1.9E: Exercises
- Page ID
- 30098
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Simplify Expressions with Square Roots
In the following exercises, simplify.
1. \(\sqrt{36}\)
- Answer
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\(\sqrt{36} = 6\)
2. \(\sqrt{4}\)
3. \(\sqrt{64}\)
- Answer
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\(\sqrt{64} = 8\)
4. \(\sqrt{169}\)
5. \(\sqrt{9}\)
- Answer
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\(\sqrt{9} = 3\)
6. \(\sqrt{16}\)
7. \(\sqrt{100}\)
- Answer
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\(\sqrt{100} = 10\)
8. \(\sqrt{144}\)
9. \(\sqrt{−4}\)
- Answer
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\(\sqrt{-4}\) is not a real number.
10. \(\sqrt{−100}\)
11. \(\sqrt{−1}\)
- Answer
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\(\sqrt{-1}\) is not a real number.
12. \(\sqrt{−121}\)
Identify Integers, Rational Numbers, Irrational Numbers, and Real Numbers
In the following exercises, write each rational number as the ratio of two integers.
13. ⓐ 5 ⓑ 3.19
- Answer
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a. \(5 = \frac{5}{1}\) b. \(3.19 = \frac{319}{100}\)
14. ⓐ 8 ⓑ 1.61
15. ⓐ −12 ⓑ 9.279
- Answer
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a. \(-12 = \frac{-12}{1}\) b. \(9.279 = \frac{9279}{1000}\)
16. ⓐ −16 ⓑ 4.399
In the following exercises, list the ⓐ rational numbers, ⓑ irrational numbers
17. {0.75, \(0.22\overline{3}\), 1.39174}
- Answer
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All of these numbers are rational.
18. {0.36, 0.94729…, \(2.52\overline{8}\)}
19. {\(0.4\overline{5}\), 1.919293…, 3.59}
- Answer
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a. \(0.4\overline{5}\) and 3.59 are rational numbers. b. 1.919293… is irrational, since it does not terminate or repeat.
20. {\(0.1\overline{3}\), 0.42982…, 1.875}
In the following exercises, identify whether each number is rational or irrational.
21. ⓐ \(\sqrt{25}\) ⓑ \(\sqrt{30}\)
- Answer
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a. \(\sqrt{25} = 5\) is rational. b. \(\sqrt{30}\) is irrational.
22. ⓐ \( \sqrt{44} \) ⓑ \( \sqrt{49} \)
23. ⓐ \( \sqrt{164} \) ⓑ \( \sqrt{169} \)
- Answer
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a. \(\sqrt{164}\) is irrational. b. \(\sqrt{169} = 13\) is rational.
24. ⓐ \( \sqrt{225} \) ⓑ \( \sqrt{216} \)
In the following exercises, identify whether each number is a real number or not a real number.
25. ⓐ \( -\sqrt{81} \) ⓑ \( \sqrt{-121} \)
- Answer
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a. \(-\sqrt{81} = -9\) is a real number. b. \(\sqrt{-121}\) is not a real number.
26. ⓐ \( -\sqrt{64} \) ⓑ \( \sqrt{-9} \)
27. ⓐ \( \sqrt{-36} \) ⓑ \( -\sqrt{144} \)
- Answer
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a. \(\sqrt{-36}\) is not a real number. b. \(-\sqrt{144} = -12\) is a real number.
28. ⓐ \( \sqrt{-49} \) ⓑ \( -\sqrt{144} \)
In the following exercises, for each set of numbers, list the ⓐ whole numbers, ⓑ integers, ⓒ rational numbers, ⓓ irrational numbers, ⓔ real numbers.
29. {−8, 0, 1.95286…, \(\tfrac{12}{5}\), \(\sqrt{36}\), 9}
- Answer
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a. whole numbers: 0, \(\sqrt{36} = 6\), 9
b. integers: -8, 0, \(\sqrt{36} = 6\), 9
c. rational numbers: -8, 0, \(\sqrt{36} = 6\), 9, \(\tfrac{12}{5}\)
d. irrational numbers: 1.95286…
e. real numbers: All of these numbers are real numbers.
