1. In a radical equation, what does it mean if a number is an extraneous solution?

2. Explain why possible solutions must be checked in radical equations.

3. Your friend tries to calculate the value [latex]-{9}^{\frac{3}{2}}[/latex] and keeps getting an ERROR message. What mistake is he or she probably making?

4. Explain why [latex]|2x+5|=-7[/latex] has no solutions.

5. Explain how to change a rational exponent into the correct radical expression.

For the following exercises, solve the rational exponent equation. Use factoring where necessary.

6. [latex]{x}^{\frac{2}{3}}=16[/latex]

7. [latex]{x}^{\frac{3}{4}}=27[/latex]

8. [latex]2{x}^{\frac{1}{2}}-{x}^{\frac{1}{4}}=0[/latex]

9. [latex]{\left(x - 1\right)}^{\frac{3}{4}}=8[/latex]

10. [latex]{\left(x+1\right)}^{\frac{2}{3}}=4[/latex]

11. [latex]{x}^{\frac{2}{3}}-5{x}^{\frac{1}{3}}+6=0[/latex]

12. [latex]{x}^{\frac{7}{3}}-3{x}^{\frac{4}{3}}-4{x}^{\frac{1}{3}}=0[/latex]

For the following exercises, solve the following polynomial equations by grouping and factoring.

13. [latex]{x}^{3}+2{x}^{2}-x - 2=0[/latex]

14. [latex]3{x}^{3}-6{x}^{2}-27x+54=0[/latex]

15. [latex]4{y}^{3}-9y=0[/latex]

16. [latex]{x}^{3}+3{x}^{2}-25x - 75=0[/latex]

17. [latex]{m}^{3}+{m}^{2}-m - 1=0[/latex]

18. [latex]2{x}^{5}-14{x}^{3}=0[/latex]

19. [latex]5{x}^{3}+45x=2{x}^{2}+18[/latex]

For the following exercises, solve the radical equation. Be sure to check all solutions to eliminate extraneous solutions.

20. [latex]\sqrt{3x - 1}-2=0[/latex]

21. [latex]\sqrt{x - 7}=5[/latex]

22. [latex]\sqrt{x - 1}=x - 7[/latex]

23. [latex]\sqrt{3t+5}=7[/latex]

24. [latex]\sqrt{t+1}+9=7[/latex]

25. [latex]\sqrt{12-x}=x[/latex]

26. [latex]\sqrt{2x+3}-\sqrt{x+2}=2[/latex]

27. [latex]\sqrt{3x+7}+\sqrt{x+2}=1[/latex]

28. [latex]\sqrt{2x+3}-\sqrt{x+1}=1[/latex]

For the following exercises, solve the equation involving absolute value.

29. [latex]|3x - 4|=8[/latex]

30. [latex]|2x - 3|=-2[/latex]

31. [latex]|1 - 4x|-1=5[/latex]

32. [latex]|4x+1|-3=6[/latex]

33. [latex]|2x - 1|-7=-2[/latex]

34. [latex]|2x+1|-2=-3[/latex]

35. [latex]|x+5|=0[/latex]

36. [latex]-|2x+1|=-3[/latex]

For the following exercises, solve the equation by identifying the quadratic form. Use a substitute variable and find all real solutions by factoring.

37. [latex]{x}^{4}-10{x}^{2}+9=0[/latex]

38. [latex]4{\left(t - 1\right)}^{2}-9\left(t - 1\right)=-2[/latex]

39. [latex]{\left({x}^{2}-1\right)}^{2}+\left({x}^{2}-1\right)-12=0[/latex]

40. [latex]{\left(x+1\right)}^{2}-8\left(x+1\right)-9=0[/latex]

41. [latex]{\left(x - 3\right)}^{2}-4=0[/latex]

For the following exercises, solve for the unknown variable.

42. [latex]{x}^{-2}-{x}^{-1}-12=0[/latex]

43. [latex]\sqrt{{|x|}^{2}}=x[/latex]

44. [latex]{t}^{25}-{t}^{5}+1=0[/latex]

45. [latex]|{x}^{2}+2x - 36|=12[/latex]

For the following exercises, use the model for the period of a pendulum, [latex]T[/latex], such that [latex]T=2\pi \sqrt{\frac{L}{g}}[/latex], where the length of the pendulum is L and the acceleration due to gravity is [latex]g[/latex].

46. If the acceleration due to gravity is [latex]9.8\mathrm{m/}{\text{s}}^{2}[/latex] and the period equals 1 s, find the length to the nearest cm (100 cm = 1 m).

47. If the gravity is [latex]32\frac{\text{ft}}{{\text{s}}^{2}}[/latex] and the period equals 1 s, find the length to the nearest in. (12 in. = 1 ft). Round your answer to the nearest in.

For the following exercises, use a model for body surface area, BSA, such that [latex]BSA=\sqrt{\frac{wh}{3600}}[/latex], where w = weight in kg and h = height in cm.

48. Find the height of a 72-kg female to the nearest cm whose [latex]BSA=1.8[/latex].

49. Find the weight of a 177-cm male to the nearest kg whose [latex]BSA=2.1[/latex].