Appendix A — Answer Key, Chapter 1: Powers of Ten and Scientific Notation

SOL 7.NS.1 · Covers textbook Chapter 1 and the companion workbook. Item numbers match the textbook and run continuously through the chapter; workbook items are the same problems, so this key serves both. Reasoning answers show an acceptable response, not the only wording.

Lesson 1.1 — Powers of 10 with Negative Exponents

Guided practice

1. $10^3 = 1000$; $10^2 = 100$; $10^1 = 10$; $10^0 = 1$
2. $10^{-1} = 0.1$; $10^{-2} = 0.01$
3. Divide by $10$.
4. $0.0001$
5. $10^{-3}$

Independent practice

6. a) $0.1$ b) $0.001$ c) $0.00001$ d) $1$
7. a) $10^{-2}$ b) $10^{-4}$ c) $10^{-6}$
8. $10$
9. $10^{-4}$, $10^{-2}$, $10^{-1}$, $10^{0}$
10. Five zeros. $10^{-6} = 0.000001$, and the $1$ sits in the sixth place, so five zeros come before it.
11. $10^{-4} = 0.0001$ meters. It takes $10\,000$ sheets to reach $1$ meter. The exponent gives this directly: $10^{-4} \times 10^{4} = 10^{0} = 1$, so the number of sheets is $10^{4} = 10\,000$. Multiplying by $10$ four times undoes dividing by $10$ four times.
12. Jonah is treating the negative sign as if it belonged to the value instead of to the exponent. The exponent tells you which operation to repeat, not which side of zero the answer lands on. Following the pattern down, $10^{0} = 1$, $10^{-1} = 0.1$, $10^{-2} = 0.01$, $10^{-3} = 0.001$. Each step divides by $10$, and dividing a positive number by $10$ keeps it positive, so $10^{-3} = 0.001$, a positive number.

Exit ticket 1.1

13. $0.01$
14. $10^{-5}$
15. The value is divided by $10$.
16. Dropping the exponent from $1$ to $0$ means dividing by $10$ once, and $10 \div 10 = 1$. So $10^{0} = 1$ comes straight out of the same pattern that produces every other row, with no special rule needed.

Lesson 1.2 — Fraction Form and Decimal Form

Guided practice

17. $10^{-1} = \dfrac{1}{10} = 0.1$
18. $10^{-4} = \dfrac{1}{10\,000} = 0.0001$
19. $10^{-2}$
20. $\dfrac{1}{1000}$
21. $6$

Independent practice

22. Completed table:

23. a) $10^{-1}$ b) $10^{-4}$ c) $10^{-6}$
24. a) $\dfrac{1}{10}$ b) $\dfrac{1}{10\,000}$ c) $\dfrac{1}{100\,000}$
25. False. The exponent counts how many tens are multiplied in the denominator, so $10^{-2} = \dfrac{1}{10 \times 10} = \dfrac{1}{100} = 0.01$. By contrast $\dfrac{1}{20} = 0.05$, which is five times larger.
26. $10^{-2}$ is greater. Written as decimals, $\dfrac{1}{1000} = 0.001$ and $10^{-2} = 0.01$, and $0.01 > 0.001$. (Equivalently, $\dfrac{1}{1000} = 10^{-3}$, and $-3 < -2$.)
27. $\dfrac{1}{1000} = 10^{-3} = 0.001$ gram. A $250$ mg tablet has mass $250 \times 0.001 = 0.25$ gram.
28. The two counts measure the same thing from two directions. A denominator of $10^{n}$ means the number has been divided by $10$ a total of $n$ times, and each division by $10$ shifts a digit exactly one place to the right in the place-value chart. So $n$ zeros in the denominator and $n$ decimal places are two ways of recording the same $n$ steps. They cannot disagree, because each one is counting divisions by $10$.

Exit ticket 1.2

29. $\dfrac{1}{1000}$ and $0.001$
30. $10^{-5}$
31. $\dfrac{1}{100}$
32. $10^{-n}$ is $1$ divided by $10^{n}$: $10^{-n} = \dfrac{1}{10^{n}}$. The negative exponent produces the reciprocal of the positive one, so the two multiply to $1$.

Lesson 1.3 — Scientific Notation and Decimal Form

Guided practice

33. $2600$
34. $0.091$
35. $5.4 \times 10^{4}$
36. $7.2 \times 10^{-3}$
37. No. The coefficient $12.5$ is not less than $10$, and scientific notation requires $1 \le a < 10$. The value is $12.5 \times 100 = 1250$, so written correctly it is $1.25 \times 10^{3}$.

