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Study Guides > MATH 1314: College Algebra

Section Exercises

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 932-{9}^{\frac{3}{2}} and keeps getting an ERROR message. What mistake is he or she probably making? 4. Explain why 2x+5=7|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. x23=16{x}^{\frac{2}{3}}=16 7. x34=27{x}^{\frac{3}{4}}=27 8. 2x12x14=02{x}^{\frac{1}{2}}-{x}^{\frac{1}{4}}=0 9. (x1)34=8{\left(x - 1\right)}^{\frac{3}{4}}=8 10. (x+1)23=4{\left(x+1\right)}^{\frac{2}{3}}=4 11. x235x13+6=0{x}^{\frac{2}{3}}-5{x}^{\frac{1}{3}}+6=0 12. x733x434x13=0{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. x3+2x2x2=0{x}^{3}+2{x}^{2}-x - 2=0 14. 3x36x227x+54=03{x}^{3}-6{x}^{2}-27x+54=0 15. 4y39y=04{y}^{3}-9y=0 16. x3+3x225x75=0{x}^{3}+3{x}^{2}-25x - 75=0 17. m3+m2m1=0{m}^{3}+{m}^{2}-m - 1=0 18. 2x514x3=02{x}^{5}-14{x}^{3}=0 19. 5x3+45x=2x2+185{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. 3x12=0\sqrt{3x - 1}-2=0 21. x7=5\sqrt{x - 7}=5 22. x1=x7\sqrt{x - 1}=x - 7 23. 3t+5=7\sqrt{3t+5}=7 24. t+1+9=7\sqrt{t+1}+9=7 25. 12x=x\sqrt{12-x}=x 26. 2x+3x+2=2\sqrt{2x+3}-\sqrt{x+2}=2 27. 3x+7+x+2=1\sqrt{3x+7}+\sqrt{x+2}=1 28. 2x+3x+1=1\sqrt{2x+3}-\sqrt{x+1}=1 For the following exercises, solve the equation involving absolute value. 29. 3x4=8|3x - 4|=8 30. 2x3=2|2x - 3|=-2 31. 14x1=5|1 - 4x|-1=5 32. 4x+13=6|4x+1|-3=6 33. 2x17=2|2x - 1|-7=-2 34. 2x+12=3|2x+1|-2=-3 35. x+5=0|x+5|=0 36. 2x+1=3-|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. x410x2+9=0{x}^{4}-10{x}^{2}+9=0 38. 4(t1)29(t1)=24{\left(t - 1\right)}^{2}-9\left(t - 1\right)=-2 39. (x21)2+(x21)12=0{\left({x}^{2}-1\right)}^{2}+\left({x}^{2}-1\right)-12=0 40. (x+1)28(x+1)9=0{\left(x+1\right)}^{2}-8\left(x+1\right)-9=0 41. (x3)24=0{\left(x - 3\right)}^{2}-4=0 For the following exercises, solve for the unknown variable. 42. x2x112=0{x}^{-2}-{x}^{-1}-12=0 43. x2=x\sqrt{{|x|}^{2}}=x 44. t25t5+1=0{t}^{25}-{t}^{5}+1=0 45. x2+2x36=12|{x}^{2}+2x - 36|=12 For the following exercises, use the model for the period of a pendulum, TT, such that T=2πLgT=2\pi \sqrt{\frac{L}{g}}, where the length of the pendulum is L and the acceleration due to gravity is gg. 46. If the acceleration due to gravity is 9.8m/s29.8\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 32fts232\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=wh3600BSA=\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=1.8BSA=1.8. 49. Find the weight of a 177-cm male to the nearest kg whose BSA=2.1BSA=2.1.

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