4 1exponents

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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

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