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).
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 โฆ.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 โฆ.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 โฆ.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 โฆ.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 โฆ.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 โฆ.x A = AN
12. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(xx)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A โฆ.x A = AN
13. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(xx)
base
exponent
Exponents
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A โฆ.x A = AN
14. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
15. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
16. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
17. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
18. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
19. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
20. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โฆ.x A = AN
21. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โ K
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A โฆ.x A = AN
22. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โ K
56
52
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A โฆ.x A = AN
23. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โ 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 โฆ.x A = AN
24. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โ 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 โฆ.x A = AN
25. Example A.
43 = (4)(4)(4) = 64
(xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
โx2 = โ(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 โ K
56
52 =
(5)(5)(5)(5)(5)(5)
(5)(5)
= 56 โ 2 = 54
We write the quantity A multiplied to itself N times as AN, i.e.
A x A x A โฆ.x A = AN
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 โ 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 โ 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 โ 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 โ 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 โ 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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ 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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, 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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
b. 3โ2
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32b. 3โ2 =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9b. 3โ2 = =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
b. 3โ2 = =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
=
1
2/5
=
b. 3โ2 = =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3โ2 = =
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3โ2 = =
a. 30 = 1
In general ( )โKa
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3โ2 = =
a. 30 = 1
In general ( )โKa
b = ( )K
b
a
d. ( )โ22
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3โ2 = =
a. 30 = 1
In general ( )โKa
b = ( )K
b
a
d. ( )โ22
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 โ 1 = A0, we obtain the 0-power Rule.
A1
A1
0-Power Rule: A0 = 1
Since = = A0 โ K = AโK, we get the negative-power Rule.
1
AK
A0
AK
Negative-Power Rule: AโK =
1
AK
Example E. Simplify
1
32
1
9
c. ( )โ12
5
=
1
2/5
= 1*
5
2
=
5
2
b. 3โ2 = =
a. 30 = 1
In general ( )โKa
b = ( )K
b
a
d. ( )โ22
5
= ( )2 =
25
4
5
2
57. e. 3โ1 โ 40 * 2โ2 =
1
3
โ 1*
1
22 = 1
3
โ 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. 3โ1 โ 40 * 2โ2 =
1
3
โ 1*
1
22 = 1
3
โ 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. 3โ1 โ 40 * 2โ2 =
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 3โ2 x4 yโ6 xโ8 y 23
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
60. e. 3โ1 โ 40 * 2โ2 =
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 3โ2 x4 yโ6 xโ8 y 23
3โ2 x4 yโ6 xโ8 y23
