Download Exponent Properties

Properties of Exponents
Definition of Exponent
• An exponent is the power p in an
expression ap.
52
• The number 5 is the base.
• The number 2 is the exponent.
• The exponent is an instruction that tells us
how many times to use the base in a
multiplication.
Examples
43 =(4)(4)(4) = 64
-34 =(-)(3)(3)(3)(3) = -81
(-2)5 =(-2)(-2)(-2)(-2)(-2)= -32
(-3/4)2 =(-3/4)(-3/4) = (9/16)
Puzzler
2
(-5)
2
-5 ?
=
(-5)(-5) = -(5)(5)
25 = -25
Multiplication with Exponents
by Definition
2
3
5
3
= (3)(3) (3)(3)(3)(3)(3)
= 37
Note 2+5=7
Property 1 for Exponents
• If a is any real
number and r and s
are integers, then
a a  a
r
s
rs
To multiply two expressions with
the same base, add exponents
and use the common base.
Examples of Property 1
x x 
2
x
4
6
2x
6
Examples of Property 1
2 2  2
3
2  32
5
2
5
4  1024
5
2  2  8  4  32
3
2
Examples of Property 1
5 5 
3
9
5
6
9
25
Power to a Power
by Definition
2
3
(3 )
=
1
1
1
((3)(3)) ((3)(3)) ((3)(3))
=
6
3
Note 3(2)=6
Property 2 for Exponents
• If a is any real number and r and s are
integers, then
a 
s
r
a
r s
A power raised to another power
is the base raised to the product
of the powers.
Examples of Property 2
3 
3
2
 3
6
One base, two exponents…
multiply the exponents.
Property 3 for Exponents
• If a and b are any real number and r is an
integer, then
ab 
r
 a b
r
Distribute the
exponent.
r
Examples of Property 3
 5x 
2
 5 x  25x
2
2
2
Examples of Property 3
 2x y 
2
5
3
(3 x y )  2 x y 3 x y
6 15
8 2
 8x y 9 x y
4
2
3
6 15 2
8
2
 (8  9)( x x )( y y )
6 8
 72x y
14 17
15
2
By the Definition of Exponents
5
x
3
x
xxxxx


xxx
Notice that
5–3=2
5
2
x
2
x
1
x
53
2

x

x
3
x
Examples
3
a
a


a
2
a
1
200
7
3
b b
3


b
4
1
b
c
100

c
100
c
Negative Exponents
3
1
xxx
x


2
5
x

x

x

x

x
x
x
3
1
x
3

5

2
Notice that


x

x
2
5
3 – 5 = -2 x
x
Definition of Negative Exponents
• If r is a positive integer, then
a
r
1 1
 r  
a
a
a0
r
Examples of Negative Exponents
1
1
2  3 
2
8
3
Notice that: Negative Exponents do
not indicate negative numbers.
Negative exponents do indicate
Reciprocals.
Examples of Negative Exponents
3
3x  6
x
6
Notice that exponent does not
touch the 3.
Property 4 for Exponents
• If a is any real number and r and s are
integers, then
r
a
r s
(
a

0)

a
s
a
To divide like bases
subtract the exponents.
Property 5 for Exponents
• If a and b are any two real numbers and r
is an integer, then
r
r
a  a
  br
b
The power of a quotient
is the quotient of the
powers.
Property 5 Example
2
25
5 5


 
2
2
y
y
y
 
2
The power of a quotient
is the quotient of the
powers.
Zero as an Exponent
2
5
25

1

2
5
25
2
5
2 2
0

1

5

5
2
5
Zero to the Zero?
2
STOP
5
2 2
0

1

5

5
2
5
2 Undefined
0
2 2
0

1

0

0
2
0
Zeros are not allowed in the
denominator. So 00 is undefined.
Examples
8  1
0
8  8
1
4  4 1 4  5
0
1
Exponent Rules
Summary
Properties
a a  a
r
s
a 
r s
Definitions
a
r
1
 r
a
a0
r s
a

 ab 
r
a a
r
r
a 1
a b
r
a
r s
a
s
a
1
0
rs
a0
a a
   r
b b
r
r
Definition: Scientific Notation
• A number is in scientific notation when it is
written as the product of a number
between 1 and 10 and an integer power of
10.
• A number written in scientific notation has
the form
r
n 10
where 1< n < 10 and r = an integer.
Write 1349.7 in scientific notation.
1349.7  1.3497 10
3
Write 1.85 x 104 in expanded
notation.
1.85  10  18500
4
4
More on Scientific Notation
5.7 10
How far does
the decimal
point move?
4
1
 5.7  4
10
1
 5.7 
10000
5.7

 0.00057
10000