The document introduces exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It provides examples of evaluating exponents like 43. It then introduces rules for exponents, including the multiply-add rule where ANAK = AN+K, and the divide-subtract rule where AN/AK = AN-K. It also covers fractional exponents by defining the 0-power rule where A0 = 1 and the negative power rule where A-K = 1/AK.

1. Exponents

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. Since = 1
A1
A1
Fractional Exponents

26. Since = 1 = A1 – 1 = A0A1
A1
Fractional Exponents

27. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Fractional Exponents

28. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Fractional Exponents

29. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since =1
AK
A0
AK
Fractional Exponents

30. Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 – K = A–K1
AK
A0
AK
Fractional Exponents

31. 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
Fractional Exponents

32. 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
Fractional Exponents

33. 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
Fractional Exponents
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

34. 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 D. Simplify.
a. 30 = 1
Fractional Exponents
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

35. 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 D. Simplify.
b. 3–2
a. 30 = 1
Fractional Exponents
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

36. 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 D. Simplify.
1
32b. 3–2 = =
a. 30 = 1
Fractional Exponents
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

37. 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 D. Simplify.
1
32
1
9b. 3–2 = =
a. 30 = 1
Fractional Exponents
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

38. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
b. 3–2 = =
a. 30 = 1
Fractional Exponents
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

39. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
b. 3–2 = =
a. 30 = 1
Fractional Exponents
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. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
Fractional Exponents
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. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
Fractional Powers
Fractional Exponents
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. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
Fractional Powers
Fractional Exponents
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
1/2-Power Rule: A½ = A.

43. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
Fractional Powers
Fractional Exponents
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
By the power-multiply rule (A½ )2 = A1
1/2-Power Rule: A½ = A.

44. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
Fractional Powers
Fractional Exponents
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
By the power-multiply rule (A½ )2 = A1 = (A)2,
1/2-Power Rule: A½ = A.

45. 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 D. Simplify.
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1
Fractional Powers
Fractional Exponents
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
By the power-multiply rule (A½ )2 = A1 = (A)2,
therefore A½ = A.
1/2-Power Rule: A½ = A.

46. Example E.
a. 16½
Fractional Exponents

47. Example E.
a. 16½ = 16
Fractional Exponents

48. Example E.
a. 16½ = 16 = 4
Fractional Exponents

49. Example E.
a. 16½ = 16 = 4
b. 9½ 25½
Fractional Exponents

50. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925
Fractional Exponents

51. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
Fractional Exponents

52. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2
Fractional Exponents

53. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3)
Fractional Exponents

54. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3
Fractional Exponents

55. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3
Fractional Exponents

56. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
Fractional Exponents

57. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½
Fractional Exponents

58. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1
Fractional Exponents

59. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1
Fractional Exponents

60. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
Fractional Exponents

61. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2
Fractional Exponents

62. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3)
Fractional Exponents

63. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3
Fractional Exponents

64. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3
Fractional Exponents

65. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
Fractional Exponents

66. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 =
A
1In general,
Fractional Exponents

67. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1 A –k/2 = ( )k
A
1In general,
Fractional Exponents

68. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1 A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Fractional Exponents

69. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Fractional Exponents

70. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3
Fractional Exponents

71. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8
3
Fractional Exponents

72. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
3
Fractional Exponents

73. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
b. 27 –2/3
3
Fractional Exponents

74. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
b. 27 –2/3 = (271/3)–2
3
Fractional Exponents

75. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
b. 27 –2/3 = (271/3)–2 = (27)–2
3
3
Fractional Exponents

76. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
b. 27 –2/3 = (271/3)–2 = (27)–2
3
3
Fractional Exponents

77. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
b. 27 –2/3 = (271/3)–2 = (27)–2 = (3)–2
3
3
Fractional Exponents

78. Example E.
a. 16½ = 16 = 4
b. 9½ 25½ =925 = 3*5 = 15
c. 43/2 = (4½ * 3) = (4½)3 = (4)3 = 23 = 8
d. 16 – ½ = (16½) –1 = (16) –1 = 4 –1 = 1/4
e. 9 –3/2 = (9 –½ * 3) = (9 ½) –3 = (9)–3 = 3–3 = 1/27
A–1/2 = and
A
1
1/k-Power Rule A1/k = A
k
A –k/2 = ( )k
A
1
Using an argument similar to the argument for ½-power rule,
we have the following rule for 1/k power.
In general,
Example F. Simplify.
a. 81/3 = 8 = 2
b. 27 –2/3 = (271/3)–2 = (27)–2 = (3)–2 = 1/32 = 1/9
3
3
Fractional Exponents