30. {−9, \(−3\tfrac{4}{9}\), \(-\sqrt{9}\), \(0.40\overline{9}\), 116, 7}
31. {\(-\sqrt{100}\), −7, \(−\tfrac{8}{3}\), −1, 0.77, \(3\tfrac{1}{4}\)}
- Answer
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a. whole numbers: None of these numbers are whole numbers.
b. integers: \(-\sqrt{100}=-10\), −7, -1
c. rational numbers: All of these numbers are rational.
d. irrational numbers: None of these numbers is irrational.
e. real numbers: All of these numbers are real numbers.
32. {−6, \(−\tfrac{5}{2}\), 0, \(0.\overline{714285}\), \(2\tfrac{1}{5}\), \(\sqrt{14}\)}
Locate Fractions on the Number Line
In the following exercises, locate the numbers on a number line.
33. \(\tfrac{3}{4}, \tfrac{8}{5}, \tfrac{10}{3}\)
- Answer
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34. \(\tfrac{1}{4}, \tfrac{9}{5}, \tfrac{11}{3}\)
35. \(\frac{3}{10}, \frac{7}{2}, \frac{11}{6}, 4\)
- Answer
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36. \(\frac{7}{10}, \frac{5}{2}, \frac{13}{8}, 3\)
37. \(\frac{2}{5}, −\frac{2}{5}\)
- Answer
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38. \(\frac{3}{4},−\frac{3}{4}\)
39. \(\frac{3}{4},−\frac{3}{4}, 1\tfrac{2}{3}, −1\tfrac{2}{3}, \frac{5}{2}, −\frac{5}{2}\)
- Answer
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40. \(\frac{1}{5}, −\frac{2}{5}, 1\tfrac{3}{4}, −1\tfrac{3}{4}, \frac{8}{3}, −\frac{8}{3}\)
In the following exercises, order each of the pairs of numbers, using < or >.
41. −1 ___ \(−\frac{1}{4}\)
- Answer
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\(−1 < −\frac{1}{4}\)
42. −1 ___ \(−\frac{1}{3}\)
43. \(−2\tfrac{1}{2}\) ___ −3
- Answer
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\(−2\tfrac{1}{2} > −3\)
44. \(−1\tfrac{3}{4}\) ___ −2
45. \(−\frac{5}{12}\) ___ \(−\frac{7}{12}\)
- Answer
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\(−\frac{5}{12} > −\frac{7}{12}\)
46. \(−\frac{9}{10}\) ___ \(−\frac{3}{10}\)
47. −3 ___ \(−\frac{13}{5}\)
- Answer
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\(−3 < −\frac{13}{5}\)
48. −4 ___ \(−\frac{23}{6}\)
Locating Decimals on the Number Line In the following exercises, locate the number on the number line.
49. 0.8
- Answer
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50. −0.9
51. −1.6
- Answer
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52. 3.1
In the following exercises, order each pair of numbers, using < or >.
53. 0.37 ___ 0.63
- Answer
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0.37 < 0.63
54. 0.86 ___ 0.69
55. 0.91 ___ 0.901
- Answer
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0.91 > 0.901
56. 0.415 ___ 0.41
57. −0.5 ___ −0.3
- Answer
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−0.5 < −0.3
58. −0.1 ___ −0.4
59. −0.62 ___ −0.619
- Answer
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−0.62 < −0.619
60. −7.31 ___ −7.3
Everyday Math
61. Field trip All the 5th graders at Lincoln Elementary School will go on a field trip to the science museum. Counting all the children, teachers, and chaperones, there will be 147 people. Each bus holds 44 people.
ⓐ How many busses will be needed?
ⓑ Why must the answer be a whole number?
ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?
62. Child care Serena wants to open a licensed child care center. Her state requires there be no more than 12 children for each teacher. She would like her child care center to serve 40 children.
ⓐ How many teachers will be needed?
ⓑ Why must the answer be a whole number?
ⓒ Why shouldn’t you round the answer the usual way, by choosing the whole number closest to the exact answer?
Writing Exercises
63. In your own words, explain the difference between a rational number and an irrational number.
64. Explain how the sets of numbers (counting, whole, integer, rational, irrationals, reals) are related to each other.
Self Check
ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objective of this section.
ⓑ On a scale of 1−10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?