​)
5 ​(64)( y​0​)
5 ​(64)( y​29​)
5 ​(64)( y​26​)
5 (64)(1)
​
5 ___
​ 64
y9
64 ​ ​
5 ​ ​ ___
y​6​
5 64
The correct solution is b.
Chapter 13 Skills Practice • 917
Lesson 13.6 Skills Practice
page 6
Simplify each expression. Show your work.
15. ​12x​4​​y​3​ ​3x​2​​y​25​
​ ​ ​2a​7​​b22
​ ​
16. ​17a​25​​b23
​12x​4​​y​3​  ​3x​2​​y​25​ 5 (12  ​3)(x​4​​​x​2​)( y​3​​y​25​)
​17a​25​​b​23​  ​2a​7​​b​22​ 5 (17  ​2)(a​25​​​a​7​)(b​23​​b​22​)
5 ​​(36)(x​412​)( y​31(25)​)
5 ​​(34)(a​2517​)(b​231(22)​)
5 ​​(36)(x​6​)( y​22​)
5 ​​(34)(a​2​)(b​25​)
​
36​x ​​6
5 ​ _____
​y2​ ​
​
34​a ​​2
5 ​ _____
​b5​ ​
18. ​(2a​5​​b​2​​​c​3​)5​ ​
17. ​(23x​4​​​y​23​)2​ ​
​(23x​4​​​y​23​)​2​ 5 ​(23)​2​​​(x​4​)​2​​​( y​23​)​2​
​(2a​5​​b​2​​​c​3​)​5​ 5 ​(2)​5​​​(a​5​)​5​​​(b​2​)​5​​​(c​3​)​5​
5 ​(9)(x)​42​​( y)​232​
5 ​(32)(a)​55​ ​(b)​25​ ​(c)​35​
5 ​9x​8​​y​26​
5 ​32a​25​​b​10​​c​15​
​ 9​x ​​ ​
5 ___
​y6​ ​
8
)(​m  ​​)(​n  ​​)
(14
3 5
3
5
​
_______
​ 28 ​  ​​___
​ ​m7​ ​  ​​​__
​ ​n2​ ​  ​​
​ 28​m7​ ​​n​
​
5
​___
2
14​m​ ​​n​ ​
35​x​8​​y​6​

20. ______
​  5 3 ​
7​x​ ​​y​ ​
( 7 )(​x  ​​)(​y  ​​)
8 y
35​x​ ​ ​y​ ​ ___
_______

​ 35 ​   ​​__
​ ​x​ ​  ​​​__
​   ​  ​
​
​ 5 ​
8
6
7​x5​ ​ ​y3​ ​
6
5
3
5 ​​(2)(m​327​)(n​522​)
​​5 (5)(x​825​)( y​623​)
5 ​​(2)(m​24​)(n​3​)
​5 5x​3​​y​3​
2​n3​ ​
​
5 ​ ____
​m4​ ​
918 • Chapter 13 Skills Practice
28​m3​ ​​n​5​
19. _______
​  7 2 ​

14​m​ ​​n​ ​
Lesson 13.6 Skills Practice
page 7
Name_________________________________________________________ Date__________________________
a​ ​​b​
​
22. ____________
​ 3​a​ ​​b​7​ 4​

​
6​a​ ​​b​9​
4
21. ​(2x​5​​​y​3​)2​ ​ ​x2​ ​​y​4​
​(2x​5​​​y​3​)​2​  ​x2​ ​​y​4​ 5 ​(2)​2​ ​​(x​5​)​2​​​( y​3​)​2​  ​x2​ ​​y​4​
5
3
(  6 )(  ​a​​ )(  ​b​​ )
5 ​(___
​ 126 ​  ) ​​(____
​ ​a​a​​ ​ ​ ) ​ ​​(____
​ ​b​b​​ ​ ​ ) ​ ​
4 5
4
5
a3​ ​​b​2
​ _____
a3​ ​
b2​
​
4
____________
​ 3​a​ ​​b​ ​  4​

​ 3  ​
​​______ ​ ​a​ ​  ​ ​

​​______
​ ​b​ ​  ​ ​  ​
​ 5 ​
6​a7​ ​​b​9​
7
5 ​(4)(x)​ ​​( y)​ ​  ​x​ ​​y​ ​
52
32
2 4
5 ​​(4)(x​ ​)( y​ ​)  ​x​ ​​y​ ​
10
6
2 4
413
512
7
9
(  )(  )
7
7
5 2​__
​ ​a7​ ​  ​​​__
​ ​b9​ ​  ​​
​a​ ​ ​b​ ​
5 ​​2(a​727​)(b​729​)
5 ​​2(a​0​)(b​22​)
5 ​2(1)(b​22​)
2  ​
5 ​ __
b2
5 ​4x​12​​y​10​
​(4​m3​ ​​n​2​)4​ ​
23. ________
​

​
8​m5​ ​
5​x2​ ​​y​4​ 6​x4​ ​y
24. __________
​

​
10​x3​ ​​y​3​
(4)4(m3)4(n2)4
(4m3n2)4 ___________
​ 5 ​

​

​​ ________
5
8m
8m5
9
5 ​​4(x​1012​)( y​614​)
2
(256)(m)3 ? 4(n2 ? 4)
5 ______________
​
​

8m5
(256)m12n8
5 __________
​
​

8m5
(  )(  )
12
​ ​(​n8​ ​)
5 ​____
​ 256
​​____
​ ​m​  ​
​
8 ​m5​ ​
​​5 (32)(m​1225​)(n​8​)
( 10 )(  x )(  y
10x y
y
5 ​( ​ 30 ​ ) ​​( ​ x  ​ )
​​ ​   ​
​
10 x (y )
y
5 3​( ​ x  ​ ) ​​( ​   ​ ) ​
x y
)
y y
5x y  6x y ____
56 ​​______ x
___________
​
​ 5 ​
​   ​
​ x 3 ​

​​_____
​  3 ​

​
3 3
2 4
4
2
214
4
4
411
___ _________
3
3
6
5
3
3
__  __
5 ​​3(x​623​)( y​523​)
5 ​3x​3​​y​2​
5 ​32m​7​​n​8​
Chapter 13 Skills Practice • 919