2-4 properties of exponents page 97 a n is called a power 거듭제곱 a is the base. 밑 n is the...
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2-4 Properties of Exponentspage 97
an is called a Power 거듭제곱a is the base. 밑n is the exponent. 지수we read it as a to the nth power or a to the n, or a to the power n
Remember::::: (page 99) Simplified Monomial•
①there are no powers of powers②each base appears once
③fractions are also simplified④there are no negative exponents
This property states that for any real num-ber x,a,b, .
Example : 9*27=243
1.Product of Powers
Properties of Exponent
This property states that for any real num-ber x,a,b, and if x is not 0, then
Example :
2. Quotient of Powers
27
This property states that for any real num-ber x,a,b, and when x is not 0, then
Example :
Negative of Exponent (this is not included in the MS2 curriculum)
= =
What is ?
Zero to zeroth power is often said to be"an indeterminate form", because it could have several different values.
Since x0 is 1 for all numbers x other than 0, it would be logical to define that 00 = 1.
But we could think of 00 also having the value 0, because zero to any power (other than the zero power) is zero.
Also, the logarithm of 00 would be 0 × infinity, which is in itself an indeterminate form. So laws of logs wouldn't work with it.
For our class, we will define it as 0
0 to any power is 0
𝟎𝟎=𝟎
TODAY : OPERATION OF POLYNOMIAL
Properties of Exponent
Power of Power
= (a x a -> x times) -> y times= a x a -> xy times
= a xy(a ) x
*When a is not 0
y
Why is it?
= = (2 x 2 x 2)==2 times= (2 x 2 x 2) (2 x 2 x 2)= 2 x 2 x 2 x 2 x 2 x 2 =64
This property states that for any real num-ber x,a,b, .
Example 1 : Example 2 : Prove example 1 :
Power of a Power
This property states that for any real num-ber x,y,a,b,
, y is not 0, and , x is not 0
Examples : ,
Power of a quotient(Negative exponents are not part of the MS2 curricu-lum)
Property Definition Examples
Product of Powers z4×z3=z4+3
76 ×7 2= 76+2
Quotient of Powers
Negative Exponent
Power of a Power
Power of a Product
Power of a Quotient
Zero Power
bab
a
xx
xx ,0
0,11
xx
xand
xx a
aaa
baba xxx
abba xx
aaa yxxy
0,10 xx
If
,0,
y
y
x
y
xa
aa
and
0,0,
yxx
yor
x
y
y
xa
aaa
4484
8235
3
5
,666
6xx
x
x
996
6 1,
3
13 b
b
3631231263232 ,444 xxx
2223333 ,822 qppqyyy
7
77
5
55
,b
a
a
b
x
y
x
y
17561928492034
180
0
Simplify 1. Using the Power of a Quotient, 2. Using the Power of a Product,
3. Using the power of a power, the answer is
Example of using proper-ties
Simplify an expression
1 Try this question first!
Definition ofnegative exponent
Definition of exponents
Divide out commonfactors
Simplify~!
Simplify an expression
Definition of exponents
Divide out commonfactors
Simplify~!
Practice – page 101
1) 2)
4) Find an expression for the area of the circle
6x4A=πr2A=π(6x4)2A=π(62x8)A=36 π x8
5) 6) 7)
Equations involving variable exponents with the same base
Since they have the same base, the expo-nents are equal….. x=y+2
36=9𝑎
𝑎=3
Homework
Page 102Make any exponent positive #15
If you do the additional exercises on pages 103-104 the answers include problems where the exponents in the question were negative so either do them with negative exponents or skip them.