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
61. e. 3โ1 โ 40 * 2โ2 =
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 3โ2 x4 yโ6 xโ8 y 23
3โ2 x4 yโ6 xโ8 y23
= 3โ2 x4 xโ8 yโ6 y23
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
62. e. 3โ1 โ 40 * 2โ2 =
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 โ 8 yโ6+23
Example F. Simplify 3โ2 x4 yโ6 xโ8 y 23
3โ2 x4 yโ6 xโ8 y23
= 3โ2 x4 xโ8 yโ6 y23
1
9
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
63. e. 3โ1 โ 40 * 2โ2 =
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 โ 8 yโ6+23
= xโ4 y17
Example F. Simplify 3โ2 x4 yโ6 xโ8 y 23
3โ2 x4 yโ6 xโ8 y23
= 3โ2 x4 xโ8 yโ6 y23
1
9
1
9
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
64. e. 3โ1 โ 40 * 2โ2 =
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 โ 8 yโ6+23
= xโ4 y17
= y17
Example F. Simplify 3โ2 x4 yโ6 xโ8 y 23
3โ2 x4 yโ6 xโ8 y23
= 3โ2 x4 xโ8 yโ6 y23
1
9
1
9
1
9x4
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
65. e. 3โ1 โ 40 * 2โ2 =
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 โ 8 yโ6+23
= xโ4 y17
= y17
=
Example F. Simplify 3โ2 x4 yโ6 xโ8 y 23
3โ2 x4 yโ6 xโ8 y23
= 3โ2 x4 xโ8 yโ6 y23
1
9
1
9
1
9x4
y17
9x4
1
3
โ 1*
1
22 = 1
3
โ 1
4
= 1
12
66. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
67. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
68. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
69. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
70. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
71. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
72. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
73. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
=
3โ2x4yโ6x2
3โ5xโ3yโ3 x6
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
74. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
=
3โ2x4yโ6x2
3โ5xโ3yโ3 x6 =
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3 3โ5xโ3x6yโ3
3โ2x4x2yโ6
75. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
=
3โ2x4yโ6x2
3โ5xโ3yโ3 x6 =
=
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3 3โ5xโ3x6yโ3
3โ2x4x2yโ6
3โ2x6yโ6
3โ5x3yโ3
76. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
=
3โ2x4yโ6x2
3โ5xโ3yโ3 x6 =
= = 3โ2 โ (โ5) x6 โ 3 yโ6 โ (โ3)
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3 3โ5xโ3x6yโ3
3โ2x4x2yโ6
3โ2x6yโ6
3โ5x3yโ3
77. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
=
3โ2x4yโ6x2
3โ5xโ3yโ3 x6 =
= = 3โ2 โ (โ5) x6 โ 3 yโ6 โ (โ3)
= 33 x3 yโ3=
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3 3โ5xโ3x6yโ3
3โ2x4x2yโ6
3โ2x6yโ6
3โ5x3yโ3
78. Exponents
Example G. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23xโ8
26 xโ3
23xโ8
26xโ3
= 23 โ 6 xโ8 โ (โ3 )
= 2โ3 xโ5
=
23
1
x5
1
* = 8x5
1
Example H. Simplify
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3
=
3โ2x4yโ6x2
3โ5xโ3yโ3 x6 =
= = 3โ2 โ (โ5) x6 โ 3 yโ6 โ (โ3)
= 33 x3 yโ3=
27 x3
(3xโ2y3)โ2 x2
3โ5xโ3(yโ1x2)3 3โ5xโ3x6yโ3
3โ2x4x2yโ6
3โ2x6yโ6
3โ5x3yโ3
y3
79. Ex. A. Write the numbers without the negative exponents and
compute the answers.
1. 2โ1 2. โ2โ2 3. 2โ3 4. (โ3)โ2 5. 3โ3
6. 5โ2 7. 4โ3 8. 1
2
( )
โ3
9. 2
3
( )
โ1
10. 3
2
( )
โ2
11. 2โ1* 3โ2 12. 2โ2+ 3โ1 13. 2* 4โ1โ 50 * 3โ1
14. 32 * 6โ1โ 6 * 2โ3 15. 2โ2* 3โ1 + 80 * 2โ1
Ex. B. Combine the exponents. Leave the answers in positive
exponentsโbut do not reciprocate the negative exponents until
the final step.
16. x3x5 17. xโ3x5 18. x3xโ5 19. xโ3xโ5
20. x4y2x3yโ4 21. yโ3xโ2 yโ4x4 22. 22xโ3xy2x32โ5
23. 32yโ152โ2x5y2xโ9 24. 42x252โ3yโ34 xโ41yโ11
25. x2(x3)5 26. (xโ3)โ5x โ6 27. x4(x3yโ5) โ3yโ8
Exponents
80. xโ8
xโ3
B. Combine the exponents. Leave the answers in positive
exponentsโbut do not reciprocate the negative exponents until
the final step.
28. x8
xโ329.
xโ8
x330. y6xโ8
xโ2y331.
x6xโ2yโ8
yโ3xโ5y232.
2โ3x6yโ8
2โ5yโ5x233.
3โ2y2x4
2โ3x3yโ234.
4โ1(x3yโ2)โ2
2โ3(yโ5x2)โ135.
6โ2 y2(x4yโ3)โ1
9โ1(x3yโ2)โ4y236.
C. Combine the exponents as much as possible.
38. 232x 39. 3x+23x 40. axโ3ax+5
41. (b2)x+1bโx+3 42. e3e2x+1eโx
43. e3e2x+1eโx
44. How would you make sense of 23 ?
2