Independent practice

38. a) $320\,000$ b) $0.000705$ c) $0.19$ d) $8\,000\,000$
39. a) $6.3 \times 10^{6}$ b) $5.8 \times 10^{-4}$ c) $4.7 \times 10^{2}$ d) $9.1 \times 10^{-6}$
40. a) Not correct — the coefficient $0.9$ is less than $1$; correctly written, $9 \times 10^{3}$. b) Correct. c) Not correct — the coefficient $10$ is not less than $10$; correctly written, $1 \times 10^{6}$. d) Correct.
41. $0.00205$
42. $n = -2$
43. $7 \times 10^{-6} = 0.000007$ meters; $6.5 \times 10^{4} = 65\,000$ people. The red blood cell measurement required moving the decimal point farther — six places, against four places for the arena.
44. Multiplying by $10$ makes a number ten times larger and shifts every digit one place to the left in the place-value chart, which shows up as the decimal point moving one place right. A positive exponent means repeated multiplication by $10$, so the point moves right that many times. A negative exponent means repeated division by $10$, which shifts digits one place right each time, so the point moves left. The direction of the point simply records whether the number is growing or shrinking.

Exit ticket 1.3

45. $0.0043$
46. $9.1 \times 10^{4}$
47. $2.6 \times 10^{-4}$
48. Compare the number to $1$. If the number is $10$ or greater, the point moves left to form the coefficient and the exponent is positive. If the number is less than $1$, the point moves right to form the coefficient and the exponent is negative. Going the other direction, a positive exponent means the answer will be large and a negative exponent means the answer will be between $0$ and $1$.

Lesson 1.4 — Comparing and Ordering Numbers in Scientific Notation

Guided practice

49. $7.4 \times 10^{5} > 7.4 \times 10^{3}$
50. $2.8 \times 10^{-4} < 6.1 \times 10^{-4}$
51. $5 \times 10^{2} = 5.0 \times 10^{2}$
52. $8.3 \times 10^{-2}$, since $-2 > -5$. In decimal form, $0.083 > 0.000083$.
53. $4.2 \times 10^{-1}$, $4.2 \times 10^{1}$, $4.2 \times 10^{3}$

Independent practice

54. a) $3.6 \times 10^{4} > 9.1 \times 10^{3}$ b) $1.5 \times 10^{-2} < 1.5 \times 10^{-1}$ c) $6.7 \times 10^{-5} > 6.07 \times 10^{-5}$ d) $2.4 \times 10^{6} > 9.9 \times 10^{5}$
55. $3.1 \times 10^{-4}$, $9.9 \times 10^{-3}$, $1.2 \times 10^{-2}$, $3.1 \times 10^{-2}$ (decimal check: $0.00031$, $0.0099$, $0.012$, $0.031$)
56. $8.44 \times 10^{3}$, $8.4 \times 10^{3}$, $8.04 \times 10^{3}$, $8.4 \times 10^{2}$ (decimal check: $8440$, $8400$, $8040$, $840$)
57. $9 \times 10^{-4}$ is least. It is the only one with exponent $-4$; the others have exponents $-3$, $-2$, and $-3$. In decimal form: $0.0009$, $0.005$, $0.049$, $0.0051$.
58. $7.9 \times 10^{-4}$ is greater. Converting the decimal, $0.00072 = 7.2 \times 10^{-4}$. The exponents match, so compare coefficients: $7.2 < 7.9$.
59. Smallest to largest: $8 \times 10^{-7}$, $2.5 \times 10^{-6}$, $9.2 \times 10^{-6}$, $1.4 \times 10^{-5}$ (particles B, A, D, C). The largest diameter is $1.4 \times 10^{-5}$ m $= 0.000014$ m.
60. Priya compared the coefficients before checking the exponents, and the exponents are not equal. Because $-3 < -2$, the number with exponent $-3$ is the smaller one no matter how large its coefficient is. The correct comparison is $9.6 \times 10^{-3} < 1.1 \times 10^{-2}$, supported by the decimal forms $0.0096$ and $0.011$.

Exit ticket 1.4

61. $6.2 \times 10^{-3} < 6.2 \times 10^{-2}$
62. $7.5 \times 10^{-2}$, $2.9 \times 10^{2}$, $7.5 \times 10^{2}$
63. $4.8 \times 10^{5}$, since $5 > 4$. In decimal form, $480\,000 > 51\,000$.
64. First compare the exponents; the number with the greater exponent is greater. Only when the exponents are equal do you compare the coefficients, digit by digit from the left.

