12.7: Multiplying Monomials
Do you know how to multiply monomials? Take a look at this one.
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
We can say that the monomial
Remember what we said about the coefficient of a variable like
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
What is the exponent? It’s a shortcut. It’s a way of writing many multiplications in a simpler way. In the monomial above,
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
The exponent truly saves a lot of space!
Let’s look at how to write an expression out into expanded form.
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.
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.
Notice that we got the same answer as doing it out the long way!
Multiply the following monomials.
Example A
Solution:
Example B
Solution:
Example C
Solution:
Now let's go back to the dilemma from the beginning of the Concept.
First, let's multiply the coefficients.
Now we can add the
Next, we multiply the
This is our answer.
Vocabulary
 Monomial
 a single term of variables, coefficients and powers.
 Coefficient
 the number part of a monomial or term.
 Variable
 the letter part of a term
 Exponent
 the little number, the power, that tells you how many times to multiply the base by itself.
 Base
 the number that is impacted by the exponent.
 Expanded Form
 write out all of the multiplication without an exponent.
Guided Practice
Here is one for you to try on your own.
Multiply the following monomials.
Solution
In this one, we can begin by multiplying the coefficients.
Now we simply put the terms together. We can't add the exponents because the terms are not alike.
This is our answer.
Video Review
Practice
Directions:Multiply the following monomials.

(5x)(6xy) 
(5x2)(−6xy) 
(−5x2y)(2xy2) 
(−5x)(−9yz) 
(18xy)(2xy2z) 
(2y4)(6y5) 
(5x3)(−5x4y3) 
(−2y5)(6y3)(2y2) 
(5xy)(−2xy)(−x2y2) 
(2ab)(6ab)(−4ab) 
7x(6xy) 
(15x2)(−10x3) 
(5x)(6xy)(−9xy5) 
(−2x3)(−4xy)(−5x2y4)  \begin{align*}(4abc)(8a)(4c)(d^2)\end{align*}
Base
When a value is raised to a power, the value is referred to as the base, and the power is called the exponent. In the expression , 32 is the base, and 4 is the exponent.Coefficient
A coefficient is the number in front of a variable.Expanded Form
Expanded form refers to a base and an exponent written as repeated multiplication.Exponent
Exponents are used to describe the number of times that a term is multiplied by itself.Power
The "power" refers to the value of the exponent. For example, is "three to the fourth power".power to a power
Power to a power is a number raised to an exponent which in turn is raised to another exponent.Variable
A variable is a symbol used to represent an unknown or changing quantity. The most common variables are a, b, x, y, m, and n.Image Attributions
Description
Learning Objectives
Here you'll multiply monomials by expanding the expression, regrouping factors and multiplying the coefficients.