79. All the following rules for operations of exponents hold for
fractional exponents N and K.
Fractional Exponents

80. All the following rules for operations of exponents hold for
fractional exponents N and K.
Multiplication Rule ANAK = AN + K
Fractional Exponents

81. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
AN
AK
Multiplication Rule ANAK = AN + K
Fractional Exponents

82. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
x(x1/3y3/2)2
x–1/2y2/3
AN
AK
Example G. Simplify the exponents.
Multiplication Rule ANAK = AN + K
Fractional Exponents

83. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
x(x1/3y3/2)2
=
x*x2/3y3
x–1/2y2/3
2*1/3 2*3/2
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3
Multiplication Rule ANAK = AN + K
Fractional Exponents

84. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
x(x1/3y3/2)2
=
x*x2/3y3
x–1/2y2/3
=
2*1/3 2*3/2
x5/3y3
1+2/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
Multiplication Rule ANAK = AN + K
Fractional Exponents

85. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
x(x1/3y3/2)2
=
x*x2/3y3
x–1/2y2/3
=
2*1/3 2*3/2
=
x5/3y3
1+2/3
x5/3 – (–1/2) y3 – 2/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
Multiplication Rule ANAK = AN + K
Fractional Exponents

86. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
x(x1/3y3/2)2
=
x*x2/3y3
x–1/2y2/3
=
2*1/3 2*3/2
=
x5/3y3
1+2/3
x5/3 – (–1/2) y3 – 2/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
= x5/3 +1/2 y7/3
Multiplication Rule ANAK = AN + K
Fractional Exponents

87. Division Rule = AN – K
Power Rule (AN)K = ANK
All the following rules for operations of exponents hold for
fractional exponents N and K.
x(x1/3y3/2)2
=
x*x2/3y3
x–1/2y2/3
=
2*1/3 2*3/2
=
x5/3y3
1+2/3
x5/3 – (–1/2) y3 – 2/3
= x13/6 y7/3
AN
AK
Example G. Simplify the exponents.
x–1/2y2/3 x–1/2y2/3
= x5/3 +1/2 y7/3
Multiplication Rule ANAK = AN + K
Fractional Exponents

88. Exercise A. Write the expressions in radicals and simplify.
1. 41/2 2. 91/2 3. 161/2 4. 251/2
6. 361/2 7. 491/2 8. 641/2 9. 811/2
10. (100x)1/2
x
4( )1/2
11.
9
25( )1/2
12.
13. (–8)1/3
14. –1
64( )1/3
16. 4–1/2 17. 9–1/2 18. 16–1/2 19. 25–1/2
20. 36–1/2 21. 49–1/2 22. 64–1/2 23. 81–1/2
15. 1251/3
x
4( ) –1/2
25.
9
25( ) –1/2
26.
27. –1
64( ) –1/3 28. 125–1/3
x
(24. )–1/2
100
29. 163/2 30. 253/2
31. –1
64( ) –2/3 32. 125–2/3
Fractional Exponents

89. Exercise B. Simplify the expressions by combining the
exponents. Interpret the problems and the results in radicals.
34. x1/2x1/433. x1/3x1/3 35. x1/2x1/3
37.36. 38.x1/3
x1/4 x1/3
x1/3
x1/2
Fractional Exponents
x1/2
Exercise C. Simplify the expressions by combining the
exponents. Leave the answers in simplified fractional
exponents.
40. x1/4x3/2x2/3
39. x1/3x4/3x1/2
45.
48.
y1/2x1/3
x–1/4
x4/3y–3/2
x1/2
41. x3/4y1/2x3/2y4/3
43. y–2/3x–1/4y–3/2x–2/342. y–1/4x5/6x4/3y1/2 44. x3/4y1/2x3/2y4/3
46.
x1/4
x–1/2
47.
x–1/4
x–1/2
49.
y–1/2x1/3
y–4/3x–3/2
50.
(8x2)1/3
(4x)3/2
51.
(27x2)–1/3
(9x)1/2
52.
(100x2)–1/2
(64x)1/3
53.
(36x3)–1/2
(64x1/2)–1/3