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Chapter 4

Chapter 4. Exponents and Polynomials. Chapter 4.1. The Rules of Exponents. The Product Rule. x a • x b. x a + b. =. 1. Multiply. a 7. a 5. •. a. a 7 + 5. a 12. •. a • a • a • a • a. a • a • a • a • a • a • a. Count how many you are multiplying.

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Chapter 4

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  1. Chapter 4 Exponents and Polynomials

  2. Chapter 4.1 The Rules of Exponents

  3. The Product Rule xa• xb xa +b =

  4. 1. Multiply. a7 a5 • a. a7 + 5 a12 • a • a • a • a • a a • a • a • a • a • a • a Count how many you are multiplying. Just add the exponents.

  5. 1. Multiply. b. w10 • w w10 + 1 w11 Add the exponents.

  6. 2. Simplify, if possible. a. x3 • x9 x3 + 9 x12 Same base, add the exponents.

  7. 2. Simplify, if possible. b. 37 • 34 37 + 4 311 Same base, add the exponents.

  8. 2. Simplify, if possible. c. a3 • b2 a3b2 Different bases, can’t use the product rule.

  9. 3. Multiply. a. (-a8)(a4) a8 + 4 (-1)(1) a12 - Multiply the coefficients. Add the exponents.

  10. 3. Multiply. b. (3y2)(-2y3) (3)(-2) y2 + 3 -6 y5 Multiply the coefficients. Add the exponents.

  11. 3. Multiply. c. (-4x3)(-5x2) (-4)(-5) x3 + 2 x5 20 Multiply the coefficients. Add the exponents.

  12. 4. Multiply. x x2 - y3 ( x 2 6 ) )( )( y y 1 3 (2)(-)(6) (y1 + 1 + 3) (x1 + 2 + 1) 2 1 -3 y5 x4 Multiply and simplify the coefficients. Add the exponents for x and then for y.

  13. The Product Rule xa• xb xa +b =

  14. The Quotient Rules xa xa xa 1 = xa –b if a > b 1. xa xb xb 2. = if b > a 3. = x0 = 1 xb –a

  15. The Quotient Rules xa = xa –b if a > b 1. xb

  16. 5. Divide. 1013 a. 107 1013 – 7 106 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 • 10 10 • 10 • 10 • 10 • 10 • 10 • 10 Count how many are crossed out. Just subtract the exponents.

  17. 5. Divide. b. x11 x x11 – 1 x10 Higher exponent in numerator. Subtract the exponents.

  18. 5. Divide. c. y18 y8 y18 – 8 y10 Higher exponent in numerator. Subtract the exponents.

  19. The Quotient Rules xa xa 1 = xa –b if a > b 1. xb xb 2. = if b > a xb –a

  20. 6. Divide. c3 a. 1 1 c4 c c4 – 3 c• c• c c• c• c• c The higher exponent is in the denominator.

  21. 6. Divide. b. 1 1031 1 1056 – 31 1056 1025 The higher exponent is in the denominator.

  22. 6. Divide. c. 1 1 z15 z6 z21 – 15 z21 The higher exponent is in the denominator.

  23. 7. Divide. x7 -7 a. 1 -21 x9 3x2 1 3 x9 – 7 Simplify. The higher exponent is in the denominator.

  24. 7. Divide. x11 15 b. -3 x4 -5 x11 – 4 1 -5x7 Simplify. The higher exponent is in the numerator.

  25. 7. Divide. x8 23 c. 1 46 x9 2x 1 2 x9 – 8 Simplify. The higher exponent is in the denominator.

  26. 8. Divide. x7 y9 a. y10 x7 y10 – 9 x7 y Can’t simplify. The higher exponent is in the denominator.

  27. 8. Divide. y6 12 x5 b. -24 x3 y8 x5 – 3 -1 2 y8 – 6 -x2 2y2 Simplify. The higher exponent is in the numerator. The higher exponent is in the denominator.

  28. The Quotient Rules xa xa xa 1 = xa –b if a > b 1. xa xb xb 2. = if b > a 3. = x0 = 1 xb –a

  29. 9. Divide. a. = 100 107 107 1 Same exponent.

  30. 9. Divide. 12 a4 b. 15 a4 4 5 Simplify. Same exponent.

  31. 10. Divide. -20 a3 b8 c4 a. -5b a3 b7 c5 28 7c b8 – 7 -5 7 c5 – 4 Simplify. Same exponent. The higher exponent is in the numerator. The higher exponent is in the denominator.

  32. 10. Divide. 5 x0 y6 b. 1 x4 y8 10 2x4y2 1 2 y8– 6 x4 Simplify. 0 exponent. The higher exponent is in the denominator.

  33. 11. Simplify. ( 3 )( ) a2 b4 -6 a b5 16a5b7 a3 -18 b9 16 a5 b7 b2 -9 Multiply. 8 a2 Add the exponents in the numerator. Simplify. Subtract the exponents.

  34. The Quotient Rules xa xa xa 1 = xa –b if a > b 1. xa xb xb 2. = if b > a 3. = x0 = 1 xb –a

  35. The Power Rules xa (xa)b xa •b = yb = (xayb)c xa •c yb•c xa • c = yb• c ( ) c

  36. 12. Simplify. a. (a4)3 a4 • 3 a12 (a4)(a4)(a4) Can write it three times. Add 4 three timesor multiply the exponents.

  37. 12. Simplify. b. (105)2 105• 2 1010 Multiply the exponents.

  38. 12. Simplify. c. (-1)15 -1 Multiply -1 an odd number (15) of times.

  39. 13. Simplify. a. (3xy)3 (3)3 x1 • 3 y1 • 3 27 x3 y3 Keep 3 in the parentheses. Multiply the exponents. Evaluate each.

  40. 13. Simplify. b. (yz)37 y1 • 37 z1 • 37 y37 z37 Multiply the exponents. Evaluate each.

  41. 13. Simplify. c. (-3x3)2 (-3)2 x3• 2 9 x6 Keep -3 in the parentheses. Multiply the exponents. Evaluate each.

  42. 14. Simplify. x a. ( )3 5 x3 (5)3 x3 125 Multiply the exponents. Keep 5 in the parentheses. Evaluate.

  43. 14. Simplify. b. ( ) 4 a 2 a ) b ( 6 a2 16 a6 b6 16 a4 b6 Evaluate. Multiply exponents. Use quotient rule and subtract exponents.

  44. 15. Simplify. -2 x3 y0 z ( )5 x z2 4 -1 x2 ( ) 5 z 2 - x10 z5 32 Work inside parentheses. Simplify and use quotient rules. Use power rule and evaluate.

  45. The Power Rules xa (xa)b xa •b = yb = (xayb)c xa •c yb•c xa • c = yb• c ( ) c

  46. Chapter 4.1 The Rules of Exponents

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