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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 $BSA=2.1$.