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### 2.1 reviews of exponents and the power functions

1. 1. Reviews of Exponents and the Power Functions
2. 2. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations.
3. 3. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN.
4. 4. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN N times
5. 5. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base
6. 6. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents
7. 7. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K
8. 8. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K For example x9x5 = x14
9. 9. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K For example x9x5 = x14 AN = AN – K Division Rule: AK
10. 10. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K For example x9x5 = x14 AN = AN – K Division Rule: AK x9 = x9–5 = x4 For example 5 x
11. 11. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K For example x9x5 = x14 AN = AN – K Division Rule: AK x9 = x9–5 = x4 For example 5 x Power Rule: (AN)K = ANK
12. 12. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K For example x9x5 = x14 AN = AN – K Division Rule: AK x9 = x9–5 = x4 For example 5 x Power Rule: (AN)K = ANK For example (x9)5 = x45
13. 13. Reviews of Exponents and the Power Functions Let’s review the basics of exponential notations. The quantity A multiplied to itself N times is written as AN. A x A x A ….x A = AN exponent N times base Rules of Exponents Multiplication Rule: ANAK =AN+K For example x9x5 = x14 AN = AN – K Division Rule: AK x9 = x9–5 = x4 For example 5 x Power Rule: (AN)K = ANK For example (x9)5 = x45 We extend the definition of exponents to negative and fractional exponents in the following manner.
14. 14. Reviews of Exponents and the Power Functions A1 = 1 Since A1
15. 15. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0 Since A1
16. 16. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1
17. 17. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1
18. 18. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 Since AK AK
19. 19. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, Since AK AK
20. 20. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A
21. 21. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k Since ( A )k = A = (A1/k )k,
22. 22. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A.
23. 23. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A.
24. 24. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last.
25. 25. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = b. 91/2 = c. 9 –3/2 =
26. 26. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = 12 = 1 9 81 b. 91/2 = c. 9 –3/2 =
27. 27. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = 12 = 1 9 81 b. 91/2 = √9 = 3 c. 9 –3/2 =
28. 28. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = 12 = 1 9 81 b. 91/2 = √9 = 3 c. 9 –3/2 =
29. 29. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = 12 = 1 9 81 pull the numerator b. 91/2 = √9 = 3 outside c. 9 –3/2 = (9½)–3
30. 30. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = 12 = 1 9 81 pull the numerator b. 91/2 = √9 = 3 outside c. 9 –3/2 = (9½)–3 = 3–3
31. 31. Reviews of Exponents and the Power Functions A1 = 1 = A1 – 1 = A0, we obtain the 0-power Rule. Since A1 0-Power Rule: A0 = 1 1 = A0 = A0 – K = A–K, we get the Negative Power Rule. Since AK AK Negative Power Rule: A–K = 1 K A k k Since ( A )k =A= (A1/k )k, hence A1/k = A. k Fractional Powers: A1/k = A. For a general fractional exponent, we interpret the operations step by step by doing the numerator of the exponent last. Example A. Simplify. a. 9–2 = 12 = 1 9 81 pull the numerator b. 91/2 = √9 = 3 outside c. 9 –3/2 = (9½)–3 = 3–3 = 13 = 1 3 27
32. 32. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical.
33. 33. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100For example, 101.22 = 10
34. 34. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100For example, 101.22 100 = 10 =( 10 )122 16.59586….
35. 35. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100For example, 101.22 = 10 =( 10 )122 16.59586…. 100For a real-number-exponent, we approximate the real numberwith fractions and use the fractional power to approximate theexact answer.
36. 36. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100 100For example, 101.22= 10 =( 10 )122 16.59586….For a real-number-exponent, we approximate the real numberwith fractions and use the fractional power to approximate theexact answer.Example B. 3.14159.. 3.1 3.14 3.141 3.1415 10
37. 37. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100 100For example, 101.22= 10 =( 10 )122 16.59586….For a real-number-exponent, we approximate the real numberwith fractions and use the fractional power to approximate theexact answer.Example B. 3.14159.. 3.1 3.14 3.141 3.1415 31 314 3141 31415 10 10 10 10 100 10 1000 10 10000
38. 38. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100 100For example, 101.22= 10 =( 10 )122 16.59586….For a real-number-exponent, we approximate the real numberwith fractions and use the fractional power to approximate theexact answer.Example B. 3.14159.. 3.1 3.14 3.141 3.1415 31 314 3141 31415 10 10 10 10 100 10 1000 10 10000 1385.45..
