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### 4 1exponents

1. 1. Exponents Back to 123a-Home
2. 2. Exponents We write “1” times the quantity “A” repeatedly N times as AN
3. 3. Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
4. 4. Example A. 43 Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
5. 5. Example A. 43 = (4)(4)(4) = 64 Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
6. 6. Example A. 43 = (4)(4)(4) = 64 (xy)2 Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
7. 7. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
8. 8. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
9. 9. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
10. 10. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
11. 11. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
12. 12. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
13. 13. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
14. 14. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Multiply-Add Rule: ANAK =AN+K Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
15. 15. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
16. 16. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
17. 17. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
18. 18. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
19. 19. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 Rules of Exponents base exponent We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
20. 20. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 Rules of Exponents Divide–Subtract Rule: AN AK = AN – K We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
21. 21. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 Rules of Exponents Divide–Subtract Rule: Example C. AN AK = AN – K 56 52 We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
22. 22. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 Rules of Exponents Divide–Subtract Rule: Example C. AN AK = AN – K 56 52 = (5)(5)(5)(5)(5)(5) (5)(5) We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
23. 23. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 Rules of Exponents Divide–Subtract Rule: Example C. AN AK = AN – K 56 52 = (5)(5)(5)(5)(5)(5) (5)(5) We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
24. 24. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiply-Add Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 = x9y13 Rules of Exponents Divide–Subtract Rule: Example C. AN AK = AN – K 56 52 = (5)(5)(5)(5)(5)(5) (5)(5) = 56 – 2 = 54 We write “1” times the quantity “A” repeatedly N times as AN, i.e. 1 x A x A x A ….x A = AN repeated N times
25. 25. Exponents Power–Multiply Rule: (AN)K = ANK
26. 26. Example D. (34)5 Exponents Power–Multiply Rule: (AN)K = ANK
27. 27. Example D. (34)5 = (34)(34)(34)(34)(34) Exponents Power–Multiply Rule: (AN)K = ANK
28. 28. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 Exponents Power–Multiply Rule: (AN)K = ANK
29. 29. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Power–Multiply Rule: (AN)K = ANK
30. 30. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 A1 A1 Power–Multiply Rule: (AN)K = ANK
31. 31. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1A1 A1 Power–Multiply Rule: (AN)K = ANK
32. 32. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0A1 A1 Power–Multiply Rule: (AN)K = ANK
33. 33. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Power–Multiply Rule: (AN)K = ANK
34. 34. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 0-Power Rule: A0 = 1, A = 0 Power–Multiply Rule: (AN)K = ANK
35. 35. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = 1 AK A0 AK 0-Power Rule: A0 = 1, A = 0 Power–Multiply Rule: (AN)K = ANK
36. 36. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K1 AK A0 AK 0-Power Rule: A0 = 1, A = 0 Power–Multiply Rule: (AN)K = ANK
37. 37. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK 0-Power Rule: A0 = 1, A = 0 Power–Multiply Rule: (AN)K = ANK
38. 38. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK
39. 39. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
40. 40. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify a. 30 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
41. 41. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
42. 42. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify b. 3–2 a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
43. 43. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32b. 3–2 = a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
44. 44. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9b. 3–2 = = a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
45. 45. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 b. 3–2 = = a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
46. 46. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = b. 3–2 = = a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
47. 47. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = 1* 5 2 = 5 2 b. 3–2 = = a. 30 = 1 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
48. 48. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = 1* 5 2 = 5 2 b. 3–2 = = a. 30 = 1 In general ( )–Ka b = ( )K b a 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
49. 49. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = 1* 5 2 = 5 2 b. 3–2 = = a. 30 = 1 In general ( )–Ka b = ( )K b a d. ( )–22 5 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
50. 50. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = 1* 5 2 = 5 2 b. 3–2 = = a. 30 = 1 In general ( )–Ka b = ( )K b a d. ( )–22 5 = ( )25 2 0-Power Rule: A0 = 1, A = 0 , A = 0 Power–Multiply Rule: (AN)K = ANK The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
51. 51. Power–Multiply Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule. A1 A1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example D. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = 1* 5 2 = 5 2 b. 3–2 = = a. 30 = 1 In general ( )–Ka b = ( )K b a d. ( )–22 5 = ( )2 = 25 4 5 2 0-Power Rule: A0 = 1, A = 0 , A = 0 The reverse of multiplication is division, so 1 x A x A x …. x A = AK A 1 x1 x A 1 x .. x A 1 = A–K repeated K times
52. 52. e. 3–1 – 40 * 2–2 = Exponents
53. 53. e. 3–1 – 40 * 2–2 = 1 3 Exponents
54. 54. e. 3–1 – 40 * 2–2 = 1 3 – 1* Exponents
55. 55. e. 3–1 – 40 * 2–2 = 1 3 – 1* 1 22 Exponents
56. 56. e. 3–1 – 40 * 2–2 = 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12 Exponents
57. 57. e. 3–1 – 40 * 2–2 = 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12 Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents.
