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9-1: Multiplying Monomials
OBJECTIVES:
You will multiply monomials and simplify expressions involving powers
of monomials.
To start the chapter, a couple of terms need to be defined.
•monomial - a number, a variable, or a product of a number and one or
more variables
Examples:
Monomials
12
11ab
q
4x3
Not Monomials
1
xyz12
3
a+b
5 - 7d
a
b
5
a2
4a
7b
•constant - either a monomial that is only a real number (no variables
attached)
-or- can refer to the real number part of the monomial
(also called coefficient later)
© William James Calhoun, 2001
9-1: Multiplying Monomials
Remember what “25” really means and why its used?
In dealing with math, there were instances where a common base (the 2)
was multiplied by itself several times, like:
2 x 2 x 2 x 2 x 2.
Rather than write that out every time, mathematicians created the
exponential notation where 25 means two multiplied by itself five times.
It is now time to put the exponential notation on variables.
In this section we will learn four rules about dealing with exponents.
© William James Calhoun, 2001
9-1: Multiplying Monomials
If “2x2x2x2x2” can be written as 25, what do you think x6 represents?
From the defining of exponents, we can see that x6 represents some
unknown number multiplied by itself six times.
In long-winded terms:
x6 = (x)(x)(x)(x)(x)(x)
This part is easy and is the same as dealing with plain old numbers.
Now, what happens when letters and powers start combining?
© William James Calhoun, 2001
9-1: Multiplying Monomials
Using the definition of what an exponent is, examine:
x5y4(4x3)
In long terms, this problems becomes:
(x)(x)(x)(x)(x)(y)(y)(y)(y)(4)(x)(x)(x).
From this, we can get all the x’s lined up with each other since order
does not matter when multiplying (commutative property) :
(x)(x)(x)(x)(x)(x)(x)(x)(y)(y)(y)(y)(4).
Now, move the constant to the front and redo the power notation
realizing there are eight x’s multiplied together and four y’s multiplied
together:
4x8y4.
Do you need to go through all those steps every time you hit one of these
problems?
Absolutely...
There is a simple rule you need to memorize which handles this.
© William James Calhoun, 2001
9-1: Multiplying Monomials
9.1.1 PRODUCT OF POWERS
For any number a, and all integers m and n,
am  an = a (m + n).
Another way to put this rule is:
“When same bases are multiplied, add their exponents.”
So, x5  x6 = x(5 + 6) = x11.
To handle the first example, remember you can change order within the
monomial.
You will want to get all constants and same bases together.
Multiply constants and add exponents of same bases.
NEVER add powers of different bases!
© William James Calhoun, 2001
9-1: Multiplying Monomials
EXAMPLE 1: Simplify each expression.
A. (3a6)(a8)
B.  8y3  3x 2 y 2   3 xy 4 


8


Not that you should do this first step all the
3a14
Just remember the rule:
Same base multiplied, add exponents, so:
a6(a8) = a(6 + 8) = a14.
This makes the problem easy.
3 2
8(-3)  (x )(x)(y3)(y2)(y4)
8
Count how
many x’s are
Use calculator multiplied
to multiply
together.
constants.
-9
x3
{
So, how many a’s are multiplied by each other?
14
So, to write this simply, we say…
{
3(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)(a)
In big problems like this, everything is
multiplied and can be written in any order.
Put all the constants and like variables
together.
{
time, but in a technical sense, this problem
reads:
Count how
many y’s are
multiplied
together.
y9
-9x3y9
© William James Calhoun, 2001
9-1: Multiplying Monomials
9.1.2 POWER OF A POWER
For any number a, and all integers m and n,
(am)n = amn.
When you have a power to a power, multiply the powers.
So, (x4)3 = x4(3) = x12.
9.1.3 POWER OF A PRODUCT
For any numbers a and b, and any integer m,
(ab)m = ambm.
This is like distribution. Everything inside gets a piece of what is on the
outside.
So, (xy)3 = x3y3.
© William James Calhoun, 2001
9-1: Multiplying Monomials
9.1.4 POWER OF A MONOMIAL
For any numbers a and b, and any integers m, n, and p,
(ambn)p = ampbnp.
This is pretty much the same as the last rule.
So, (x2y)3 = x2(3)y1(3) = x6y3.
When you are asked to simplify an expression, you must rewrite it so:
(1) there are no powers of powers left,
(2) each base appears only once (no repeats on letters), and
(3) all fractions are reduced.
© William James Calhoun, 2001
9-1: Multiplying Monomials
EXAMPLE 2: Simplify (2a4b)3[(-2b)3]2.
First thing is use the Power of
Products rule.
Use the Power of a Power rule.
3x2
23 (a 4 )3 (b3 ) (2b)6
Take care of the constants
and use Power of a Power
rule.
Use the Power of Products rule.
4x3
(8) (a12)(b3) (-2)6 b6
Also handle the (-2)6.
Get constants and variables
together.
(8)(64)(a12)(b3)(b6)
Twelve a’s.
8x64
Nine b’s.
And a partridge in a pear tree.
(512) (a12) (b9)
512a12b9
© William James Calhoun, 2001
9-1: Multiplying Monomials
HOMEWORK
Page 499
#17 - 35 odd
© William James Calhoun, 2001