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Mathematics LibreTexts

8.1E: Exercises

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    Simplifying Expressions with Roots

    In the following exercises, simplify.

    1. a. \(\sqrt{64}\) b. \(-\sqrt{81}\)


    a. \(8\) b. \(-9\)

    2. a. \(\sqrt{169}\) b. \(-\sqrt{100}\)

    3. a. \(\sqrt{196}\) b. \(-\sqrt{1}\)


    a. \(14\) b. \(-1\)

    4. a. \(\sqrt{144}\) b. \(-\sqrt{121}\)

    5. a. \(\sqrt{\frac{4}{9}}\) b. \(-\sqrt{0.01}\)


    a. \(\frac{2}{3}\) b. \(-0.1\)

    6. a. \(\sqrt{\frac{64}{121}}\) b. \(-\sqrt{0.16}\)

    7. a. \(\sqrt{-121}\) b. \(-\sqrt{289}\)


    a. not a real number b. \(-17\)

    8. a. \(-\sqrt{400}\) b. \(\sqrt{-36}\)

    9. a. \(-\sqrt{225}\) b. \(\sqrt{-9}\)


    a. \(-15\) b. not a real number

    10. a. \(\sqrt{-49}\) b. \(-\sqrt{256}\)

    11. a. \(\sqrt[3]{216}\) b. \(\sqrt[4]{256}\)


    a. \(6\) b. \(4\)

    12. a. \(\sqrt[3]{27}\) b. \(\sqrt[4]{16}\) c. \(\sqrt[5]{243}\)

    13. a. \(\sqrt[3]{512}\) b. \(\sqrt[4]{81}\) c. \(\sqrt[5]{1}\)


    a. \(8\) b. \(3\) b. \(1\)

    14. a. \(\sqrt[3]{125}\) b. \(\sqrt[4]{1296}\) c. \(\sqrt[5]{1024}\)

    15. a. \(\sqrt[3]{-8}\) b. \(\sqrt[4]{-81}\) c. \(\sqrt[5]{-32}\)


    a. \(-2\) b. not a real number c. \(-2\)

    16. a. \(\sqrt[3]{-64}\) b. \(\sqrt[4]{-16}\) c. \(\sqrt[5]{-243}\)

    17. a. \(\sqrt[3]{-125}\) b. \(\sqrt[4]{-1296}\) c. \(\sqrt[5]{-1024}\)


    a. \(-5\) b. not a real number c. \(-4\)

    18. a. \(\sqrt[3]{-512}\) b. \(\sqrt[4]{-81}\) c. \(\sqrt[5]{-1}\)

    In the following exercises, estimate each root by giving the interval of two consecutive whole numbers in which the root lies.

    19. a. \(\sqrt{70}\) b. \(\sqrt[3]{71}\)


    a. \(8<\sqrt{70}<9\) b. \(4<\sqrt[3]{71}<5\)

    20. a. \(\sqrt{55}\) b. \(\sqrt[3]{119}\)

    21. a. \(\sqrt{200}\) b. \(\sqrt[3]{137}\)


    a. \(14<\sqrt{200}<15\) b. \(5<\sqrt[3]{137}<6\)

    22. a. \(\sqrt{172}\) b. \(\sqrt[3]{200}\)

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

    23. a. \(\sqrt{19}\) b. \(\sqrt[3]{89}\) c. \(\sqrt[4]{97}\)


    a. \(\approx 4.36\) b. \(\approx 4.46\) c. \(\approx 3.14\)

    24. a. \(\sqrt{21}\) b. \(\sqrt[3]{93}\) c. \(\sqrt[4]{101}\)

    25. a. \(\sqrt{53}\) b. \(\sqrt[3]{147}\) c. \(\sqrt[4]{452}\)


    a. \(\approx 7.28\) b. \(\approx 5.28\) c. \(\approx 4.61\)

    26. a. \(\sqrt{47}\) b. \(\sqrt[3]{163}\) c. \(\sqrt[4]{527}\)

    Simplify Variable Expressions with Roots

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

    27. a. \(\sqrt[5]{u^{5}}\) b. \(\sqrt[8]{v^{8}}\)


    a. \(u\) b. \(|v|\)

