10.2E: Exercises
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- Aug 24, 2020
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Simplifying Expressions with Roots
In the following exercises, simplify.
1. a. √64 b. −√81
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a. 8 b. −9
2. a. √169 b. −√100
3. a. √196 b. −√1
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a. 14 b. −1
4. a. √144 b. −√121
5. a. √49 b. −√0.01
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a. 23 b. −0.1
6. a. √64121 b. −√0.16
7. a. √−121 b. −√289
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a. not a real number b. −17
8. a. −√400 b. √−36
9. a. −√225 b. √−9
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a. −15 b. not a real number
10. a. √−49 b. −√256
11. a. 3√216 b. 4√256
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a. 6 b. 4
12. a. 3√27 b. 4√16 c. 5√243
13. a. 3√512 b. 4√81 c. 5√1
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a. 8 b. 3 b. 1
14. a. 3√125 b. 4√1296 c. 5√1024
15. a. 3√−8 b. 4√−81 c. 5√−32
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a. −2 b. not a real number c. −2
16. a. 3√−64 b. 4√−16 c. 5√−243
17. a. 3√−125 b. 4√−1296 c. 5√−1024
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a. −5 b. not a real number c. −4
18. a. 3√−512 b. 4√−81 c. 5√−1
In the following exercises, estimate each root by giving the interval of two consecutive whole numbers in which the root lies.
19. a. √70 b. 3√71
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a. 8<√70<9 b. 4<3√71<5
20. a. √55 b. 3√119
21. a. √200 b. 3√137
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a. 14<√200<15 b. 5<3√137<6
22. a. √172 b. 3√200
In the following exercises, approximate each root and round to two decimal places.
23. a. √19 b. 3√89 c. 4√97
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a. ≈4.36 b. ≈4.46 c. ≈3.14
24. a. √21 b. 3√93 c. 4√101
25. a. √53 b. 3√147 c. 4√452
- Answer
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a. ≈7.28 b. ≈5.28 c. ≈4.61
26. a. √47 b. 3√163 c. 4√527
Simplify Variable Expressions with Roots
In the following exercises, simplify using absolute values as necessary.
27. a. 5√u5 b. 8√v8
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a. u b. |v|
28. a. 3√a3 b. 9√b9
29. a. 4√y4 b. 7√m7
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a. |y| b. m
30. a. 8√k8 b. 6√p6
31. a. √x6 b. √y16
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a. |x3| b. y8
32. a. √a14 b. √w24
33. a. √x24 b. √y22
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a. x12 b. |y11|
34. a. √a12 b. √b26
35. a. 3√x9 b. 4√y12
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a. x3 b. |y3|
36. a. 5√a10 b. 3√b27
37. a. 4√m8 b. 5√n20
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a. m2 b. n4
38. a. 6√r12 b. 3√s30
39. a. √49x2 b. −√81x18
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a. 7|x| b. −9|x9|
40. a. √100y2 b. −√100m32
41. a. √121m20 b. −√64a2
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a. 11m10 b. −8|a|
42. a. √81x36 b. −√25x2
43. a. 4√16x8 b. 6√64y12
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a. 2x2 b. 2y2
44. a. 3√−8c9 b. 3√125d15
45. a. 3√216a6 b. 5√32b20
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a. 6a2 b. 2b4
46. a. 7√128r14 b. 4√81s24
47. a. √144x2y2 b. √169w8y10 c. 3√8a51b6
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a. 12|xy| b. 13w4|y5| c. 2a17b2
48. a. √196a2b2 b. √81p24q6 c. 3√27p45q9
49. a. √121a2b2 b. √9c8d12 c. 3√64x15y66
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a. 11|ab| b. 3c4d6 c. 4x5y22
50. a. √225x2y2z2 b. √36r6s20 c. 3√125y18z27
Writing Exercises
51. Why is there no real number equal to √−64?
- Answer
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Since the square of any real number is positive, it's not possible for a real number to square to −64.
52. What is the difference between 92 and √9?
53. Explain what is meant by the nth root of a number.
- Answer
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If you raise this root to the nth power, it will give you back the original number (under the radical).
54. Explain the difference of finding the nth root of a number when the index is even compared to when the index is odd.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

b. If most of your checks were:
…confidently. Congratulations! You have achieved the objectives in this section. Reflect on the study skills you used so that you can continue to use them. What did you do to become confident of your ability to do these things? Be specific.
…with some help. This must be addressed quickly because topics you do not master become potholes in your road to success. In math every topic builds upon previous work. It is important to make sure you have a strong foundation before you move on. Who can you ask for help? Your fellow classmates and instructor are good resources. Is there a place on campus where math tutors are available? Can your study skills be improved?
…no - I don’t get it! This is a warning sign and you must not ignore it. You should get help right away or you will quickly be overwhelmed. See your instructor as soon as you can to discuss your situation. Together you can come up with a plan to get you the help you need.