39. 39. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100For example, = 10 =( 10 )122 16.59586…. 101.22 100For a real-number-exponent, we approximate the real numberwith fractions and use the fractional power to approximate theexact answer.Example B. 3.14159.. 3.1 3.14 3.141 3.1415 31 314 3141 31415 10 10 10 10 100 10 1000 10 10000 1385.45..It’s important to know that all the decimal values representingsuch expressions are approximations.
40. 40. Reviews of Exponents and the Power FunctionsFor a decimal-exponent, we write the decimal as a fraction andview the exponent as a radical. 122 100 100For example, 101.22 = 10 =( 10 )122 16.59586….For a real-number-exponent, we approximate the real numberwith fractions and use the fractional power to approximate theexact answer.Example B. 3.14159.. 3.1 3.14 3.141 3.1415 31 314 3141 31415 10 10 10 10 100 10 1000 10 10000 1385.45..It’s important to know that all the decimal values representingsuch expressions are approximations. All the decimal outputsof such expressions are only finitely many digits of an infinitelylong expansions with no seemingly discernable patterns.
41. 41. Power EquationsThe solution to the equationx 3 = –8 isx=
42. 42. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.
43. 43. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.Using fractional exponent notation, we write these steps as
44. 44. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.Using fractional exponent notation, we write these steps asif x3 = –8 thenx = (–8)1/3 = –2.
45. 45. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.Using fractional exponent notation, we write these steps asif x3 = –8 then The reciprocal powerx = (–8)1/3 = –2.
46. 46. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.Using fractional exponent notation, we write these steps asif x3 = –8 then The reciprocal powerx = (–8)1/3 = –2.(Rational) Power equations are equations of the type xP/Q = c.To solve them, we take the reciprocal power,
47. 47. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.Using fractional exponent notation, we write these steps asif x3 = –8 then The reciprocal powerx = (–8)1/3 = –2.(Rational) Power equations are equations of the type xP/Q = c.To solve them, we take the reciprocal power, that is,if xP/Q = c, The reciprocal powerthen x = ( ) cQ/P
48. 48. Power EquationsThe solution to the equationx 3 = –8 is 3x = √–8 = –2.Using fractional exponent notation, we write these steps asif x3 = –8 then The reciprocal powerx = (–8)1/3 = –2.(Rational) Power equations are equations of the type xP/Q = c.To solve them, we take the reciprocal power, that is,if xP/Q = c, The reciprocal powerthen x = ( ) cQ/PNote that xP/Q may not exist, or that we that we may getboth ( ) xP/Q as solutions.
49. 49. Power EquationsExample C. Solve for the real solutions.a. x3 = 64b. x2 = 64c. x2 = –64d. x –2/3 = 64
50. 50. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or thatb. x2 = 64c. x2 = –64d. x –2/3 = 64
51. 51. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.b. x2 = 64c. x2 = –64d. x –2/3 = 64
52. 52. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64c. x2 = –64d. x –2/3 = 64
53. 53. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64 x = 641/2c. x2 = –64d. x –2/3 = 64
54. 54. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64 x = 641/2 or actually that x = √64 = 8.c. x2 = –64d. x –2/3 = 64
55. 55. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64 x = 641/2 or actually that x = √64 = 8.We note that both 8 are solutions.c. x2 = –64d. x –2/3 = 64
56. 56. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64 x = 641/2 or actually that x = √64 = 8.We note that both 8 are solutions.c. x2 = –64 x = (–64)1/2 which is UDF as the return on most calculators.d. x –2/3 = 64
57. 57. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64 x = 641/2 or actually that x = √64 = 8.We note that both 8 are solutions.c. x2 = –64 x = (–64)1/2 which is UDF as the return on most calculators.d. x –2/3 = 64 x = 64–3/2
58. 58. Power EquationsExample C. Solve for the real solutions.a. x3 = 64 x = 641/3 or that 3 x = √64 = 4.We note that this is the only solution.b. x2 = 64 x = 641/2 or actually that x = √64 = 8.We note that both 8 are solutions.c. x2 = –64 x = (–64)1/2 which is UDF as the return on most calculators.d. x –2/3 = 64 x = 64–3/2 x = (√64)–3 = 8–3 = 1/512.
