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
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
46. 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
Let’s extend exponent notation to other types of exponents
such as A0 or A–1.
Non–Positive–Whole–Number Exponents
!Note: (AN ± BM)K = ANK ± BMK, e.g. (2 + 3)2 = 22 + 32.
47. 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
Let’s extend exponent notation to other types of exponents
such as A0 or A–1. 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).
Non–Positive–Whole–Number Exponents
!Note: (AN ± BM)K = ANK ± BMK, e.g. (2 + 3)2 = 22 + 32.
48. 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
Let’s extend exponent notation to other types of exponents
such as A0 or A–1. 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). But 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.
49. Exponents
We extract the meaning of A0 or A–1 by contemplating the
consequences of the above operational rules.
50. Exponents
Since = 1
A1
A1
We extract the meaning of A0 or A–1 by contemplating the
consequences of the above operational rules.
51. Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.
A1
A1
We extract the meaning of A0 or A–1 by contemplating the
consequences of the above operational rules.
52. 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 contemplating the
consequences of the above operational rules.
53. 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 contemplating the
consequences of the above operational rules.
It's a major intellectual leap for one to
accept that this symbolic system requires
that A0 = 1, and to reject the more intuitive
translation of A0 as “0 copy of A”
which leads to the seemingly
obvious but wrong answer A0 = 0.
54. 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 contemplating the
consequences of the above operational rules.
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
0-Power Rule: A0 = 1, A = 0
We extract the meaning of A0 or A–1 by contemplating the
consequences of the above operational rules.
56. 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 contemplating the
consequences of the above operational rules.
57. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AKA–K =
58. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
59. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
60. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
=
61. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
62. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
63. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
64. 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 contemplating the
consequences of the above operational rules.
and in general that
1
AK
The “negative” of an exponents mean to reciprocate the base.
A–K =
70. 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.
71. 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.
72. 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
73. 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
74. 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
75. 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
76. 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
77. 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
78. 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
79. Exponents
Example I. Simplify using the rules for exponents.
Leave the answer in positive exponents only.
23x–8
26 x–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
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 )
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
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
84. 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
85. 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
86. 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
87. 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
88. 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
89. 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
90. 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
91. 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
92. 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
93. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
(divide–subtract)
(divide–subtract)
94. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2,
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
(divide–subtract)
(divide–subtract)
95. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
(divide–subtract)
(divide–subtract)
96. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
1/n - Power Rule: A1/n = A , the nth root of A.
n
(divide–subtract)
(divide–subtract)
97. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
Example B.
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
1/n - Power Rule: A1/n = A , the nth root of A.
n
c. 641/3 =
b. 81/3 =
a. 641/2 =
(divide–subtract)
(divide–subtract)
98. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
Example B.
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
1/n - Power Rule: A1/n = A , the nth root of A.
n
c. 641/3 =
b. 81/3 =
a. 641/2 = 64 = 8
(divide–subtract)
(divide–subtract)
99. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
Example B.
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
1/n - Power Rule: A1/n = A , the nth root of A.
n
c. 641/3 =
b. 81/3 = 8 = 2
3
a. 641/2 = 64 = 8
(divide–subtract)
(divide–subtract)
100. 0-power Rule: A0 = 1 (A=0)
=
1
Ak
1
Ak
A0
Ak
Special Exponents
½ - Power Rule: A½ = A , the square root of A,
because (A½)2 = A = (A)2, so A½ = A
Example B.
because 1 = = A1–1 = A0A1
A1
Negative Power Rule: A–k =
because = A0–k = A–k
1/n - Power Rule: A1/n = A , the nth root of A.
n
c. 641/3 = 64 = 4
3
b. 81/3 = 8 = 2
3
a. 641/2 = 64 = 8
(divide–subtract)
(divide–subtract)
102. Special Exponents
By the power–multiply rule, the fractional exponent
A
k
n±
(A )n
1
is
take the nth root of A
103. Special Exponents
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
104. Special Exponents
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
105. Special Exponents
a. 9 –3/2 =
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 =
106. Special Exponents
a. 9 –3/2 = (9 ½ * –3)
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 =
107. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 =
108. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 =
109. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 =
110. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 =
111. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 = (271/3)-2 = (27)-23
112. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 =
3
113. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 =
b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9
3
114. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 = (161/4)-3 = (16)-34
b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9
3
115. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-34
b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9
3
116. Special Exponents
a. 9 –3/2 = (9 ½ * –3) = (9½)–3 = (9)–3 = 3–3 = 1/27
To calculate a fractional power: extract the root first,
then raise the root to the numerator–power.
Example C. Find the root, then raise the root to the
numerator–power.
