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Mathematics LibreTexts

8.6E: Exercises

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    30899
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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

    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

    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.

    1. 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.
    2. 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.
    3. 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.
    4. 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.
    5. 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.
    6. 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

    57. \(8.7\) feet

    59. \(4.7\) seconds

    61. \(72\) feet

    Exercise SET D: Writing Exercises

    1. Explain why an equation of the form \(\sqrt{x}+1=0\) has no solution.
      1. Solve the equations \(\sqrt{r+4}-r+2=0\).
      2. Explain why one of the "solutions" that was found was not actually a solution to the equation.
    Answer

    63. Answers will vary.

    Self Check

    a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

    The table has 4 columns and 4 rows. The first row is a header row with the headers “I can…”, “Confidently”, “With some help.”, and “No – I don’t get it!”. The first column contains the phrases “Solve radical equations”, “solve radical equations with two radicals”, and “use radicals in applications”. The other columns are left blank so the learner can indicate their level of understanding.
    Figure 8.6.42

    b. After reviewing this checklist, what will you do to become confident for all objectives?

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