The document discusses exponents and the rules for working with them. Exponents describe the number of times a base is multiplied by itself. The main rules covered are: - The multiply-add rule: When the same base has two exponents, you add the exponents and multiply the results, written as ANAK = AN+K - The divide-subtract rule: When dividing exponents with the same base, you subtract the exponents and divide the results, written as AN/AK = AN-K Examples are provided to demonstrate calculating exponents using these rules.

1. Exponents

2. In the notation
23
Exponents

3. In the notation
23this is the base
Exponents

4. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Exponents

5. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
= 8
Exponents

6. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
Exponents

7. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
Exponents

8. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
Exponents

9. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
a. 3(4) b. 34 c. 43
Exponents

10. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
= 12
a. 3(4) b. 34 c. 43
Exponents

11. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
= 12 = 3*3*3*3
a. 3(4) b. 34 c. 43
Exponents

12. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
= 12 = 3*3*3*3
= 9 9*
a. 3(4) b. 34 c. 43
Exponents

13. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
= 12 = 3*3*3*3
= 9 9*
= 81
a. 3(4) b. 34 c. 43
Exponents

14. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4
a. 3(4) b. 34 c. 43
Exponents

15. In the notation
= 2 * 2 * 223this is the base
this is the exponent,
or the power, which is
the number of repetitions.
Example A. Calculate the following.
The Base-1 Rule: 1 any power = 1 * 1 * ..*1 = 1.
The Blank-1Power : The expression x (with blank power) is x1,
so 2 = 21, 7 = 71, etc.., i.e. we have one copy of x.
We say that “2 to the power 3 is 8” or
that “2 to the 3rd power is 8.”
= 8
= 12 = 3*3*3*3
= 9 9*
= 81
= 4 * 4 * 4
a. 3(4) b. 34 c. 43
= 16 * 4
= 64
Exponents

16. 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

17. Example B.
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

18. Example B.
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

19. Example B.
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

20. Example B.
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

21. Example B.
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

22. Example B.
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

23. Example B.
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

24. Example B.
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

25. Example B.
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

26. Example B.
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

27. Example B.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
–x2 = –(xx)
base
exponent
Exponents
Multiply-Add 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

28. Example B.
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 C .
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

29. Example B.
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 C .
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

30. Example B.
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 C .
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

31. Example B.
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 C .
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

32. Example B.
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 C .
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

33. Example B.
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 C .
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 the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
Example D .
56
52 =

34. Example B.
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 C .
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 the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
Example D .
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)

35. Example B.
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 C .
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 the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
Example D .
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
= 56 – 2

36. Example B.
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 C .
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 the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
Example D .
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
= 56 – 2 = 54

37. Power-Multiply Rule: (AN)K = ANK
Example E. (34)5 = (34)(34)(34)(34)(34)
= 34+4+4+4+4
= 34*5 = 320
Exponents

38. Power-Multiply Rule: (AN)K = ANK
Example E. (34)5 = (34)(34)(34)(34)(34)
= 34+4+4+4+4
= 34*5 = 320
Exponents
Power-Distribute Rule: (ANBM)K = ANK BMK
Example F. (34 23)5 = (34 23)(34 23)(34 23)(34 23)(34 23)
= 34+4+4+4+4 2 3+3+3+3+3
= 34*5 23*5
= 320 215

39. Power-Multiply Rule: (AN)K = ANK
Example E. (34)5 = (34)(34)(34)(34)(34)
= 34+4+4+4+4
= 34*5 = 320
Exponents
Power-Distribute Rule: (ANBM)K = ANK BMK
Example F. (34 23)5 = (34 23)(34 23)(34 23)(34 23)(34 23)
= 34+4+4+4+4 2 3+3+3+3+3
= 34*5 23*5
= 320 215
!Note: (AN ± BM)K = ANK ± BMK, e.g. (2 + 3)2 = 22 + 32.

40. Power-Multiply Rule: (AN)K = ANK
Example E. (34)5 = (34)(34)(34)(34)(34)
= 34+4+4+4+4
= 34*5 = 320
Exponents
Power-Distribute Rule: (ANBM)K = ANK BMK
Example F. (34 23)5 = (34 23)(34 23)(34 23)(34 23)(34 23)
= 34+4+4+4+4 2 3+3+3+3+3
= 34*5 23*5
= 320 215
The positive–whole–number exponent specifies a tangible
number of copies of the base to be multiplied (e.g. A2 = A x A,
2 copies of A). Let’s extend exponent notation to other types
of exponents such as A0 or A–1. However A0 does not mean
there is “0” copy of A, or that A–1 is “–1” copy of A.
Non–Positive–Whole–Number Exponents
!Note: (AN ± BM)K = ANK ± BMK, e.g. (2 + 3)2 = 22 + 32.

41. Exponents
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

42. Exponents
Since = 1
A1
A1
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

43. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

44. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

45. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since =
1
A
A0
A1
0-Power Rule: A0 = 1, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

46. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
0-Power Rule: A0 = 1, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

47. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
0-Power Rule: A0 = 1, A = 0
, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.

48. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
0-Power Rule: A0 = 1, A = 0
, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AKA–K =

49. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
0-Power Rule: A0 = 1, A = 0
, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =

50. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
Example J. Simplify
c. ( )–12
5
=
b. 3–2 =
a. 30 =
0-Power Rule: A0 = 1, A = 0
, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =

51. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
Example J. Simplify
c. ( )–12
5
b. 3–2 =
a. 30 = 1
0-Power Rule: A0 = 1, A = 0
, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
=

52. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
Example J. Simplify
1
32
1
9
c. ( )–12
5
=
b. 3–2 = =
a. 30 = 1
0-Power Rule: A0 = 1, A = 0
, A = 0
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =

53. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
Example J. 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
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =

54. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
Example J. 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
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =

55. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
Since = = A0 – 1 = A–1, we get the negative-power rule.
1
A
A0
A1
Negative-Power Rule: A–1 =
1
A
Example J. 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
We extract the meaning of A0 or A–1 by examining the
consequences of the above rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =

56. e. 3–1 – 40 * 2–2 =
Exponents

57. e. 3–1 – 40 * 2–2 =
1
3
Exponents

58. e. 3–1 – 40 * 2–2 =
1
3
– 1*
Exponents

59. e. 3–1 – 40 * 2–2 =
1
3
– 1*
1
22
Exponents

60. e. 3–1 – 40 * 2–2 =
1
3
– 1*
1
22 = 1
3
– 1
4
= 1
12
Exponents

61. 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.

62. 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.

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.
Example H. Simplify 3–2 x4 y–6 x–8 y 23
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.
Example H. 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

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.
Example H. 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

66. 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 H. 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

67. 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 H. 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

68. 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 H. 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

69. 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 H. 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

70. Exponents
Example I. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3

71. Exponents
Example I. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3

72. Exponents
Example I. 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 )

73. Exponents
Example I. 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

74. Exponents
Example I. 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

75. Exponents
Example I. 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 J. Simplify (3x–2y3)–2 x2
3–5x–3(y–1x2)3

76. Exponents
Example I. 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 J. Simplify (3x–2y3)–2 x2
3–5x–3(y–1x2)3
(3x–2y3)–2 x2
3–5x–3(y–1x2)3

77. Exponents
Example I. 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 J. 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

78. Exponents
Example I. 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 J. 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

79. Exponents
Example I. 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 J. 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

80. Exponents
Example I. 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 J. 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

81. Exponents
Example I. 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 J. 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

82. Exponents
Example I. 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 J. 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

83. Exponents
Example I. 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 J. 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