Chapter 1 Review

Part A — Powers of 10 with negative exponents (7.NS.1a)

65. a) $0.01$ b) $0.00001$ c) $1$
66. $10^{-4}$
67. $10^{-2}$, $10^{-3}$ (values $0.01$ and $0.001$)
68. The value is divided by $10$. Going from $10^{0}$ to $10^{-1}$ is one such step, so $10^{-1} = 1 \div 10 = 0.1$, which is less than $1$.

Part B — Fraction form and decimal form (7.NS.1b)

69. $\dfrac{1}{1000}$ and $0.001$
70. $10^{-5}$
71. a) $0.1$ b) $0.000001$
72. $\dfrac{1}{1000}$ is greater. It equals $0.001$, while $10^{-4} = 0.0001$. Written as powers of ten, $\dfrac{1}{1000} = 10^{-3}$ and $-3 > -4$.

Part C — Scientific notation and decimals (7.NS.1c)

73. a) $54\,000$ b) $0.00306$
74. a) $7.2 \times 10^{5}$ b) $4.5 \times 10^{-4}$
75. The coefficient $62$ is not less than $10$, and scientific notation requires $1 \le a < 10$. Its value is $62 \times 1000 = 62\,000$, so correctly written it is $6.2 \times 10^{4}$.
76. $n = -2$
77. $0.0005$ meters

Part D — Comparing and ordering (7.NS.1d)

78. a) $2.7 \times 10^{4} < 2.7 \times 10^{5}$ b) $6.3 \times 10^{-3} > 6.03 \times 10^{-3}$
79. $8.8 \times 10^{-4}$, $4.5 \times 10^{-3}$, $1.9 \times 10^{-2}$, $4.5 \times 10^{-2}$ (decimal check: $0.00088$, $0.0045$, $0.019$, $0.045$)
80. $3.3 \times 10^{6}$, $3.03 \times 10^{6}$, $1.1 \times 10^{6}$, $9.7 \times 10^{5}$ (decimal check: $3\,300\,000$, $3\,030\,000$, $1\,100\,000$, $970\,000$)

Part E — Mixed application and reasoning

81. Ascending: $4.07 \times 10^{-7}$ m, $4.7 \times 10^{-7}$ m, $5.1 \times 10^{-7}$ m, $6.2 \times 10^{-7}$ m. Every exponent is $-7$, so step one of the rule settled nothing and the coefficients did all the work. Comparing $4.07$ with $4.7$ needs care: writing $4.7$ as $4.70$ shows that $4.07 < 4.70$.
82. $0.000094 = 9.4 \times 10^{-5}$, which is less than $9.4 \times 10^{-4}$. The coefficients are identical, so the comparison rests entirely on the exponents, and they differ by one power of ten — $9.4 \times 10^{-4}$ is ten times as large.
83. A negative exponent counts divisions by $10$. In $10^{-6}$ you have divided $1$ by $10$ six times; in $10^{-2}$ only twice. More divisions leave less, so $10^{-6} = 0.000001$ is less than $10^{-2} = 0.01$. The comparison $6 > 2$ is about how many times you divided, not about how much is left.
84. The rule guarantees exactly one nonzero digit in front of the decimal point, which makes the scientific-notation form of a positive number unique. Without it, $470\,000$ could be written as $4.7 \times 10^{5}$, $47 \times 10^{4}$, or $0.47 \times 10^{6}$, all correct in value. A specific problem: the two-step comparison rule would break, because comparing exponents first would say $47 \times 10^{4}$ is smaller than $1 \times 10^{5}$, when in fact $470\,000 > 100\,000$.

Workbook-only items

Page 2, ladder table. $1000$; $100$; $10$; $1$; $0.1$; $0.01$; $0.001$

Page 2, fill in the blanks. Each time the exponent goes down by 1, the value is divided by 10. Each time the exponent goes up by 1, the value is multiplied by 10. A negative exponent does not make the number negative. Every power of ten is greater than 0.

Page 3, label the columns. thousands $10^{3}$; hundreds $10^{2}$; tens $10^{1}$; ones $10^{0}$; tenths $10^{-1}$; hundredths $10^{-2}$; thousandths $10^{-3}$

Page 4, item 12 pattern chain. $10^{0} = 1$ → $10^{-1} = 0.1$ → $10^{-2} = 0.01$ → $10^{-3} = 0.001$

Page 10, label the parts. In $6.3 \times 10^{-5}$, the coefficient is $6.3$ and the power of ten is $10^{-5}$.

Page 10, YES or NO. $4.8 \times 10^{3}$ — YES. $0.7 \times 10^{2}$ — NO, the coefficient is less than $1$. $12.5 \times 10^{4}$ — NO, the coefficient is not less than $10$. $9.99 \times 10^{-6}$ — YES.