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# 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. $$\sqrt{216}$$ b. $$\sqrt{256}$$

a. $$6$$ b. $$4$$

12. a. $$\sqrt{27}$$ b. $$\sqrt{16}$$ c. $$\sqrt{243}$$

13. a. $$\sqrt{512}$$ b. $$\sqrt{81}$$ c. $$\sqrt{1}$$

a. $$8$$ b. $$3$$ b. $$1$$

14. a. $$\sqrt{125}$$ b. $$\sqrt{1296}$$ c. $$\sqrt{1024}$$

15. a. $$\sqrt{-8}$$ b. $$\sqrt{-81}$$ c. $$\sqrt{-32}$$

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

16. a. $$\sqrt{-64}$$ b. $$\sqrt{-16}$$ c. $$\sqrt{-243}$$

17. a. $$\sqrt{-125}$$ b. $$\sqrt{-1296}$$ c. $$\sqrt{-1024}$$

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

18. a. $$\sqrt{-512}$$ b. $$\sqrt{-81}$$ c. $$\sqrt{-1}$$

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. $$\sqrt{71}$$

a. $$8<\sqrt{70}<9$$ b. $$4<\sqrt{71}<5$$

20. a. $$\sqrt{55}$$ b. $$\sqrt{119}$$

21. a. $$\sqrt{200}$$ b. $$\sqrt{137}$$

a. $$14<\sqrt{200}<15$$ b. $$5<\sqrt{137}<6$$

22. a. $$\sqrt{172}$$ b. $$\sqrt{200}$$

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

23. a. $$\sqrt{19}$$ b. $$\sqrt{89}$$ c. $$\sqrt{97}$$

a. $$\approx 4.36$$ b. $$\approx 4.46$$ c. $$\approx 3.14$$

24. a. $$\sqrt{21}$$ b. $$\sqrt{93}$$ c. $$\sqrt{101}$$

25. a. $$\sqrt{53}$$ b. $$\sqrt{147}$$ c. $$\sqrt{452}$$

a. $$\approx 7.28$$ b. $$\approx 5.28$$ c. $$\approx 4.61$$

26. a. $$\sqrt{47}$$ b. $$\sqrt{163}$$ c. $$\sqrt{527}$$

### Simplify Variable Expressions with Roots

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

27. a. $$\sqrt{u^{5}}$$ b. $$\sqrt{v^{8}}$$

a. $$u$$ b. $$|v|$$

28. a. $$\sqrt{a^{3}}$$ b. $$\sqrt{b^{9}}$$

29. a. $$\sqrt{y^{4}}$$ b. $$\sqrt{m^{7}}$$

a. $$|y|$$ b. $$m$$

30. a. $$\sqrt{k^{8}}$$ b. $$\sqrt{p^{6}}$$

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. $$\sqrt{x^{9}}$$ b. $$\sqrt{y^{12}}$$

a. $$x^{3}$$ b. $$|y^{3}|$$

36. a. $$\sqrt{a^{10}}$$ b. $$\sqrt{b^{27}}$$

37. a. $$\sqrt{m^{8}}$$ b. $$\sqrt{n^{20}}$$

a. $$m^{2}$$ b. $$n^{4}$$

38. a. $$\sqrt{r^{12}}$$ b. $$\sqrt{s^{30}}$$

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. $$\sqrt{16 x^{8}}$$ b. $$\sqrt{64 y^{12}}$$

a. $$2x^{2}$$ b. $$2y^{2}$$

44. a. $$\sqrt{-8 c^{9}}$$ b. $$\sqrt{125 d^{15}}$$

45. a. $$\sqrt{216 a^{6}}$$ b. $$\sqrt{32 b^{20}}$$

a. $$6a^{2}$$ b. $$2b^{4}$$

46. a. $$\sqrt{128 r^{14}}$$ b. $$\sqrt{81 s^{24}}$$

47. a. $$\sqrt{144 x^{2} y^{2}}$$ b. $$\sqrt{169 w^{8} y^{10}}$$ c. $$\sqrt{8 a^{51} b^{6}}$$

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. $$\sqrt{27 p^{45} q^{9}}$$

49. a. $$\sqrt{121 a^{2} b^{2}}$$ b. $$\sqrt{9 c^{8} d^{12}}$$ c. $$\sqrt{64 x^{15} y^{66}}$$

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. $$\sqrt{125 y^{18} z^{27}}$$

### 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.