    28. a. \(\sqrt[3]{a^{3}}\) b. \(\sqrt[9]{b^{9}}\)

    29. a. \(\sqrt[4]{y^{4}}\) b. \(\sqrt[7]{m^{7}}\)


    a. \(|y|\) b. \(m\)

    30. a. \(\sqrt[8]{k^{8}}\) b. \(\sqrt[6]{p^{6}}\)

    31. a. \(\sqrt{x^{6}}\) b. \(\sqrt{y^{16}}\)


    a. \(|x^{3}|\) b. \(y^{8}\)

    32. a. \(\sqrt{a^{14}}\) b. \(\sqrt{w^{24}}\)

    33. a. \(\sqrt{x^{24}}\) b. \(\sqrt{y^{22}}\)


    a. \(x^{12}\) b. \(|y^{11}|\)

    34. a. \(\sqrt{a^{12}}\) b. \(\sqrt{b^{26}}\)

    35. a. \(\sqrt[3]{x^{9}}\) b. \(\sqrt[4]{y^{12}}\)


    a. \(x^{3}\) b. \(|y^{3}|\)

    36. a. \(\sqrt[5]{a^{10}}\) b. \(\sqrt[3]{b^{27}}\)

    37. a. \(\sqrt[4]{m^{8}}\) b. \(\sqrt[5]{n^{20}}\)


    a. \(m^{2}\) b. \(n^{4}\)

    38. a. \(\sqrt[6]{r^{12}}\) b. \(\sqrt[3]{s^{30}}\)

    39. a. \(\sqrt{49 x^{2}}\) b. \(-\sqrt{81 x^{18}}\)


    a. \(7|x|\) b. \(-9|x^{9}|\)

    40. a. \(\sqrt{100 y^{2}}\) b. \(-\sqrt{100 m^{32}}\)

    41. a. \(\sqrt{121 m^{20}}\) b. \(-\sqrt{64 a^{2}}\)


    a. \(11m^{10}\) b. \(-8|a|\)

    42. a. \(\sqrt{81 x^{36}}\) b. \(-\sqrt{25 x^{2}}\)

    43. a. \(\sqrt[4]{16 x^{8}}\) b. \(\sqrt[6]{64 y^{12}}\)


    a. \(2x^{2}\) b. \(2y^{2}\)

    44. a. \(\sqrt[3]{-8 c^{9}}\) b. \(\sqrt[3]{125 d^{15}}\)

    45. a. \(\sqrt[3]{216 a^{6}}\) b. \(\sqrt[5]{32 b^{20}}\)


    a. \(6a^{2}\) b. \(2b^{4}\)

    46. a. \(\sqrt[7]{128 r^{14}}\) b. \(\sqrt[4]{81 s^{24}}\)

    47. a. \(\sqrt{144 x^{2} y^{2}}\) b. \(\sqrt{169 w^{8} y^{10}}\) c. \(\sqrt[3]{8 a^{51} b^{6}}\)


    a. \(12|x y|\) b. \(13 w^{4}\left|y^{5}\right|\) c. \(2 a^{17} b^{2}\)

    48. a. \(\sqrt{196 a^{2} b^{2}}\) b. \(\sqrt{81 p^{24} q^{6}}\) c. \(\sqrt[3]{27 p^{45} q^{9}}\)

    49. a. \(\sqrt{121 a^{2} b^{2}}\) b. \(\sqrt{9 c^{8} d^{12}}\) c. \(\sqrt[3]{64 x^{15} y^{66}}\)


    a. \(11|ab|\) b. \(3c^{4}d^{6}\) c. \(4x^{5}y^{22}\)

    50. a. \(\sqrt{225 x^{2} y^{2} z^{2}}\) b. \(\sqrt{36 r^{6} s^{20}}\) c. \(\sqrt[3]{125 y^{18} z^{27}}\)

    Writing Exercises

    51. Why is there no real number equal to \(\sqrt{-64}\)?


    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 \(9^{2}\) and \(\sqrt{9}\)?

    53. Explain what is meant by the \(n^{th}\) root of a number.


    If you raise this root to the \(n^{th}\) power, it will give you back the original number (under the radical).

    54. Explain the difference of finding the \(n^{th}\) 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.