Pre-TestUnit7:RealNumbersKEY
No calculator necessary. Please do not use a calculator.
Convert the following fraction to a decimal or decimal to a fraction. (5 pts; 3 pts for correct set-up/work, 2 pts for
correct answer)
1. 0. 45
2.
0. 27
Identify if the given number is rational or irrational and explain how you know. (5 pts; 2 pts for correct answer,
3 pts for explanation)
3. √25
4. √2
5. 6. 0.45
Rational
Irrational
Irrational
Rational
Evaluate the following roots. (5 pts; no partial credit)
7. √49
8. √−27
7
−3
Approximate the square roots to one decimal place. (5 pts; 2 pts for whole number accuracy, 1 pt if within 0.1)
9. √20
10. √8
≈ 4.5
≈ 2.8
Compare the following irrational numbers using < or >. (5 pts; no partial credit)
11. √21 > √19
12. −√17 < −√15
List the following numbers in order from least to greatest. (5 pts; no partial credit)
13. √7, , 3, 4.15,
√7, 3, , 4.15,
14. √27, √23, 5,
√23, 5, √27,
1
Match the given number with the letter that approximates that number’s position on the number line. (5 pts; no
partial credit)
A
E
B
D
C
2.0 2.5 3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5
15. √14
C
16. √5
A
17. √40
E
Estimate the value of the expressions to the nearest whole number. (5 pts; 3 pts for correct √ approximations)
18. 2√35
≈ 12
19. 5 + √50
≈ 12
Answer the following question. (5 pts; partial credit at teacher discretion)
20. A calculator displays the following number in its display: 0.37 which does not fill the display screen. Is this
enough to determine whether this number is rational or irrational? In at least one complete sentence, explain
why.
Yes, it is rational because the repeating digit is a zero at the end (otherwise known as a terminating decimal). This
means it can be turned into a fraction.
2
Unit 7 Homework Key
Lesson 7.1
Identify which of the following numbers are rational or irrational and explain how you know.
1. √25
Rational
Integer
2. √24
Irrational
No repeat
3. −√36
Rational
Integer
4. −√64
Rational
Integer
5. −√27
Irrational
No repeat
6.
Rational
Fraction
7. 0.45
Rational
Repeats 0
8. 0. 2
Rational
Repeats
9. √49
Rational
Integer
10. √18
Irrational
No repeat
11. −√10
Irrational
No repeat
12.
Rational
Fraction
14. 0. 42
Rational
Repeats
15. 0.39
Rational
Repeats 0
16. −√100
Rational
Integer
17. −√16
Rational
Integer
18. −√43
Irrational
No repeat
13. Rational
Fraction
19. If the number 0.77 is displayed on a calculator that can only display ten digits, do we know whether it is
rational or irrational? In one complete sentence explain why.
Rational because it terminates (or repeats the digit zero).
20. If the number 0.123456789 is displayed on a calculator that can only display ten digits, do we know whether
it is rational or irrational? In one complete sentence explain why.
We don’t know. The decimal could repeat later making it rational or could never repeat making it irrational.
21. If the number 0.987098709 is displayed on a calculator that can only display ten digits, do we know whether
it is rational or irrational? In one complete sentence explain why.
Our best guess is that it is rational because there appears to be a pattern, but we can’t be sure.
22. If the number 0.425364758 is displayed on a calculator that can only display ten digits, do we know whether
it is rational or irrational? In one complete sentence explain why.
We don’t know. The decimal could repeat later making it rational or could never repeat making it irrational.
3
Lesson 7.2
Convert the following fractions to repeating decimals.
1.
0.46
7.
0.916
2.
0. 6
8.
0. 3
3.
0. 7
9.
0.83
4.
5.
0. 30
10.
6.
0. 1
11.
0. 45
0. 18
12.
0.16
0.38
Convert the following repeating decimals to fractions.
13. 0. 2
14. 0.15
15. 0.36
16. 0.48
17. 1.23
18. 1. 5
!"1 19. 0. 81
20. 0. 35
21. 0.215
22. 0.123
23. 1. 16
24. 3. 25
!"1 !"3 4
!"1 Lesson 7.3
Find both square roots of the given numbers.
1. 49
2. 64
3. 25
4. 16
5. 1
6. 121
±7
±8
±5
±4
±1
±11
7. 9
8. 196
9. 625
10. 4
11. 36
12. 81
±3
±14
±25
±2
±6
±9
Evaluate the following roots giving the principal root.
13. √81
14. −√100
15. √36
16. √−4
9
−10
6
$!%!&'()!$
17. √144
18. −√225
19. −√169
20. √400
12
−15
−13
20
21. −√900
22. √−27
23. √125
24. √1
−30
−3
5
1
25. √−1
26. √−64
27. √216
28. √8
−1
−4
6
2
29. √−1000
30. √27
31. − √27
32. − √1
−10
3
−3
−1
5
Approximate the following irrational numbers to the nearest whole number.
33. √28
34. √14
35. −√39
36. −√56
37. −√77
38. √18
≈5
≈4
≈ −6
≈ −7
≈ −9
≈4
39. √2
40. √41
41. √21
42. −√65
43. −√12
44. −√120
≈1
≈6
≈5
≈ −8
≈ −3
≈ −11
45. √8
46. √13
47. √32
48. √47
49. −√99
50. −√5
≈3
≈4
≈6
≈7
≈ −10
≈ −2
Approximate the following irrational numbers to one decimal place.
