Download 7-1 Multiplying Monomials

Notes 7-1
Multiplying Monomials
I. What is a monomial?
Discuss.
A monomial is a number, a variable, or a product
of numbers and variables with whole-number
exponents. Monomials that are real numbers are
called constants.
A. Identifying monomials
Expression Monomial? Reason
-5
Yes
p+q
No
x
Yes
-5 is a real number
and an example of a
constant
The expression
involves the addition,
not the product, of
two variables
Single variables are
monomials
B. Examples: Identifying
monomials
 a. mn2
Yes
 b. 3x2 + 5x + 7
No
 c. 0.05ab
Yes
 d. -19x +5
No
 e. -19x
Yes
Multiplying monomials is often used when
comparing a characteristic of several
items, such as acidity of different fruits.
It is also used when determining the
probability of something, like guessing the
correct answer on a test or winning the
lottery (Chapter 12).
Today, you will learn three new properties
that will help you multiply monomials.
II. Products of Powers
Products of powers with the same base can be
found by writing each power as a repeated
multiplication.
am  an = (a  a  …  a)  (a  a  …  a)
m factors
n factors
= a  a  …  a = am+n
m + n factors
KEY CONCEPT
Product of Powers
Words: To multiply two powers that have the
same base, add their exponents.
Symbols: For any number a and all integers
m and n, am • an = am + n
Example: a4 • a12 = a4 + 12 or a16
Remember!
A number or variable written without an exponent
actually has an exponent of 1.
10 = 101
y = y1
Simplify.
A.
Since the powers have the
same base, keep the base
and add the exponents.
Your turn!
Simplify.
a.
Since the powers have
the same base, keep
the base and add the
exponents.
Simplify.
B.
Group powers with the
same base together.
Add the exponents of
powers with the same
base.
Your turn!
 b. a2b6a4b9
a2a4b6b9
a2
+ 4b 6 + 9
a6b15
Group powers with the same base
together.
Add the exponents of powers with
the same base.
Multiply.
C. (6y3)(3y5)
(6y3)(3y5)
(6  3)(y3  y5)
Group factors with like bases
together.
18y8
Multiply.
D. (3mn2) (9m2n)
(3mn2)(9m2n)
(3  9)(m  m2)(n2  n)
27m3n3
Group factors with like bases
together.
Multiply.
Your turn!
Multiply.
c. (3x3)(6x2)
(3x3)(6x2)
(3  6)(x3  x2)
18x5
Group factors with like bases
together.
Multiply.
d. (2r2t)(5t3)
(2r2t)(5t3)
(2  5)(r2)(t3  t)
10r2t4
Group factors with like bases
together.
Multiply.
Again…
When multiplying powers with the same base,
keep the base and add the exponents.
x2  x3 = x2+3 = x5
III. Power of a Power
To find a power of a power, you can use the meaning
of exponents.
= am  am  …  am
n factors
= a  a … a  a  a … a  … a  a … a = amn
m factors
m factors
m factors
n groups of m factors
KEY CONCEPT
Power of a Power
Words: To find the power of a power, multiply
the exponents.
Symbols: For any number a and all integers
m and n, (am)n = am • n
Example: (k5)9 = k5 • 9 or k45
Simplify.
Use the Power of a Power Property.
Simplify.
Your turn!
Simplify.
Use the Power of a Power Property.
Simplify.
Simplify
B. [(32)3]2
(32•3)2
(36)2
36•2
312
Power of a Power
Simplify
Power of a Power
Simplify
You Try!
Simplify
B. [(22)2]4
(22•2)4
(24)4
24•4
216
Power of a Power
Simplify
Power of a Power
Simplify
Powers of products can be found by using the
meaning of an exponent.
(ab)n = ab  ab  …  ab
n factors
= a  a  …  a  b  b  …  b = anbn
n factors
n factors
KEY CONCEPT
Power of a Product
Words: To find the power of a product, find
the power of each factor and multiply.
Symbols: For all numbers a and b and any
integer m, (ab)m = ambm
Example: (-2xy)3 = (-2)3x3y3 or -8x3y3
Simplify.
A.
Use the Power of a Product Property.
Simplify.
B.
Use the Power of a Product Property.
Simplify.
Caution!
In Example 4A, the negative sign is
not part of the base. –(2y)2 = –1(2y)2
Classwork
Workbook
Section 7-1 Page 87