Post on 13-Apr-2017
transcript
Exponents
In the notation
23
Exponents
In the notation
23this is the base
Exponents
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
Exponents
In the notation
= 2 * 2 * 223this is the base
this is the exponent, or the power, which is the number of repetitions.
= 8
Exponents
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
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
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
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
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
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
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
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
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
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
base
exponent
ExponentsWe write the quantity A multiplied to itself N times as AN, i.e.
A x A x A ….x A = AN
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 =
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)
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
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
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-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
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.
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.
ExponentsWe extract the meaning of A0 or A–1 by examining the consequences of the above rules.
Exponents
Since = 1A1
A1
We extract the meaning of A0 or A–1 by examining the consequences of the above rules.
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.
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.
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.
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.
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.
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 =
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 =
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 =
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 =
=
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 =
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 =
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 =
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 =
e. 3–1 – 40 * 2–2 =
Exponents
e. 3–1 – 40 * 2–2 = 1 3
Exponents
e. 3–1 – 40 * 2–2 = 1 3 – 1*
Exponents
e. 3–1 – 40 * 2–2 = 1 3 – 1* 1
22
Exponents
e. 3–1 – 40 * 2–2 = 1 3 – 1* 1
22 = 1 3
– 1 4 = 1
12
Exponents
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.
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.
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
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
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
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
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
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
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
ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.
23x–8
26 x–3
ExponentsExample I. Simplify using the rules for exponents. Leave the answer in positive exponents only.
23x–8
26 x–3
23x–8
26x–3
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 )
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
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
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
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
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
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
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
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
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
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
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