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Be able to multiply monomials.
Be able to simplify expressions involving
powers of monomials.
 6b 
5a 2a
3
4
2
4 
 c
5 
3
Monomials - a number, a variable, or a product of a
number and one or more variables.
Constant – A monomial that is a real number.
Power – An expression in the form xn.
Base – In an expression of the form xn, the base is x.
Exponent – In an expression of the form xn, the
exponent is n.
Power
y
8
Base
 3
 
 2
4
Exponent
Definitions
Product of Powers
For any number a, and all integers m and n, am * an = am+n.
(a3 ) (a4 ) = a3+4 = a7
Power of a Power
For any number a, and all integers m and n, (am) n = amn .
(am )n = amn.
Product of a Product
For all numbers a and b, and any integer m, (ab)m = am bm .
(2*x)2 = 22 x2
Definitions
Power of a Monomial
For all numbers a and b, and all integers m, n, and p,
(ammn)p = ampbnp.
(22x3)4 = 22*4x3*4 = 28x12
Quotient of Powers
For all integers m and n, and any nonzero number a,
a7
7 5
a .
5
a
am
 a mn .
n
a
Definitions
Zero Exponent
For any nonzero number a, a0 = 1.
40 = 1
Negative Exponents
1
n
a

.
For any nonzero number a and any integer n,
n
a
1
x  3
x
3
Writing Using Exponents
Rewrite the following expressions using exponents.
xxxx y y
The variables, x and y, represent the bases. The
number of times each base occurs will be the value of
the exponent.
xxxx y yx y
4 2
2 2 2 2  2
    
3 3 3 3  3
4
Writing Out Expressions with
Exponents
Write out each expression the long way.
8a b  8  a  a  a  b  b
3 2
xy
4
 xy  xy  xy  xy
The exponent tells how many times the base occurs. If
the exponent is outside the parentheses, then the
exponent belongs with each number and/or variable
inside the parentheses.
Simplify the following expression:
(5a2)(a5).
Step 1: Write out the expressions the long way
or in expanded form.
5a a  5  a  a  a  a  a  a  a
2
5
Step 2: Rewrite using exponents.
5  a  a  a  a  a  a  a  5a
For any number a, and all integers m and n,
am • an = am+n
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Simplify the following:
x 
3 4
First, write the expression in expanded form.

x
3 4
 x3  x3  x3  x3
However,
x3  x3  x3  x3  x  x  x  x  x  x  x  x  x  x  x  x
Therefore,
 
x
3 4
 x12
Note:
3 x 4 = 12.
For any number, a, and all integers m and n,
a 
m n
a
mn
(xy)5
Simplify:
xy5  xy  xy  xy  xy  xy
 ( x  x  x  x  x )( y  y  y  y  y)
 x5y5
For all numbers a and b, and any integer m,
 ab
m
a b
m m
Simplify:
4 4
5
4
4
6
5+ 6
11
Apply the
Product of
Powers property
3 6
(a )
a
a
3 6
18
Apply the
Power of a
Power Property.
3x y
3 x  y 
4
3
3
4 3
27 x 12 y 3
Apply the Power
of a Product
Property and
Simplify.
3
1.
r t r t 
4 5
7 3
3
2.
 
 1 3
 w  w2
2 
4
Problem 1
r t r t  r  r t  t 
 r
t 
4 5
7 3
4
4+ 7
7
5
5+ 3
r t
11 8
Group like terms.
Apply the Product of Powers Property.
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