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- 1. Exponents Back to 123a-Home
- 2. Exponents We write “1” times the quantity “A” repeatedly N times as AN
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. Exponents Power–Multiply Rule: (AN)K = ANK
- 26. Example D. (34)5 Exponents Power–Multiply Rule: (AN)K = ANK
- 27. Example D. (34)5 = (34)(34)(34)(34)(34) Exponents Power–Multiply Rule: (AN)K = ANK
- 28. Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 Exponents Power–Multiply Rule: (AN)K = ANK
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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 consolidating 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 consolidating 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 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. 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. 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. 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. 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. 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. 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. Exponents Example F. Simplify using the rules for exponents. Leave the answer in positive exponents only. 23x–8 26 x–3
- 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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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