The document discusses exponents and rules for exponents. It defines exponents as representing repeated multiplication, where AN means multiplying A by itself N times. It then provides examples to illustrate four rules for exponents: 1) The multiply-add rule states that ANAK = AN+K. 2) The divide-subtract rule states that AN/AK = AN-K. 3) The power-multiply rule states that (AN)K = ANK. 4) It also defines rules for exponents of 0 and negative numbers.

1. Exponents
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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