## Section Exercises

1. What does it mean when a radical does not have an index? Is the expression equal to the radicand? Explain.

2. Where would radicals come in the order of operations? Explain why.

3. Every number will have two square roots. What is the principal square root?

4. Can a radical with a negative radicand have a real square root? Why or why not?

For the following exercises, simplify each expression.

5. $\sqrt{256}$

6. $\sqrt{\sqrt{256}}$

7. $\sqrt{4\left(9+16\right)}$

8. $\sqrt{289}-\sqrt{121}$

9. $\sqrt{196}$

10. $\sqrt{1}$

11. $\sqrt{98}$

12. $\sqrt{\frac{27}{64}}$

13. $\sqrt{\frac{81}{5}}$

14. $\sqrt{800}$

15. $\sqrt{169}+\sqrt{144}$

16. $\sqrt{\frac{8}{50}}$

17. $\frac{18}{\sqrt{162}}$

18. $\sqrt{192}$

19. $14\sqrt{6}-6\sqrt{24}$

20. $15\sqrt{5}+7\sqrt{45}$

21. $\sqrt{150}$

22. $\sqrt{\frac{96}{100}}$

23. $\left(\sqrt{42}\right)\left(\sqrt{30}\right)$

24. $12\sqrt{3}-4\sqrt{75}$

25. $\sqrt{\frac{4}{225}}$

26. $\sqrt{\frac{405}{324}}$

27. $\sqrt{\frac{360}{361}}$

28. $\frac{5}{1+\sqrt{3}}$

29. $\frac{8}{1-\sqrt{17}}$

30. $\sqrt[4]{16}$

31. $\sqrt[3]{128}+3\sqrt[3]{2}$

32. $\sqrt[5]{\frac{-32}{243}}$

33. $\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$

34. $3\sqrt[3]{-432}+\sqrt[3]{16}$

For the following exercises, simplify each expression.

35. $\sqrt{400{x}^{4}}$

36. $\sqrt{4{y}^{2}}$

37. $\sqrt{49p}$

38. ${\left(144{p}^{2}{q}^{6}\right)}^{\frac{1}{2}}$

39. ${m}^{\frac{5}{2}}\sqrt{289}$

40. $9\sqrt{3{m}^{2}}+\sqrt{27}$

41. $3\sqrt{a{b}^{2}}-b\sqrt{a}$

42. $\frac{4\sqrt{2n}}{\sqrt{16{n}^{4}}}$

43. $\sqrt{\frac{225{x}^{3}}{49x}}$

44. $3\sqrt{44z}+\sqrt{99z}$

45. $\sqrt{50{y}^{8}}$

46. $\sqrt{490b{c}^{2}}$

47. $\sqrt{\frac{32}{14d}}$

48. ${q}^{\frac{3}{2}}\sqrt{63p}$

49. $\frac{\sqrt{8}}{1-\sqrt{3x}}$

50. $\sqrt{\frac{20}{121{d}^{4}}}$

51. ${w}^{\frac{3}{2}}\sqrt{32}-{w}^{\frac{3}{2}}\sqrt{50}$

52. $\sqrt{108{x}^{4}}+\sqrt{27{x}^{4}}$

53. $\frac{\sqrt{12x}}{2+2\sqrt{3}}$

54. $\sqrt{147{k}^{3}}$

55. $\sqrt{125{n}^{10}}$

56. $\sqrt{\frac{42q}{36{q}^{3}}}$

57. $\sqrt{\frac{81m}{361{m}^{2}}}$

58. $\sqrt{72c}-2\sqrt{2c}$

59. $\sqrt{\frac{144}{324{d}^{2}}}$

60. $\sqrt[3]{24{x}^{6}}+\sqrt[3]{81{x}^{6}}$

61. $\sqrt[4]{\frac{162{x}^{6}}{16{x}^{4}}}$

62. $\sqrt[3]{64y}$

63. $\sqrt[3]{128{z}^{3}}-\sqrt[3]{-16{z}^{3}}$

64. $\sqrt[5]{1,024{c}^{10}}$

65. A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating $\sqrt{90,000+160,000}$. What is the length of the guy wire?

66. A car accelerates at a rate of $6-\frac{\sqrt{4}}{\sqrt{t}}{\text{ m/s}}^{2}$ where t is the time in seconds after the car moves from rest. Simplify the expression.

For the following exercises, simplify each expression.

67. $\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-{2}^{\frac{1}{2}}$

68. $\frac{{4}^{\frac{3}{2}}-{16}^{\frac{3}{2}}}{{8}^{\frac{1}{3}}}$

69. $\frac{\sqrt{m{n}^{3}}}{{a}^{2}\sqrt{{c}^{-3}}}\cdot \frac{{a}^{-7}{n}^{-2}}{\sqrt{{m}^{2}{c}^{4}}}$

70. $\frac{a}{a-\sqrt{c}}$

71. $\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}$

72. $\left(\frac{\sqrt{250{x}^{2}}}{\sqrt{100{b}^{3}}}\right)\left(\frac{7\sqrt{b}}{\sqrt{125x}}\right)$

73. $\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$