Review of Exponents Quiz Review 2 Solution Key

HPC Review of Exponents - Quiz Review Name: Key Simplify 1. (-5)^2 = 25 2. -6^2 = -36 3. -5^-3 = -1/125 4. (2^-2)^3 = 1/64 5. (a^-6)^-1/6 = a 6. (2^-3 + 3^-2)^-2 = 13^2/36 7. (a^-2 + b^-2)^-1 = a^2b^2/(a^2 + b^2) 8. (36^1/2)^-1/2 = 1/6 9. (32x^10)^-1/5 = 1/(2x^2) 10. (4x^3)^1/2 + (4x^3)^-3/2 = 2x^(3/2) + 1/(4x^9/2) 11. w^x * w^y * w^z = w^(x+y+z) 12. 6a^2 - 3a^-4 / 2a^-4 = 3a^6 - 3/2 13. (3n^2)^-3 * (3n^5)^7 = 81n^29 14. x^(1/2) * x^(-3/2) = x^-1 15. (8n^3)^(1/3) - (2n^4)^(1/2) = 2n - 2n^2 16. (√3x)^-2 = 1/(3x) 17. (√a * √b)/(√a^3 * b^2/3) = 1/(a^(1/3) * b^(1/6)) 18. x^(1/2) * x^(-1/2) = 1 19. -8x^1/2 + 8x^-1/2 = -8√x + 8/√x 20. -x^(1/2)y - 5xy^(1/2) = -xy^(1/2) - 5xy^(1/2) = -6xy^(1/2) 21. (4c^3)^-2 * (4c^2)^3 = 4 22. (5^-1 - 5^-2) / (5 - 1 + 5^-2) = 2/3 23. (4x^-3 - 16√x^3) / (-2x^3) = -2/x^6 + 8/x^(9/2) 24. (9x^4)^(1/2) * (27x^2)^(4/3) = 3x^5 Solve. 25. 2^(2x+1) - 17*2^x + 8 = 0 Let u = 2^x 2u^2 - 17u + 8 = 0 (2u - 1)(u - 8) = 0 u = 1/2, 8 2^x = 1/2, 2^x = 8 x = -1, 3 26. 4^(2x) - 10*4^x + 16 = 0 Let u = 4^x u^2 - 10u + 16 = 0 (u - 8)(u - 2) = 0 u = 8, 2 4^x = 8, 4^x = 2 x = 3/2, 1/2 27. 2^(2x+1) - 9*2^x + 4 = 0 Let u = 2^x 2u^2 - 9u + 4 = 0 (2u - 1)(u - 4) = 0 u = 1/2, 4 2^x = 1/2, 2^x = 4 x = -1, 2 28. 3^(2x) - 26*3^x - 9 = 0 Let u = 3^x u^2 - 26u - 9 = 0 (3u + 1)(u - 9) = 0 u = -1/3, u = 9 3^x = -1/3, 3^x = 9 x = 2 29. 2^(6-x) = 4^(2+x) 2^(6-x) = (2^2)^(2+x) 2^(6-x) = 2^(4+2x) 6 - x = 4 + 2x 2 = 3x x = 2/3 30. 8^(x-1) = 2^(x+1) (2^3)^(x-1) = 2^(x+1) 2^(3x-3) = 2^(x+1) 3x - 3 = x + 1 2x = 4 x = 2 31. 3^(2x) = 81 3^(2x) = 3^4 2x = 4 x = 2 32. x^(x^2) = 256 x^(x^2) = 4^4 x^2 = 4 x = ±2 33. 5(x-3)^-2 = 125 (x-3)^(-2) = 25 x - 3 = 25^(-1/2) x - 3 = ±1/5 x = 16/5, 14/5 34. -3(x+6)^-2 + 1 = -26 -3(x+6)^(-2) = -27 (x+6)^(-2) = 9 x + 6 = 9^(-1/2) x + 6 = ±1/3 x = -17/3, -19/3 35. √(x^3) = 1024 x^(3/2) = 1024 x = (1024)^(2/3) = 64 36. (5x)^-4 = 100000000 5x = (10^-8)^(-1/4) 5x = ±100 x = ±20 37. (x+1)^(-1/3) = -32 x + 1 = (-32)^(-3/5) x + 1 = 1/8 x = -7/8 38. 1/(a^(3x)) = 4/6^x 4 * 8^(3x) = 8^x 2^2 * (2^3)^(3x) = 2^x 2^(2+9x) = 2^x 2 + 9x = x 8 = -8x x = -1 41. Find an exponential function of the form f(x) = ab^x with the given information. a. f(0) = 3; f(-3) = 8/3 3 = ab^0 3 = a 8/3 = 3b^-3 b^-3 = 8/9 b = (8/9)^(-1/3) b = 3/2 y = 3(3/2)^x b. f(0) = 4; f(4) = 1/4 4 = ab^0 4 = a 1/4 = 4b^4 b^4 = 1/16 b = 1/2 (since b > 0) y = 4(1/2)^x 42. At a 6.5% APR, approximately how long will it take to double an investment? Rule of 72: 72 / 6.5 = 11.08 years 43. At a 4.25% APR, approximately how long will it take to double an investment? Rule of 72: 72 / 4.25 = 16.94 years 44. The cost of a pen has been increasing 4% each year. It now sells for $75. a. What will it sell for in 5 years? 