10.7E: Exercises
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Practice Makes Perfect
Exercise SET A: Solve Radical Equations
In the following exercises, solve.
1. \(\sqrt{5 x-6}=8\)
2. \(\sqrt{4 x-3}=7\)
3. \(\sqrt{5 x+1}=-3\)
4. \(\sqrt{3 y-4}=-2\)
5. \(\sqrt[3]{2 x}=-2\)
6. \(\sqrt[3]{4 x-1}=3\)
7. \(\sqrt{2 m-3}-5=0\)
8. \(\sqrt{2 n-1}-3=0\)
9. \(\sqrt{6 v-2}-10=0\)
10. \(\sqrt{12 u+1}-11=0\)
11. \(\sqrt{4 m+2}+2=6\)
12. \(\sqrt{6 n+1}+4=8\)
13. \(\sqrt{2 u-3}+2=0\)
14. \(\sqrt{5 v-2}+5=0\)
15. \(\sqrt{u-3}+3=u\)
16. \(\sqrt{v-10}+10=v\)
17. \(\sqrt{r-1}=r-1\)
18. \(\sqrt{s-8}=s-8\)
19. \(\sqrt[3]{6 x+4}=4\)
20. \(\sqrt[3]{11 x+4}=5\)
21. \(\sqrt[3]{4 x+5}-2=-5\)
22. \(\sqrt[3]{9 x-1}-1=-5\)
23. \((6 x+1)^{\frac{1}{2}}-3=4\)
24. \((3 x-2)^{\frac{1}{2}}+1=6\)
25. \((8 x+5)^{\frac{1}{3}}+2=-1\)
26. \((12 x-5)^{\frac{1}{3}}+8=3\)
27. \((12 x-3)^{\frac{1}{4}}-5=-2\)
28. \((5 x-4)^{\frac{1}{4}}+7=9\)
29. \(\sqrt{x+1}-x+1=0\)
30. \(\sqrt{y+4}-y+2=0\)
31. \(\sqrt{z+100}-z=-10\)
32. \(\sqrt{w+25}-w=-5\)
33. \(3 \sqrt{2 x-3}-20=7\)
34. \(2 \sqrt{5 x+1}-8=0\)
35. \(2 \sqrt{8 r+1}-8=2\)
36. \(3 \sqrt{7 y+1}-10=8\)
- Answer
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1. \(m=14\)
3. no solution
5. \(x=-4\)
7. \(m=14\)
9. \(v=17\)
11. \(m=\frac{7}{2}\)
13. no solution
15. \(u=3, u=4\)
17. \(r=1, r=2\)
19. \(x=10\)
21. \(x=-8\)
23. \(x=8\)
25. \(x=-4\)
27. \(x=7\)
29. \(x=3\)
31. \(z=21\)
33. \(x=42\)
35. \(r=3\)
Exercise SET B: Solve Radical Equations with Two Radicals
In the following exercises, solve.
37. \(\sqrt{3 u+7}=\sqrt{5 u+1}\)
38. \(\sqrt{4 v+1}=\sqrt{3 v+3}\)
39. \(\sqrt{8+2 r}=\sqrt{3 r+10}\)
40. \(\sqrt{10+2 c}=\sqrt{4 c+16}\)
41. \(\sqrt[3]{5 x-1}=\sqrt[3]{x+3}\)
42. \(\sqrt[3]{8 x-5}=\sqrt[3]{3 x+5}\)
43. \(\sqrt[3]{2 x^{2}+9 x-18}=\sqrt[3]{x^{2}+3 x-2}\)
44. \(\sqrt[3]{x^{2}-x+18}=\sqrt[3]{2 x^{2}-3 x-6}\)
45. \(\sqrt{a}+2=\sqrt{a+4}\)
46. \(\sqrt{r}+6=\sqrt{r+8}\)
47. \(\sqrt{u}+1=\sqrt{u+4}\)
48. \(\sqrt{x}+1=\sqrt{x+2}\)
49. \(\sqrt{a+5}-\sqrt{a}=1\)
50. \(-2=\sqrt{d-20}-\sqrt{d}\)
51. \(\sqrt{2 x+1}=1+\sqrt{x}\)
52. \(\sqrt{3 x+1}=1+\sqrt{2 x-1}\)
53. \(\sqrt{2 x-1}-\sqrt{x-1}=1\)
54. \(\sqrt{x+1}-\sqrt{x-2}=1\)
55. \(\sqrt{x+7}-\sqrt{x-5}=2\)
56. \(\sqrt{x+5}-\sqrt{x-3}=2\)
- Answer
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37. \(u=3\)
39. \(r=-2\)
41. \(x=1\)
43. \(x=-8, x=2\)
45. \(a=0\)
47. \(u=\frac{9}{4}\)
49. \(a=4\)
51. \(x=0\: x=4\)
53. \(x=1\: x=5\)
55. \(x=9\)
Exercise SET C: Use Radicals in Applications
In the following exercises, solve. Round approximations to one decimal place.
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Landscaping
Reed wants to have a square garden plot in his backyard. He has enough compost to cover an area of \(75\) square feet. Use the formula \(s=\sqrt{A}\) to find the length of each side of his garden. Round your answer to the nearest tenth of a foot.
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Landscaping
Vince wants to make a square patio in his yard. He has enough concrete to pave an area of \(130\) square feet. Use the formula \(s=\sqrt{A}\) to find the length of each side of his patio. Round your answer to the nearest tenth of a foot.
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Gravity
A hang glider dropped his cell phone from a height of \(350\) feet. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the cell phone to reach the ground.
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Gravity
A construction worker dropped a hammer while building the Grand Canyon skywalk, \(4000\) feet above the Colorado River. Use the formula \(t=\frac{\sqrt{h}}{4}\) to find how many seconds it took for the hammer to reach the river.
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Accident investigation
The skid marks for a car involved in an accident measured \(216\) feet. Use the formula \(s=\sqrt{24d}\) to find the speed of the car before the brakes were applied. Round your answer to the nearest tenth.
- Accident investigation An accident investigator measured the skid marks of one of the vehicles involved in an accident. The length of the skid marks was \(175\) feet. Use the formula \(s=\sqrt{24d}\) to find the speed of the vehicle before the brakes were applied. Round your answer to the nearest tenth.
- Answer
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57. \(8.7\) feet
59. \(4.7\) seconds
61. \(72\) feet
Exercise SET D: Writing Exercises
- Explain why an equation of the form \(\sqrt{x}+1=0\) has no solution.
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- Solve the equations \(\sqrt{r+4}-r+2=0\).
- Explain why one of the "solutions" that was found was not actually a solution to the equation.
- Answer
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63. Answers will vary.
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After reviewing this checklist, what will you do to become confident for all objectives?