Chapter 7: Exponential Functions 7.1 & 7.2 Laws of Exponents Zero and Negative Exponents.
Laws of Exponents - Worcester ALC of Exponents.pdf · The Laws of Exponents. Exponents {53 Power...
Transcript of Laws of Exponents - Worcester ALC of Exponents.pdf · The Laws of Exponents. Exponents {53 Power...
The Laws of Exponents
Exponents
{ 35Power
base
exponent
3 3 means that is the exponentialform of tExample:
he number125 5 5
.125=
53 means 3 factors of 5 or 5 x 5 x 5
The Laws of Exponents:#1: Exponential form: The exponent of a power indicates how many times the base multiplies itself.
n
n times
x x x x x x x x−
= ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅!!!!!!!!!
3Example: 5 5 5 5= ⋅ ⋅
n factors of x
#2: Multiplying Powers: If you are multiplying Powers with the same base, KEEP the BASE & ADD the EXPONENTS!
m n m nx x x +⋅ =
So, I get it! When you multiply Powers, you add the exponents!
5122222 93636
=
==× +
#3: Dividing Powers: When dividing Powers with the same base, KEEP the BASE & SUBTRACT the EXPONENTS!
mm n m n
n
xx x x
x−= ÷ =
So, I get it!
When you divide Powers, you subtract the exponents!
16
2222 4262
6
=
== −
Try these:
=× 22 33.1
=× 42 55.2
=× 25.3 aa
=× 72 42.4 ss
=−×− 32 )3()3(.5
=× 3742.6 tsts
=4
12
.7ss
=5
9
33.8
=44
812
.9tsts
=54
85
436.10
baba
=× 22 33.1=× 42 55.2=× 25.3 aa
=× 72 42.4 ss
=−×− 32 )3()3(.5
=× 3742.6 tsts
8133 422 ==+
725 aa =+
972 842 ss =×× +
SOLUTIONS
642 55 =+
243)3()3( 532 −=−=− +
793472 tsts =++
=4
12
.7ss
=5
9
33.8
=44
812
.9tsts
=54
85
436.10
baba
SOLUTIONS
8412 ss =−
8133 459 ==−
4848412 tsts =−−
35845 9436 abba =×÷ −−
#4: Power of a Power: If you are raising a Power to an exponent, you multiply the exponents!
( )nm mnx x=
So, when I take a Power to a power, I multiply the exponents
52323 55)5( == ×
#5: Product Law of Exponents: If the product of the bases is powered by the same exponent, then the result is a multiplication of individual factors of the product, each powered by the given exponent.
( )n n nxy x y= ⋅So, when I take a Power of a Product, I apply the exponent to all factors of the product.
222)( baab =
#6: Quotient Law of Exponents: If the quotient of the bases is powered by the same exponent, then the result is both numerator and denominator , each powered by the given exponent.
n n
n
x xy y
⎛ ⎞=⎜ ⎟
⎝ ⎠
So, when I take a Power of a Quotient, I apply the exponent to all parts of the quotient.
8116
32
32
4
44
==⎟⎠
⎞⎜⎝
⎛
Try these:
( ) =523.1( ) =43.2 a
( ) =322.3 a
( ) =23522.4 ba
=− 22 )3(.5 a
( ) =342.6 ts
=⎟⎠
⎞⎜⎝
⎛5
.7ts
=⎟⎟⎠
⎞⎜⎜⎝
⎛2
5
9
33.8
=⎟⎟⎠
⎞⎜⎜⎝
⎛2
4
8
.9rtst
=⎟⎟⎠
⎞⎜⎜⎝
⎛2
54
85
436.10
baba
( ) =523.1( ) =43.2 a
( ) =322.3 a
( ) =23522.4 ba
=− 22 )3(.5 a
( ) =342.6 ts
SOLUTIONS
10312a
6323 82 aa =×
6106104232522 1622 bababa ==×××
( ) 4222 93 aa =×− ×
1263432 tsts =××
=⎟⎠
⎞⎜⎝
⎛5
.7ts
=⎟⎟⎠
⎞⎜⎜⎝
⎛2
5
9
33.8
=⎟⎟⎠
⎞⎜⎜⎝
⎛2
4
8
.9rtst
=⎟⎟⎠
⎞⎜⎜⎝
⎛2
54
85
43610
baba
SOLUTIONS
( ) 62232223 8199 babaab == ×
2
8224
rts
rst
=⎟⎟⎠
⎞⎜⎜⎝
⎛
( ) 824 33 =
5
5
ts
#7: Negative Law of Exponents: If the base is powered by the negative exponent, then the base becomes reciprocal with the positive exponent.
1mmxx
− =So, when I have a Negative Exponent, I switch the base to its reciprocal with a Positive Exponent.
Ha Ha!
If the base with the negative exponent is in the denominator, it moves to the numerator to lose its negative sign!
9331
1251
515
22
33
==
==
−
−
and
#8: Zero Law of Exponents: Any base powered by zero exponent equals one.
0 1x =
1)5(
1
15
0
0
0
=
=
=
aandaand
So zero factors of a base equals 1. That makes sense! Every power has a coefficient of 1.
Try these:
( ) =022.1 ba=× −42.2 yy
( ) =−15.3 a
=×− 72 4.4 ss
( ) =−− 4323.5 yx
( ) =042.6 ts
=⎟⎟⎠
⎞⎜⎜⎝
⎛−122.7
x
=⎟⎟⎠
⎞⎜⎜⎝
⎛−2
5
9
33.8
=⎟⎟⎠
⎞⎜⎜⎝
⎛−2
44
22
.9tsts
=⎟⎟⎠
⎞⎜⎜⎝
⎛−2
54
5
436.10baa
SOLUTIONS
( ) =022.1 ba=× −42.2 yy
( ) =−15.3 a
=×− 72 4.4 ss
( ) =−− 4323.5 yx
( ) =042.6 ts
1
22 1
yy =−
5
1a
54s
( ) 12
81284
813
yx
yx =−−
1
=⎟⎟⎠
⎞⎜⎜⎝
⎛−122.7
x
=⎟⎟⎠
⎞⎜⎜⎝
⎛−2
5
9
33.8
=⎟⎟⎠
⎞⎜⎜⎝
⎛−2
44
22
.9tsts
=⎟⎟⎠
⎞⎜⎜⎝
⎛−2
54
5
436.10baa
SOLUTIONS
44 1 xx
=⎟⎠
⎞⎜⎝
⎛−
( ) 8824
3133 == −−
( ) 44222 tsts =−−−
2
101022
819
abba =−−