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# 1exponents

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• 1. Exponents Frank Ma © 2011
• 2. Exponents We write the quantity A multiplied to itself N times as AN,
• 3. Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 4. base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 5. Example A. 43 base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 6. Example A. 43 = (4)(4)(4) = 64 base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 7. Example A. 43 = (4)(4)(4) = 64 (xy)2 base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 8. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 9. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 10. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 11. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 12. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 13. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 14. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Rules of Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 15. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication Rule: ANAK =AN+K Rules of Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 16. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 Rules of Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 17. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) Rules of Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 18. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 Rules of Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 19. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication Rule: ANAK =AN+K Example B. a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57 b. x5y7x4y6 = x5x4y7y6 Rules of Exponents We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 20. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication 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 We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 21. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication 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 Division Rule: AN AK = AN – K We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 22. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication 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 Division Rule: Example C. AN AK = AN – K 56 52 We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 23. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication 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 Division Rule: Example C. AN AK = AN – K 56 52 = (5)(5)(5)(5)(5)(5) (5)(5) We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 24. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication 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 Division Rule: Example C. AN AK = AN – K 56 52 = (5)(5)(5)(5)(5)(5) (5)(5) We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 25. Example A. 43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2 xy2 = (x)(yy) –x2 = –(xx) base exponent Exponents Multiplication 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 Division Rule: Example C. AN AK = AN – K 56 52 = (5)(5)(5)(5)(5)(5) (5)(5) = 56 – 2 = 54 We write the quantity A multiplied to itself N times as AN, i.e. A x A x A ….x A = AN
• 26. Power Rule: (AN)K = ANK Exponents
• 27. Power Rule: (AN)K = ANK Example D. (34)5 Exponents A1
• 28. Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34) Exponents
• 29. Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 Exponents
• 30. Power Rule: (AN)K = ANK Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320 Exponents
• 31. Power 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 A1
• 32. Power 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 – 1A1 A1
• 33. Power 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 = A0A1 A1
• 34. Power 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
• 35. Power 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 0-Power Rule: A0 = 1
• 36. Power 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 0-Power Rule: A0 = 1 Since = 1 AK A0 AK
• 37. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K1 AK A0 AK
• 38. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK
• 39. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK
• 40. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify a. 30
• 41. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify a. 30 = 1
• 42. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify b. 3–2 a. 30 = 1
• 43. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify 1 32b. 3–2 = a. 30 = 1
• 44. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify 1 32 1 9b. 3–2 = = a. 30 = 1
• 45. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify 1 32 1 9 c. ( )–12 5 b. 3–2 = = a. 30 = 1
• 46. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = b. 3–2 = = a. 30 = 1
• 47. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. Simplify 1 32 1 9 c. ( )–12 5 = 1 2/5 = 1* 5 2 = 5 2 b. 3–2 = = a. 30 = 1
• 48. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. 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
• 49. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. 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
• 50. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. 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
• 51. Power 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 0-Power Rule: A0 = 1 Since = = A0 – K = A–K, we get the negative-power Rule. 1 AK A0 AK Negative-Power Rule: A–K = 1 AK Example E. 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
• 52. e. 3–1 – 40 * 2–2 = Exponents
• 53. e. 3–1 – 40 * 2–2 = 1 3 Exponents
• 54. e. 3–1 – 40 * 2–2 = 1 3 – 1* Exponents
• 55. e. 3–1 – 40 * 2–2 = 1 3 – 1* 1 22 Exponents
• 56. e. 3–1 – 40 * 2–2 = 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12 Exponents
• 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 collecting exponents, we do not reciprocate the negative exponents.
• 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 collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first.
• 59. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. Example F. Simplify 3–2 x4 y–6 x–8 y 23 1 3 – 1* 1 22 = 1 3 – 1 4 = 1 12
• 60. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. Example F. 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. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting exponents, we do not reciprocate the negative exponents. Instead we add or subtract them using the multiplication and division rules first. Example F. 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. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting 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 F. 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. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting 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 F. 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. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting 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 F. 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. e. 3–1 – 40 * 2–2 = Exponents Although the negative power means to reciprocate, for problems of collecting 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 F. 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. Exponents Example G. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3
• 67. Exponents Example G. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3 23x–8 26x–3
• 68. Exponents Example G. 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. Exponents Example G. 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. Exponents Example G. 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. Exponents Example G. 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 H. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3
• 72. Exponents Example G. 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 H. Simplify (3x–2y3)–2 x2 3–5x–3(y–1x2)3 (3x–2y3)–2 x2 3–5x–3(y–1x2)3
• 73. Exponents Example G. 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 H. 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. Exponents Example G. 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 H. 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. Exponents Example G. 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 H. 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. Exponents Example G. 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 H. 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. Exponents Example G. 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 H. 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. Exponents Example G. 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 H. 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. Ex. 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 Ex. 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 Exponents
• 80. 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 much 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