75(1.03)^5 = $86.95 b. What did it sell for 5 years ago? 75(1.03)^(-5) = $64.70 45. The cost of a computer has been decreasing 12% every five years. It now sells for $2425. a. What will it sell for in 15 years? 2425(1-0.12)^(15/5) = $1125.96 b. What will it sell for in 3 years? 2425(1-0.12)^(3/5) = $2245.96 46. A radioactive material has a half-life of 20 years. If 18 grams are present now, a. How much will be present in 80 years? 18(1/2)^(80/20) 1   47. An investment of $500 is made at a 7.25% APR. Find the amount present after: a. 1 year, compounded semi-annually: 500(1 + 0.0725/2)^(2*1) = $536.91 b. 3 years, compounded quarterly: 500(1 + 0.0725/4)^(4*3) = $620.27 c. 3 years, compounded continuously: 500(e)^(0.0725*3) = $621.48 48. After a period of a year, an investment of $300 has increased to $334.39. The interest has been compounded quarterly. a. What is the effective annual yield? (334.39 - 300) / 300 = 11.4633% b. What is the APR on the investment? 334.39 = 300(1 + r/4)^(4*1) => r = 11.0011% 49. What is the effective annual yield with an APR of 9% in which interest is compounded: a. Daily? (1 + 0.09/365)^(365*1) - 1 = 0.094162 ≈ 9.4162% b. Continuously? e^(0.09*1) - 1 = 0.094174 ≈ 9.4174% 50. If the price of an item has increased $45 over the last two years and it now costs $200, what is the average percent increase per year? 200 = 155(1+r)^2 => r ≈ 13.59% 51. A gallon of gas was $1.99 two years ago. Now it costs $2.19. To the nearest tenth of a percent, what has been the annual rate of increase? 2.19 = 1.99(1+r)^2 => r ≈ 4.9% 52. A bacteria colony triples every 5 days. The initial amount of bacteria is 4.5 mg. How much will there be in 16 days? A = A₀(3)^(t/5) 4.5(3)^(16/5) ≈ 151.36 mg **Factor** 53. (x-3)^(3/2) - x(x-3)^(1/2) (x-3)^(1/2)[(x-3) - x] = 3(x-3)^(1/2) 54. (3x-2)^(2/3) - 4(3x-2)^(-1/3) (3x-2)^(-1/3)[(3x-2) - 4] = (3x-2)^(-1/3)(3x-6) = 3(x-2)(3x-2)^(-1/3) 55. (x²+1)^(-3/2) - (x²+1)^(-1/2) (x²+1)^(-1/2)[(x²+1)^(-1) - 1] = -(x²+1)^(-1/2)(x²) 56. (x²-5)^(1/2) - x²(x²-5)^(-1/2) (x²-5)^(-1/2)[(x²-5) - x²] = -5(x²-5)^(-1/2)