The document discusses exponents and rules for exponents. It defines exponents as representing the quantity A multiplied by itself N times, written as AN. It then provides examples to illustrate the multiplication rule (ANAK = AN+K), division rule (AN/AK = AN-K), power rule ((AN)K = ANK), 0-power rule (A0 = 1), and negative power rule (A-K = 1/AK).

1. Exponents
Frank Ma © 2011

2. Exponents
We write the quantity A multiplied to itself N times as AN,

3. Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

4. base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

5. Example A.
43
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

6. Example A.
43 = (4)(4)(4) = 64
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

7. Example A.
43 = (4)(4)(4) = 64
(xy)2
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

8. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

9. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

10. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

11. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

12. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

13. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

14. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

15. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

16. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

17. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5)
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

18. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

19. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication Rule: ANAK =AN+K
Example B.
a. 5354 = (5*5*5)(5*5*5*5) = 53+4 = 57
b. x5y7x4y6 = x5x4y7y6
Rules of Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

20. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication 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
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

21. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication 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
Division Rule:
AN
AK = AN – K
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

22. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication 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
Division Rule:
Example C.
AN
AK = AN – K
56
52
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

23. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication 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
Division Rule:
Example C.
AN
AK = AN – K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

24. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication 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
Division Rule:
Example C.
AN
AK = AN – K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

25. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiplication 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
Division Rule:
Example C.
AN
AK = AN – K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
= 56 – 2 = 54
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN

26. Power Rule: (AN)K = ANK
Exponents

27. Power Rule: (AN)K = ANK
Example D. (34)5
Exponents
A1

28. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34)
Exponents

29. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4
Exponents

30. Power Rule: (AN)K = ANK
Example D. (34)5 = (34)(34)(34)(34)(34) = 34+4+4+4+4 = 34*5 = 320
Exponents

31. Power 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
A1

32. Power 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 – 1A1
A1

33. Power 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 = A0A1
A1

34. Power 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

35. Power 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
0-Power Rule: A0 = 1

36. Power 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
0-Power Rule: A0 = 1
Since =
1
AK
A0
AK

37. Power 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
0-Power Rule: A0 = 1
Since = = A0 – K1
AK
A0
AK

38. Power 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
0-Power Rule: A0 = 1
Since = = A0 – K = A–K, we get the negative-power Rule.
1
AK
A0
AK

39. Power 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
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

40. Power 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
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 E. Simplify
a. 30

41. Power 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
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 E. Simplify
a. 30 = 1

42. Power 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
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 E. Simplify
b. 3–2
a. 30 = 1

43. Power 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
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 E. Simplify
1
32b. 3–2 =
a. 30 = 1

44. Power 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
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 E. Simplify
1
32
1
9b. 3–2 = =
a. 30 = 1

45. Power 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
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 E. Simplify
1
32
1
9
c. ( )–12
5
b. 3–2 = =
a. 30 = 1

46. Power 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
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 E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
=
b. 3–2 = =
a. 30 = 1

47. Power 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
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 E. Simplify
1
32
1
9
c. ( )–12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3–2 = =
a. 30 = 1

48. Power 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
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 E. 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

49. Power 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
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 E. 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

50. Power 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
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 E. 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

51. Power 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
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 E. 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

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 collecting 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 collecting 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 collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
Example F. 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 collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
Example F. 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 collecting exponents, we do not reciprocate
the negative exponents. Instead we add or subtract them
using the multiplication and division rules first.
Example F. 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 collecting 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 F. 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 collecting 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 F. 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 collecting 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 F. 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 collecting 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 F. 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 G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3

67. Exponents
Example G. 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 G. 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 G. 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 G. 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 G. 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 H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3

72. Exponents
Example G. 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 H. Simplify
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
(3x–2y3)–2 x2
3–5x–3(y–1x2)3

73. Exponents
Example G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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 G. 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 H. 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. Ex. 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
Ex. 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
Exponents

80. 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 much 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