51. √30
52. √10
53. −√40
54. −√17
55. √101
56. √7
≈ 5.5
≈ 3.2
≈ −6.3
≈ −4.1
≈ 10.0
≈ 2.6
57. √3
58. √90
59. √35
60. −√11
61. −√22
62. √61
≈ 1.7
≈ 9.5
≈ 5.9
≈ −3.3
≈ −4.7
≈ 7.8
63. √50
64. √6
65. √67
66. √140
67. −√55
68. −√45
≈ 7.1
≈ 2.4
≈ 8.2
≈ 11.8
≈ −7.4
≈ −6.7
6
Lesson 7.4
Place a point on the number line given for each of the following irrational numbers.
1. Point A: √2
2. Point B: √17
1.0 1.5 2.0
A
6. Point V: √26
E
3. Point C: √11
2.5 3.0
D C 3.5
7. Point W: √88
5.0 5.5 6.0
V Z
Y
4.0
B 4.5
4. Point D: √8
5.0 5.5
8. Point X: √77
6.5 7.0
7.5 8.0
5. Point E: √5
9. Point Y: √37
10. Point Z: √30
8.5 9.0 9.5
X
W
Name the point on the number line associated with each irrational number.
11. √50
12. √103
13. √62
14. √90
15. √37
E
A
D
C
B
B
E
D
C
A
6.0 6.5 7.0 7.5 8.0 8.5 9.0 9.5 10.0 10.5
16. √7
17. √22
18. √34
19. √38
20. √15
B
A
D
C
E
B
2.0 2.5 3.0
E
A
3.5 4.0 4.5 5.0
7
D
C
5.5 6.0 6.5
Compare the following numbers using < or >.
21. √32 > 5.1
25. √99 >
22. √38 <
√42
26. √17 < 4.5
29. √16 > 3.9
30. √2 <
23. √17 <
27.
>
24. √49 < 7.1
28. √12 < √21
√65
31. √50 <
32. √9 < 3.01
List the following numbers in order from least to greatest.
33. √16,4.2,
√16, 4.2,
√24, 5.1, √33
35. √100, √110,
,
34. √24, √33,5.1
36. 9.4,
√80, 9.4,
√100, √110
37. √35, √32,√37,
√32, √35,√37,
√10,3.5, √15,
√9, √12, √15, 4.3
40. √39, √25,5.3, √26,
41. √12, √15,4.3, √9,
,
, √80
38. √10,3.5, √15,
39. √65, √60,8.5,
√60, √65, 8.5,
√25, √26, 5.3,
, √39
42. √49, √63,7.3, √38,
√38, √49, 7.3,
8
, √63
Lesson 7.5
Estimate the following expressions to the nearest whole number.
1. √8 + √18
2. 11 − √80
3. 4√48
4. 3√24 + 3
5. 2√35 − 3√8
≈7
≈2
≈ 28
≈ 18
≈3
6. √14 + √26
7. √120 − 7
8. 2√63
9. 4√15 − 5
10. 2√66 − 3√5
≈9
≈4
≈ 16
≈ 11
≈ 10
11. √9 + √10
12. 20 − √102
13. 2√15
14. 3√15 + 1
15. 4√24 − 3√3
≈6
≈ 10
≈8
≈ 13
≈ 14
16. √14 + √34
17. √105 − 9
18. 5√26
19. 2√83 − 8
20. 3√17 − 2√1
≈ 10
≈1
≈ 25
≈ 10
≈ 10
21. √47 + √8
22. 8 − √48
23. 7√10
24. 4√5 + 9
25. 3√24 − 5√5
≈ 10
≈1
≈ 21
≈ 17
≈5
26. √65 + √63
27. √100 − 2
28. 6√5
29. 2√26 − 3
30. 4√26 − 3√4
≈ 16
=8
≈ 12
≈7
≈ 14
9
ReviewUnit7:RealNumbersKEY
No calculator necessary. Please do not use a calculator.
Unit 7 Goals
• Know that numbers that are not rational are called irrational. Understand informally that every number has a
decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert to a
decimal expansion which repeats eventually into a rational number. (8.NS.1)
• Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them
approximately on a number line diagram, and estimate the value of expressions. (8.NS.2)
Convert the following fractions into decimals or decimals into lowest terms fractions.
2. 0.13
0. 5
1.
3. 0.28
4. 0. 16
Identify if the given number is rational or irrational and explain how you know.
5. √21
6. 0. 4
7. √0.04
8. − Irrational
Rational
Rational
Rational
9. √81
10. −√4
11. √−1000
12. √125
9
−2
−10
5
Evaluate the following roots.
Approximate the square roots to one decimal place.
13. √84
14. −√27
15. −√50
16. √95
≈ 9.2
≈ −5.2
≈ −7.1
≈ 9.7
19. √15 > √13
20. > √5
Compare the following numbers using < or >.
17. √20 < 5.1
18. − > −√2
10
List the following numbers in order from least to greatest.
21. √8, , 9, 3.1,
22. √35, √40, 6.01,
, √8, 3.1, , 9
√35, 6.01, √40,
23. √3, 1.1, 2,
, 1.1, √3, 2
Match the given number with the letter that approximates that number’s position on the number line.
A
E
B
D
C
3.0 3.5 4.0 4.5 5.0 5.5 6.0 6.5 7.0 7.5
24. √10
25. √40
26. √18
27. √21
A
D
B
C
Estimate the value of the expressions to the nearest whole number.
28. 3√15
29. 4 + √35
≈ 12
≈ 10
Answer the following questions.
30. A calculator displays the following number in its display: 0.4153 which does not fill the display screen. Is this
enough to determine whether this number is rational or irrational? In at least one complete sentence, explain
why.
Rational because it terminates (or repeats the digit zero).
11