Multiplying Monomials
Jen Kershaw
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Printed: May 12, 2015
AUTHOR
Jen Kershaw
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Chapter 1. Multiplying Monomials
C HAPTER
1
Multiplying Monomials
Here you’ll multiply monomials by expanding the expression, regrouping factors and multiplying the coefficients.
Do you know how to multiply monomials? Take a look at this one.
(6y3 )(8y5 )(−1xy)
This Concept will show you how to multiply monomial expressions.
Guidance
You have already seen exponents, but let’s review the definitions of some words so that we can get started. First,
let’s look at some of the parts of a term.
In the monomial above, the 7 is called the coefficient , the x is the variable , and the 3 is the exponent .
We can say that the monomial 7x3 has a power of 3 or is to the 3rd power.
Remember what we said about the coefficient of a variable like x—if there is no visible coefficient, then the
coefficient is an unwritten 1. You could write “1x” but it is not necessary. Similarly, if there is no exponent on
a coefficient or variable, then you can think of it as having an unwritten exponent of 1. So 7 could be written as 71 .
The constant 7 then, is to the 1st power.
Also, the exponent is applied to the constant, variable, or quantity that is directly to its left. That value is called the
base . In the monomial above, the base is x. The exponent, in this case, is not applied to the 7 because it is not
directly to the left of the exponent.
What is the exponent ? It’s a shortcut. It’s a way of writing many multiplications in a simpler way. In the monomial
above, 7x3 , the 3 indicates that the variable x is multiplied by itself three times.
7x3 = 7 · x · x · x
You can see the amount of space that is saved by using the exponent. When we write all of the multiplications
instead of using the exponent, it is called the expanded form . You can see that it is, indeed, expanded—it takes
much more space to write. Imagine if the exponent were greater, like 7x27 .
7x27 = 7 · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x · x
The exponent truly saves a lot of space!
Let’s look at how to write an expression out into expanded form.
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(7x3 )(4x5 )
= (7 · x · x · x)(4 · x · x · x · x · x)
Here we have written the expression out into expanded form.
Now, use the commutative property of multiplication to change the order of the factors so that similar factors are
next to each other.
= 7·4·x·x·x·x·x·x·x·x
= 28x
8
Parentheses disappear.
Multiply coefficients 7 and 4.
Use exponent as a shortcut for the variables.
This may seem like a cumbersome long way to work, but it is accurate!
We could do this one another way too.
When we multiply the expressions, we multiply the coefficients, but we add the exponents. Take a look.
(7x3 )(4x5 )
7 × 4 = 28
3+5 = 8
28x8
Notice that we got the same answer as doing it out the long way!
Multiply the following monomials.
Example A
(6x2 )(4x4 )
Solution: 14x6
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Chapter 1. Multiplying Monomials
Example B
(2x3 )(4x9 )
Solution: 8x12
Example C
(6y3 )(8y5 )
Solution: 48y8
Now let’s go back to the dilemma from the beginning of the Concept.
(6y3 )(8y5 )(−1xy)
First, let’s multiply the coefficients.
6 × 8 × −1 = −48
Now we can add the x because there isn’t another x to multiply with this one.
−48x
Next, we multiply the y0 s and we do this by adding the exponents.
−48xy9
This is our answer.
Guided Practice
Here is one for you to try on your own.
Multiply the following monomials.
(−6x3 )(8y5 )
Solution
In this one, we can begin by multiplying the coefficients.
−6 × 8 = −48
Now we simply put the terms together. We can’t add the exponents because the terms are not alike.
−48x3 y5
This is our answer.
Video Review
MEDIA
Click image to the left or use the URL below.
URL: http://www.ck12.org/flx/render/embeddedobject/63320
Multiplying Monomials
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Explore More
Directions: Multiply the following monomials.
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
11.
12.
13.
14.
15.
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(5x)(6xy)
(5x2 )(−6xy)
(−5x2 y)(2xy2 )
(−5x)(−9yz)
(18xy)(2xy2 z)
(2y4 )(6y5 )
(5x3 )(−5x4 y3 )
(−2y5 )(6y3 )(2y2 )
(5xy)(−2xy)(−x2 y2 )
(2ab)(6ab)(−4ab)
7x(6xy)
(15x2 )(−10x3 )
(5x)(6xy)(−9xy5 )
(−2x3 )(−4xy)(−5x2 y4 )
(−4abc)(−8a)(−4c)(d 2 )