44 exponents
TRANSCRIPT
![Page 1: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/1.jpg)
Exponents
![Page 2: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/2.jpg)
In the notation
23
Exponents
![Page 3: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/3.jpg)
In the notation
23this is the base
Exponents
![Page 4: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/4.jpg)
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
Exponents
![Page 5: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/5.jpg)
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
= 8
Exponents
![Page 6: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/6.jpg)
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
![Page 7: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/7.jpg)
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
![Page 8: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/8.jpg)
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
![Page 9: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/9.jpg)
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
![Page 10: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/10.jpg)
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
= 12a. 3(4) b. 34 c. 43
Exponents
![Page 11: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/11.jpg)
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*3a. 3(4) b. 34 c. 43
Exponents
![Page 12: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/12.jpg)
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
![Page 13: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/13.jpg)
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
![Page 14: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/14.jpg)
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 * 4a. 3(4) b. 34 c. 43
Exponents
![Page 15: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/15.jpg)
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 * 4a. 3(4) b. 34 c. 43
= 16 * 4= 64
Exponents
![Page 16: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/16.jpg)
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 17: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/17.jpg)
Example B.43
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 18: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/18.jpg)
Example B.43 = (4)(4)(4) = 64
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 19: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/19.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 20: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/20.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy)
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 21: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/21.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 22: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/22.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 23: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/23.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy)
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 24: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/24.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 25: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/25.jpg)
Example B.43 = (4)(4)(4) = 64 (xy)2= (xy)(xy) = x2y2
xy2 = (x)(yy) –x2 = –(xx)
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
![Page 26: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/26.jpg)
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
![Page 27: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/27.jpg)
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
![Page 28: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/28.jpg)
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
![Page 29: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/29.jpg)
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
![Page 30: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/30.jpg)
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
![Page 31: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/31.jpg)
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
![Page 32: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/32.jpg)
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
![Page 33: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/33.jpg)
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 =
![Page 34: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/34.jpg)
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)
![Page 35: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/35.jpg)
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
![Page 36: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/36.jpg)
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
![Page 37: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/37.jpg)
Power-Multiply Rule: (AN)K = ANK
Example E. (34)5 = (34)(34)(34)(34)(34)
= 34+4+4+4+4
= 34*5 = 320
Exponents
![Page 38: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/38.jpg)
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
![Page 39: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/39.jpg)
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.
![Page 40: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/40.jpg)
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.
![Page 41: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/41.jpg)
ExponentsWe extract the meaning of A0 or A–1 by examining the consequences of the above rules.
![Page 42: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/42.jpg)
Exponents
Since = 1A1
A1
We extract the meaning of A0 or A–1 by examining the consequences of the above rules.
![Page 43: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/43.jpg)
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.
![Page 44: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/44.jpg)
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.
![Page 45: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/45.jpg)
Exponents
Since = 1 = A1 – 1 = A0, we obtain the 0-power Rule.A1
A1
Since =1A
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.
![Page 46: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/46.jpg)
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.1A
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.
![Page 47: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/47.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
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.
![Page 48: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/48.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
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 that1AKA–K =
![Page 49: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/49.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
![Page 50: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/50.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
Example J. Simplify
c. ( )–1 2 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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
![Page 51: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/51.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
Example J. Simplify
c. ( )–1 2 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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
=
![Page 52: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/52.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
Example J. Simplify
1 32
1 9
c. ( )–1 2 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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
![Page 53: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/53.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
Example J. Simplify
1 32
1 9
c. ( )–1 2 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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
![Page 54: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/54.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
Example J. Simplify
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
In general ( )–K
a b = ( )K b
a d. ( )–2 2
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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
![Page 55: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/55.jpg)
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.1A
A0
A1
Negative-Power Rule: A–1 = 1A
Example J. Simplify
1 32
1 9
c. ( )–1 2 5 = 1
2/5 = 1* 5 2 = 5
2
b. 3–2 = =a. 30 = 1
In general ( )–K
a b = ( )K b
a d. ( )–2 2
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 that1AK
The “negative” of an exponents mean to reciprocate the base. A–K =
![Page 56: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/56.jpg)
e. 3–1 – 40 * 2–2 =
Exponents
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e. 3–1 – 40 * 2–2 = 1 3
Exponents
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e. 3–1 – 40 * 2–2 = 1 3 – 1*
Exponents
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e. 3–1 – 40 * 2–2 = 1 3 – 1* 1
22
Exponents
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e. 3–1 – 40 * 2–2 = 1 3 – 1* 1
22 = 1 3
– 1 4 = 1
12
Exponents
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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.
![Page 62: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/62.jpg)
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.
![Page 63: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/63.jpg)
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
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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
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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
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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
![Page 67: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/67.jpg)
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
![Page 68: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/68.jpg)
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
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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
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ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.
23x–8
26 x–3
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ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
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ExponentsExample 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 )
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ExponentsExample 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
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ExponentsExample 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
= 231
x51
* = 8x51
![Page 75: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/75.jpg)
ExponentsExample 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
= 231
x51
* = 8x51
Example J. Simplify (3x–2y3)–2 x2
3–5x–3(y–1x2)3
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ExponentsExample 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
= 231
x51
* = 8x51
Example J. Simplify (3x–2y3)–2 x2
3–5x–3(y–1x2)3
(3x–2y3)–2 x2
3–5x–3(y–1x2)3
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ExponentsExample 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
= 231
x51
* = 8x51
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
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ExponentsExample 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
= 231
x51
* = 8x51
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
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ExponentsExample 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
= 231
x51
* = 8x51
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
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ExponentsExample 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
= 231
x51
* = 8x51
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
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ExponentsExample 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
= 231
x51
* = 8x51
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
![Page 82: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/82.jpg)
ExponentsExample 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
= 231
x51
* = 8x51
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
![Page 83: 44 exponents](https://reader031.vdocuments.site/reader031/viewer/2022022414/589d370f1a28abd17d8b5a13/html5/thumbnails/83.jpg)
ExponentsExample 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
= 231
x51
* = 8x51
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