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 $-{9}^{\frac{3}{2}}$ and keeps getting an ERROR message. What mistake is he or she probably making?

4. Explain why $|2x+5|=-7$ 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. ${x}^{\frac{2}{3}}=16$

7. ${x}^{\frac{3}{4}}=27$

8. $2{x}^{\frac{1}{2}}-{x}^{\frac{1}{4}}=0$

9. ${\left(x - 1\right)}^{\frac{3}{4}}=8$

10. ${\left(x+1\right)}^{\frac{2}{3}}=4$

11. ${x}^{\frac{2}{3}}-5{x}^{\frac{1}{3}}+6=0$

12. ${x}^{\frac{7}{3}}-3{x}^{\frac{4}{3}}-4{x}^{\frac{1}{3}}=0$

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

13. ${x}^{3}+2{x}^{2}-x - 2=0$

14. $3{x}^{3}-6{x}^{2}-27x+54=0$

15. $4{y}^{3}-9y=0$

16. ${x}^{3}+3{x}^{2}-25x - 75=0$

17. ${m}^{3}+{m}^{2}-m - 1=0$

18. $2{x}^{5}-14{x}^{3}=0$

19. $5{x}^{3}+45x=2{x}^{2}+18$

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

20. $\sqrt{3x - 1}-2=0$

21. $\sqrt{x - 7}=5$

22. $\sqrt{x - 1}=x - 7$

23. $\sqrt{3t+5}=7$

24. $\sqrt{t+1}+9=7$

25. $\sqrt{12-x}=x$

26. $\sqrt{2x+3}-\sqrt{x+2}=2$

27. $\sqrt{3x+7}+\sqrt{x+2}=1$

28. $\sqrt{2x+3}-\sqrt{x+1}=1$

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

29. $|3x - 4|=8$

30. $|2x - 3|=-2$

31. $|1 - 4x|-1=5$

32. $|4x+1|-3=6$

33. $|2x - 1|-7=-2$

34. $|2x+1|-2=-3$

35. $|x+5|=0$

36. $-|2x+1|=-3$

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

37. ${x}^{4}-10{x}^{2}+9=0$

38. $4{\left(t - 1\right)}^{2}-9\left(t - 1\right)=-2$

39. ${\left({x}^{2}-1\right)}^{2}+\left({x}^{2}-1\right)-12=0$

40. ${\left(x+1\right)}^{2}-8\left(x+1\right)-9=0$

41. ${\left(x - 3\right)}^{2}-4=0$

For the following exercises, solve for the unknown variable.

42. ${x}^{-2}-{x}^{-1}-12=0$

43. $\sqrt{{|x|}^{2}}=x$

44. ${t}^{25}-{t}^{5}+1=0$

45. $|{x}^{2}+2x - 36|=12$

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

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

47. If the gravity is $32\frac{\text{ft}}{{\text{s}}^{2}}$ 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 $BSA=\sqrt{\frac{wh}{3600}}$, where w = weight in kg and h = height in cm.

48. Find the height of a 72-kg female to the nearest cm whose $BSA=1.8$.

49. Find the weight of a 177-cm male to the nearest kg whose $BSA=2.1$.