59. 59. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1
60. 60. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8
61. 61. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4
62. 62. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2
63. 63. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.
64. 64. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.We note that actually both 8 are solutions.
65. 65. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.We note that actually both 8 are solutions.The general power equations are equations of the type xr = c.
66. 66. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.We note that actually both 8 are solutions.The general power equations are equations of the type xr = c.To solve them, we take the reciprocal power,
67. 67. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.We note that actually both 8 are solutions.The general power equations are equations of the type xr = c.To solve them, we take the reciprocal power, that is,if xr = c, The reciprocal powerthen x = ( ) c1/r (These are calculator problems.)
68. 68. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.We note that actually both 8 are solutions.The general power equations are equations of the type xr = c.To solve them, we take the reciprocal power, that is,if xr = c, The reciprocal powerthen x = ( ) c1/r (These are calculator problems.)Example D. Solve for the real solutions.x1/√2 = 3
69. 69. Power EquationsFor the linear form of the power equations, we solve for thepower term first , then apply the reciprocal power to find x.e. 2x2/3 – 7 = 1 2x2/3 = 8 x2/3 = 4 x = 43/2 x = (√4)3 = 8.We note that actually both 8 are solutions.The general power equations are equations of the type xr = c.To solve them, we take the reciprocal power, that is,if xr = c, The reciprocal powerthen x = ( ) c1/r (These are calculator problems.)ExampleD. Solve for the real solutions.x1/√2 = 3 x = 3√2 ≈ 4.73
70. 70. Fractional ExponentsThe following problems are from math123 b.Exercise A. Write the expressions in radicals and simplify.(no calculator) 1. 41/2 2. 9 1/2 3. 161/2 4. 251/2 5. 36 1/2 6. 36 –1/2 7. 491/2 8. 641/2 9. 811/2 x 1/2 11. ( 4 ) 9 1/2 12. ( 25 )10. (100x)1/213. (–8)1/3 14. ( –1 ) 1/3 64 15. 1251/316. 4–1/2 17. 9–1/2 18. 16–1/2 19. 25–1/220. 36–1/2 21. 49–1/2 22. 64–1/2 23. 81–1/224.( x )–1/2 25. ( x ) –1/2 26. ( 9 ) –1/2 100 4 2527. ( –1 ) –1/3 28. 125–1/3 29. 163/2 30. 253/2 6431. ( –1 ) –2/3 64 32. 125–2/3
71. 71. Power EquationsExercise B. (no calculator) 3. x = –8 3 1. x3 = 8 2. x3 = –1 4. x3 = –64 5. x2 = 64 6. x-2 = 9 7. x–2 = 1/25 8. x–2 = 4/9 9. x–3 = –1/64Exercise C. (no calculator)10. x1/2 = 4 11. x1/3 = –2 12. x1/2 = 413. x–1/2 = 4 14. x–1/2 = 4 15. x–1/3 = –216. x3/2 = 8 17. x2/3 = 9 18. x–2/3 = 419. x–3/2 = 1/8 20. x–3/2 = 1/64 21. x–2/3 = 1/64 22. x–3/2 = 1/64Exercise D. (with calculator) 23. x0.23 = 3.5 24. x–1.72 = 6.77 25. 5 + 2x2.1 = 6.4 26. –5.1x0.23 + 3.2 = –3.5 27. 1.2 + 5.6x√3.2 = 7.8