58. 58. e. 3–1 – 40 * 2–2 = 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12 Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.
59. 59. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. Example E. Simplify 3–2 x4 y–6 x–8 y 23 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
60. 60. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. Example E. Simplify 3–2 x4 y–6 x–8 y 23 3–2 x4 y–6 x–8 y23 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
61. 61. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. Example E. Simplify 3–2 x4 y–6 x–8 y 23 3–2 x4 y–6 x–8 y23 = 3–2 x4 x–8 y–6 y23 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
62. 62. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. = x4 – 8 y–6+23 Example E. Simplify 3–2 x4 y–6 x–8 y 23 3–2 x4 y–6 x–8 y23 = 3–2 x4 x–8 y–6 y23 1 9 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
63. 63. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. = x4 – 8 y–6+23 = x–4 y17 Example E. Simplify 3–2 x4 y–6 x–8 y 23 3–2 x4 y–6 x–8 y23 = 3–2 x4 x–8 y–6 y23 1 9 1 9 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
64. 64. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. = x4 – 8 y–6+23 = x–4 y17 = y17 Example E. Simplify 3–2 x4 y–6 x–8 y 23 3–2 x4 y–6 x–8 y23 = 3–2 x4 x–8 y–6 y23 1 9 1 9 1 9x4 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
65. 65. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of consolidating exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. = x4 – 8 y–6+23 = x–4 y17 = y17 = Example E. Simplify 3–2 x4 y–6 x–8 y 23 3–2 x4 y–6 x–8 y23 = 3–2 x4 x–8 y–6 y23 1 9 1 9 1 9x4 y17 9x4 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
66. 66. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3
67. 67. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3
68. 68. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 )
69. 69. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5
70. 70. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1
71. 71. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3
72. 72. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–5x–3(y–1x2)3
73. 73. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 (3x–2y3)–2 x2 3–5x–3(y–1x2)3
74. 74. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 = (3x–2y3)–2 x2 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6
75. 75. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 = = (3x–2y3)–2 x2 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6 3–2x6y–6 3–5x3y–3
76. 76. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 = = = 3–2 – (–5) x6 – 3 y–6 – (–3) (3x–2y3)–2 x2 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6 3–2x6y–6 3–5x3y–3
77. 77. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 = = = 3–2 – (–5) x6 – 3 y–6 – (–3) = 33 x3 y–3= (3x–2y3)–2 x2 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6 3–2x6y–6 3–5x3y–3
78. 78. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 = = = 3–2 – (–5) x6 – 3 y–6 – (–3) = 33 x3 y–3= 27 x3 (3x–2y3)–2 x2 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6 3–2x6y–6 3–5x3y–3 y3
79. 79. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3 = 23 – 6 x–8 – (–3 ) = 2–3 x–5 = 23 1 x5 1 * = 8x5 1 Example G. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 = 3–2x4y–6x2 3–5x–3y–3 x6 = = = 3–2 – (–5) x6 – 3 y–6 – (–3) = 33 x3 y–3= 27 x3 (3x–2y3)–2 x2 3–5x–3(y–1x2)3 3–5x–3x6y–3 3–2x4x2y–6 3–2x6y–6 3–5x3y–3 y3
80. 80. Exercise. A. Write the numbers without the negative exponents and compute the answers. 1. 2–1 2. –2–2 3. 2–3 4. (–3)–2 5. 3–3 6. 5–2 7. 4–3 8. 1 2 ( ) –3 9. 2 3 ( ) –1 10. 3 2 ( ) –2 11. 2–1* 3–2 12. 2–2+ 3–1 13. 2* 4–1– 50 * 3–1 14. 32 * 6–1– 6 * 2–3 15. 2–2* 3–1 + 80 * 2–1 B. Combine the exponents. Leave the answers in positive exponents–but do not reciprocate the negative exponents until the final step. 16. x3x5 17. x–3x5 18. x3x–5 19. x–3x–5 20. x4y2x3y–4 21. y–3x–2 y–4x4 22. 22x–3xy2x32–5 23. 32y–152–2x5y2x–9 24. 42x252–3y–34 x–41y–11 25. x2(x3)5 26. (x–3)–5x –6 27. x4(x3y–5) –3y–8
81. 81. x–8 x–3 B. Combine the exponents. Leave the answers in positive exponents–but do not reciprocate the negative exponents until the final step. 28. x8 x–329. x–8 x330. y6x–8 x–2y331. x6x–2y–8 y–3x–5y232. 2–3x6y–8 2–5y–5x233. 3–2y2x4 2–3x3y–234. 4–1(x3y–2)–2 2–3(y–5x2)–135. 6–2 y2(x4y–3)–1 9–1(x3y–2)–4y236. C. Combine the exponents as far as possible. 38. 232x 39. 3x+23x 40. ax–3ax+5 41. (b2)x+1b–x+3 42. e3e2x+1e–x 43. e3e2x+1e–x 44. How would you make sense of 23 ? 2