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6.3.2: Homework

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    Reading Questions

    1. What is a radical equation?
    2. What is the primary strategy for solving a radical equation containing one radical?
    3. Why is it crucial to check your solutions when solving radical equations, especially if you've raised both sides to an even power? What kind of solutions might arise?
    4. If, after isolating an even-indexed radical, you find it is equal to a negative number (e.g., \(\sqrt{x} = -2\)), what can you conclude about the real solutions to the equation?
    5. When solving an equation with a rational exponent, like \((3x-2)^{1/4} + 3 = 5\), what is the first step after isolating the term with the rational exponent?
    6. If an equation contains two radicals, what is the general approach to solving it?
    7. When solving an equation like \(x - 2\sqrt{x} - 8 = 0\), what substitution could you make to transform it into a quadratic equation?
    8. What does the term "extraneous solution" mean in the context of solving radical equations?
    9. If you are solving an equation and raise both sides to an odd power (e.g., cubing both sides), is it still mandatory to check for extraneous solutions for the same reasons as with even powers?
    10. Describe the general steps for solving a radical equation as outlined in the "How To: Solve a Radical Equation" box.

    Homework

    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. \((6 x+1)^{1/2}-3=4\)
    24. \((3 x-2)^{1/2}+1=6\)
    25. \((8 x+5)^{1/3}+2=-1\)
    26. \((12 x-5)^{1/3}+8=3\)
    27. \((12 x-3)^{1/4}-5=-2\)
    28. \((5 x-4)^{1/4}+7=9\)
    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. \(3 \sqrt{7 y+1}-10=8\)

    In the following exercises, solve.

    1. \(\sqrt{3 u+7}=\sqrt{5 u+1}\)
    2. \(\sqrt{4 v+1}=\sqrt{3 v+3}\)
    3. \(\sqrt{8+2 r}=\sqrt{3 r+10}\)
    4. \(\sqrt{10+2 c}=\sqrt{4 c+16}\)
    5. \(\sqrt[3]{5 x-1}=\sqrt[3]{x+3}\)
    6. \(\sqrt[3]{8 x-5}=\sqrt[3]{3 x+5}\)
    7. \(\sqrt[3]{2 x^{2}+9 x-18}=\sqrt[3]{x^{2}+3 x-2}\)
    8. \(\sqrt[3]{x^{2}-x+18}=\sqrt[3]{2 x^{2}-3 x-6}\)
    9. \(\sqrt{a}+2=\sqrt{a+4}\)
    10. \(\sqrt{r}+6=\sqrt{r+8}\)
    11. \(\sqrt{u}+1=\sqrt{u+4}\)
    12. \(\sqrt{x}+1=\sqrt{x+2}\)
    13. \(\sqrt{a+5}-\sqrt{a}=1\)
    14. \(-2=\sqrt{d-20}-\sqrt{d}\)
    15. \(\sqrt{2 x+1}=1+\sqrt{x}\)
    16. \(\sqrt{3 x+1}=1+\sqrt{2 x-1}\)
    17. \(\sqrt{2 x-1}-\sqrt{x-1}=1\)
    18. \(\sqrt{x+1}-\sqrt{x-2}=1\)
    19. \(\sqrt{x+7}-\sqrt{x-5}=2\)
    20. \(\sqrt{x+5}-\sqrt{x-3}=2\)

    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.

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