Loading [MathJax]/jax/output/HTML-CSS/jax.js
Skip to main content
Library homepage
 

Text Color

Text Size

 

Margin Size

 

Font Type

Enable Dyslexic Font
Mathematics LibreTexts

10.2E: Exercises

( \newcommand{\kernel}{\mathrm{null}\,}\)

Simplifying Expressions with Roots

In the following exercises, simplify.

1. a. 64 b. 81

Answer

a. 8 b. 9

2. a. 169 b. 100

3. a. 196 b. 1

Answer

a. 14 b. 1

4. a. 144 b. 121

5. a. 49 b. 0.01

Answer

a. 23 b. 0.1

6. a. 64121 b. 0.16

7. a. 121 b. 289

Answer

a. not a real number b. 17

8. a. 400 b. 36

9. a. 225 b. 9

Answer

a. 15 b. not a real number

10. a. 49 b. 256

11. a. 3216 b. 4256

Answer

a. 6 b. 4

12. a. 327 b. 416 c. 5243

13. a. 3512 b. 481 c. 51

Answer

a. 8 b. 3 b. 1

14. a. 3125 b. 41296 c. 51024

15. a. 38 b. 481 c. 532

Answer

a. 2 b. not a real number c. 2

16. a. 364 b. 416 c. 5243

17. a. 3125 b. 41296 c. 51024

Answer

a. 5 b. not a real number c. 4

18. a. 3512 b. 481 c. 51

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. 371

Answer

a. 8<70<9 b. 4<371<5

20. a. 55 b. 3119

21. a. 200 b. 3137

Answer

a. 14<200<15 b. 5<3137<6

22. a. 172 b. 3200

In the following exercises, approximate each root and round to two decimal places.

23. a. 19 b. 389 c. 497

Answer

a. 4.36 b. 4.46 c. 3.14

24. a. 21 b. 393 c. 4101

25. a. 53 b. 3147 c. 4452

Answer

a. 7.28 b. 5.28 c. 4.61

26. a. 47 b. 3163 c. 4527

Simplify Variable Expressions with Roots

In the following exercises, simplify using absolute values as necessary.

27. a. 5u5 b. 8v8

Answer

a. u b. |v|

28. a. 3a3 b. 9b9

29. a. 4y4 b. 7m7

Answer

a. |y| b. m

30. a. 8k8 b. 6p6

31. a. x6 b. y16

Answer

a. |x3| b. y8

32. a. a14 b. w24

33. a. x24 b. y22

Answer

a. x12 b. |y11|

34. a. a12 b. b26

35. a. 3x9 b. 4y12

Answer

a. x3 b. |y3|

36. a. 5a10 b. 3b27

37. a. 4m8 b. 5n20

Answer

a. m2 b. n4

38. a. 6r12 b. 3s30

39. a. 49x2 b. 81x18

Answer

a. 7|x| b. 9|x9|

40. a. 100y2 b. 100m32

41. a. 121m20 b. 64a2

Answer

a. 11m10 b. 8|a|

42. a. 81x36 b. 25x2

43. a. 416x8 b. 664y12

Answer

a. 2x2 b. 2y2

44. a. 38c9 b. 3125d15

45. a. 3216a6 b. 532b20

Answer

a. 6a2 b. 2b4

46. a. 7128r14 b. 481s24

47. a. 144x2y2 b. 169w8y10 c. 38a51b6

Answer

a. 12|xy| b. 13w4|y5| c. 2a17b2

48. a. 196a2b2 b. 81p24q6 c. 327p45q9

49. a. 121a2b2 b. 9c8d12 c. 364x15y66

Answer

a. 11|ab| b. 3c4d6 c. 4x5y22

50. a. 225x2y2z2 b. 36r6s20 c. 3125y18z27

Writing Exercises

51. Why is there no real number equal to 64?

Answer

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

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.

This table has 4 rows and 4 columns. The first row is a header row and it labels each column. The first column header is “I can…”, the second is “Confidently”, the third is “With some help”, and the fourth is “No, I don’t get it”. Under the first column are the phrases “simplify expressions with roots.”, “estimate and approximate roots”, and “simplify variable expressions with roots”. The other columns are left blank so that the learner may indicate their mastery level for each topic.

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.


This page titled 10.2E: Exercises is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Chau D Tran.

Support Center

How can we help?