1. Chapter 4 Exponents and Polynomials

2. The Rules of Exponents Chapter 4.1

3. The Product Rule x a x b = x a + b

4. 1. Multiply. a. a 12 a a a a a a a a a a a a a 7 + 5 a 5 a 7 Count how many you are multiplying. Just add the exponents.

5. 1. Multiply. b.w 10 w w 10 + 1 w 11 Add the exponents.

6. 2. Simplify, if possible. a.x 3 x 9 x 3 + 9 x 12 Same base, add the exponents.

7. 2. Simplify, if possible. b.3 7 3 4 3 7 + 4 3 11 Same base, add the exponents.

8. 2. Simplify, if possible. c.a 3 b 2 a3b2a3b2 Different bases, can’t use the product rule.

9. 3. Multiply. a.( - a 8 )(a 4 ) - a 8 + 4 a 12 Multiply the coefficients. Add the exponents. ( - 1)(1)

10. 3. Multiply. b.(3y 2 )( - 2y 3 ) - 6 y 2 + 3 y 5 Multiply the coefficients. Add the exponents. (3)( - 2)

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

12. ( (2)( -  )(6) - 3 (x 1 + 2 + 1 ) (y 1 + 1 + 3 ) 1 2 1 3 4. Multiply. x 4 y 5 Multiply and simplify the coefficients. Add the exponents for x and then for y. 2 )( y x 6 y x2x2 -- y3y3 x )

13. x a x b = x a + b The Product Rule

14. xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 xaxa xaxa = x0x0 = 1 1. 2. 3. The Quotient Rules

15. xbxb xaxa = x a – b if a > b 1.

16. a. 10 6 10 13 10 7 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10 13 – 7 5. Divide. Count how many are crossed out. Just subtract the exponents.

17. b. x 11 – 1 x 10 x x 11 5. Divide. Higher exponent in numerator. Subtract the exponents.

18. c. y 18 – 8 y 10 y8y8 y 18 5. Divide. Higher exponent in numerator. Subtract the exponents.

19. The Quotient Rules xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 1. 2.

20. a. c4c4 c3c3 c 4 – 3 1 c 1 c c c c c c c 6. Divide. The higher exponent is in the denominator.

21. b. 10 56 10 31 10 56 – 31 1 10 25 1 6. Divide. The higher exponent is in the denominator.

22. c. z 21 z 15 z 21 – 15 1 z6z6 1 6. Divide. The higher exponent is in the denominator.

23. a. -21 -7 3 1 3x 2 1 x 9 – 7 7. Divide. Simplify. x7x7 x9x9 The higher exponent is in the denominator.

24. b. -3 15 1 -5 x 11 – 4 7. Divide. Simplify. x 11 x4x4 The higher exponent is in the numerator. - 5x 7

25. c. 46 23 2 1 x 9 – 8 7. Divide. Simplify. x8x8 x9x9 The higher exponent is in the denominator. 2x 1

26. a. y 10 x7x7 x 7 y x7x7 y 10 – 9 8. Divide. y9y9 The higher exponent is in the denominator. Can’t simplify.

27. b. -24 12 2 -1 2y 2 -x 2 x 5 – 3 y 8 – 6 8. Divide. x5x5 y6y6 x3x3 y8y8 The higher exponent is in the numerator. Simplify. The higher exponent is in the denominator.

28. The Quotient Rules xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 xaxa xaxa = x0x0 = 1 1. 2. 3.

29. a. 1 10 7 = 10 0 9. Divide. Same exponent.

30. b. 15 12 5 4 9. Divide. a4a4 a4a4 Simplify. Same exponent.

31. a. 28 -20 7 -5 7c -5b b 8 – 7 c 5 – 4 10. Divide. a3a3 b8b8 c4c4 a3a3 b7b7 c5c5 The higher exponent is in the numerator. Simplify. The higher exponent is in the denominator. Same exponent.

32. x0x0 b. 10 5 2 1 x 4 y 8 – 6 10. Divide. y6y6 x4x4 y8y8 The higher exponent is in the denominator. Simplify. 0 exponent. 2x 4 y 2 1

33. 16a 5 b 7 ( -18 a 3 16 b 9 -9 8 a 2 b 2 11. Simplify. )( ab5b5 -6 3a2a2 b4b4 ) Simplify. Multiply. Subtract the exponents. Add the exponents in the numerator. a 5 b 7

34. The Quotient Rules xbxb xaxa = x a – b if a > b xbxb xaxa = if b > a x b – a 1 xaxa xaxa = x0x0 = 1 1. 2. 3.

35. The Power Rules (x a ) b = x a b (x a y b ) c = x a c ( ) ybyb xaxa c x a c y b c =

36. a.(a 4 ) 3 a 4 3 a 12 (a 4 )(a 4 )(a 4 ) 12. Simplify. Can write it three times. Add 4 three times or multiply the exponents.

37. b.(10 5 ) 2 10 5 2 10 12. Simplify. Multiply the exponents.

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

39. a.(3xy) 3 (3) 3 27 x 1 3 y 1 3 13. Simplify. Multiply the exponents. Keep 3 in the parentheses. Evaluate each. x3x3 y3y3

40. y 1 37 z 1 37 Multiply the exponents. Evaluate each. y 37 z 37 b.(yz) 37

41. ( - 3) 2 9 x 3 2 13. Simplify. Multiply the exponents. Keep - 3 in the parentheses. Evaluate each. x6x6 c.( - 3x 3 ) 2

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

43. b. ( ( 4 a ) 2 a b ) 6 Multiply exponents. Evaluate. Use quotient rule and subtract exponents. 16 a 2 a 6 b 6 16 a 4 b 6

44. ( ) 5 4 -2 15. Simplify. x3x3 y0y0 z x z2z2 Simplify and use quotient rules. Work inside parentheses. Use power rule and evaluate. 2 -1 x 2 z ( ) 5 32 -x 10 z 5

45. The Power Rules (x a ) b = x a b (x a y b ) c = x a c ( ) ybyb xaxa c x a c y b c =

46. The Rules of Exponents Chapter 4.1