By the power–multiply rule, the fractional exponent
A
k
n±
(A ) kn ±1
is
take the nth root of A
then raise the
root to ±k power
c. 16 -3/4 = (161/4)-3 = (16)-3 = (2)-3 = 1/23 = 1/84
b. 27 -2/3 = (271/3)-2 = (27)-2 = (3)-2 = 1/32 = 1/9
3
118. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
119. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
We use the multiply–add, divide–subtract, and power–
multiply rules to collect fractional exponents.
120. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
We use the multiply–add, divide–subtract, and power–
multiply rules to collect fractional exponents.
x*(x1/3y3/2)2
x–1/2y2/3
=
Example D. Simplify by combining the exponents.
121. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
We use the multiply–add, divide–subtract, and power–
multiply rules to collect fractional exponents.
x*(x1/3y3/2)2
x–1/2y2/3
=
x*x2/3y3
x–1/2y2/3
power–multiply rule
1/3*2 3/2*2
Example D. Simplify by combining the exponents.
122. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
We use the multiply–add, divide–subtract, and power–
multiply rules to collect fractional exponents.
x*(x1/3y3/2)2
x–1/2y2/3
=
x*x2/3y3
x–1/2y2/3
= x–1/2y2/3
x5/3y3
Example D. Simplify by combining the exponents.
power–multiply rule
1/3*2 3/2*2
multiply–add rule
1 + 2/3
123. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
We use the multiply–add, divide–subtract, and power–
multiply rules to collect fractional exponents.
x*(x1/3y3/2)2
x–1/2y2/3
=
x*x2/3y3
x–1/2y2/3
= x–1/2y2/3
=
x5/3y3
x5/3 – (–1/2) y3 – 2/3
Example D. Simplify by combining the exponents.
power–multiply rule
1/3*2 3/2*2
multiply–add rule
1 + 2/3
divide–subtract rule
124. a.16–½ =
Fractional Powers
b. 43/2 =
Your turn: calculate the root, then raise the root to the
numerator–power.
Ans: ¼, 8
We use the multiply–add, divide–subtract, and power–
multiply rules to collect fractional exponents.
x*(x1/3y3/2)2
x–1/2y2/3
=
x*x2/3y3
x–1/2y2/3
= x–1/2y2/3
=
x5/3y3
x5/3 – (–1/2) y3 – 2/3
= x13/6 y7/3
Example D. Simplify by combining the exponents.
power–multiply rule
1/3*2 3/2*2
multiply–add rule
1 + 2/3
divide–subtract rule
126. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
127. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 =
a. 53 or (5 )3 =
b. 9a2 =
128. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 =
a. 53 or (5 )3 = 53/2
b. 9a2 =
129. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 =
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2
130. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 =
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
131. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 = (9 + a2)1/2
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
132. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
133. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
d. Express a2 (a ) as one radical.3 4
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
134. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
a2 a = a2/3a1/43 4
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
d. Express a2 (a ) as one radical.3 4
135. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
a2 a = a2/3a1/4 = a11/123 4
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
d. Express a2 (a ) as one radical.3 4
136. Fractional Powers
Often it’s easier to manipulate radical–expressions
using the fractional exponent notation.
a2 a = a2/3a1/4 = a11/12 = a11
3 4 12
To write a radical in fractional exponent,
assuming a is defined, we have that:k
an = ( a )n → a
k k k
n
Example E. Write the following expressions using
fractional exponents then simplify if possible.
c. 9 + a2 = (9 + a2)1/2 ≠ (3 + a ).
a. 53 or (5 )3 = 53/2
b. 9a2 = (9a2)1/2 = 3a
d. Express a2 (a ) as one radical.3 4
137. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
138. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 =
b. 16–0.75 =
c. 30.4 =
139. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2
b. 16–0.75 =
c. 30.4 =
140. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3
b. 16–0.75 =
c. 30.4 =
141. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 =
c. 30.4 =
142. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4
c. 30.4 =
143. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3
4
c. 30.4 =
144. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8
4
c. 30.4 =
145. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8
4
c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)
5
146. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8
4
c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)
5
Working with real numbers and interpreting decimal
exponents as fractions causes problems if the base
is negative.
147. Decimal Powers
We write decimal–exponents as fractional exponents,
then we extract roots and raise powers as indicated
by the fractional exponents.
Example F. Write the following decimal exponents as
fractional exponents then simplify, if possible.
a. 9–1.5 = 9 –3/2 = (9)–3 = 3–3 = 1/27
b. 16–0.75 = 16 –3/4 = (16)–3 = (2)–3 = 1/8
4
c. 30.4 = 32/5 = (4)2 ≈ 1.55 (by calculator)
5
Working with real numbers and interpreting decimal
exponents as fractions causes problems if the base
is negative. For example, (–32)0.2 can be viewed as
(–32)1/5 = –32 = –2, or as (–32)2/10 = (–32)2 which is
not defined.
5 10
