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Sections 8-3, 8-4, & 8-5 Properties of Exponents
SPI 11D: use exponents to simplify a monomial written in expanded form
Objectives:
• Apply the rules of exponents
Monomial:
• an algebraic expression consisting of only one term
Example of some monomials: 12, y, -5x2y, ½x2
Exponent:
• a quantity representing the power to which some other
quantity is raised
• shorthand way to show products of like factors
2 • 2 • 2 = 23
Exponent
Factors Base
Property of Exponents
Multiplying powers with the Same Base
For every non-zero number a and integers m and n,
am ∙ an = a m + n
If multiplying terms with like bases, add the exponents.
Rewrite each expression using exponents.
2
4  4  4  441( 2)
 43
4
1
6 x  x  2 x  12 x
2
3
2  31
 12x 6
Property of Exponents
When terms have more than one base, combine only those
powers with the same base
Simplify each expression.
3a  b  a
4
4
2
 3a
4  ( 2 )
 3a  b
2
4
18
5x  2 y  3x  15x
4
8
b
4
2y
 30 x y
9
4
4
Property of Exponents
Negative Exponents
Always remove negative signs from the exponents
For every non-zero number a and integer n,
1
1
n
n
a  n

a
a
a n
If term with negative exponent is on top, put on bottom and
remove negative sign from exponent, and vice versa.
Simplify each expression:
4 yx
3
4y
 3
x
4
3xy  3xy4 w4
4
w
Property of Exponents
Raising a Power to a Power
Always remove negative signs from the exponents
For every non-zero number a and integers m and n,
a 
n m
 a mn
Simplify each expression:
18
3 6
x 
x
c c
5

3 2
5 6
c c
c
5  ( 6 )
c
1
1

c
Property of Exponents
Raising a Product to a Power
Always remove negative signs from the exponents
For every non-zero number a and integer n,
ab 
n
a b
n
2n 
n
2 3
 23 n6  8n6

 2 4 c 6
Simplify each expression:
3x 
3 6
3 x
6
18
 729x
18
2
2
c
3 2
16
 6
c
Property of Exponents
Dividing powers with the Same Base
For every non-zero number a and integers m and n,
am
mn

a
an
If dividing terms with like bases, subtract the exponents.
Simplify each expression
6
a
6 3

a
a3
 a9
a 6  a 614
a14
 a 8
1
 8
a
Property of Exponents
Multiplying Numbers in Scientific Notation
Scientific Notation:
• shorthand to write very large or very small numbers
• written in the form with base of 10 (2 10 )( 4 10 )
3
5
• all the exponent rules apply
Simplify the expression and write in scientific notation.
(2 103 )(4 105 )  (2  4)(103 105 )
 8  10 3 5
8
 8 10
Exponents and Physical Science
An object has a mass of 102 kg. The expression
102 • (3  108)2 describes the amount of resting energy in
joules the object contains. Simplify the expression and write
in scientific notation.
102 • (3  108)2 = 102 • 32 • (108)2
= 102 • 32 • 1016
= 32 • 102 • 1016
= 32 • 102 + 16
= 9  1018