In this explainer, we will learn how to multiply monomials involving single and multiple variables.
Algebraic expressions allow us to represent unknown values. This means that manipulating and simplifying algebraic expressions can help us rewrite or simplify these unknown values expressed in terms of some variables.
In this explainer, we will focus on simplifying the product of monomial terms. To do this, let’s start by recalling the definition of a monomial.
Definition: Monomial
A monomial is a single algebraic term where every variable is raised to a nonnegative integer power.
For example, is a monomial since it is a single term and the variable is raised to a nonnegative integer power; . Similarly, is a monomial since it is a single term and every variable is raised to a nonnegative integer power.
We want to multiply two monomials. We can do this by considering an example. Let’s say we want to multiply by . We have
Remember that represents a number, so we can use here the commutative and associative properties of multiplication to rewrite this product as follows:
The product of the coefficients is . Similarly, we can remember that the repeated multiplication can be written as .
Hence, we have shown that
We evaluated this product of monomials by multiplying the coefficients and the factors of separately.
When we multiplied the factors of in this expression, we saw that had 2 factors of and has 1 factor of . We can add the number of factors together to get the number of factors of in the product:
We can note that this process will work in general, since we add the number of factors to determine the power. This is called the product rule for powers, and we can prove this result following the same reasoning.
Let and and be nonnegative integers. We can multiply by by expanding the powers and then adding the number of factors of :
We have shown the following result.
Rule: Product Rule for Exponents
For any rational number and nonnegative integers and , we have
It is important to understand here why the product rule is true only if the two powers have the same base. Let’s look at an example: . We can write and here, we see that we are stuck; we cannot simplify this expression because the bases are different (2 and 5).
This holds true for any base. This means it will also hold true if we have a variable as a base.
In our first example, we will apply the product rule to simplify the product of two monomials.
Example 1: Applying the Product Rule
Simplify .
Answer
We want to simplify this product of two powers. We see that the base is the same in both factors and both powers are nonnegative integers. Therefore, we can apply the product rule, which tells us that for any rational number and nonnegative integers and , we have
In other words, the product of two powers that have the same base can be expressed as a power of this base with an exponent equal to the sum of the exponents.
Hence, we add the exponents to get
In our next example, we will find the product of two monomials with variables raised to powers greater than 1.
Example 2: Multiplying Monomials Using Laws of Exponents
Simplify .
Answer
In this question, we want to simplify the product of two monomials. To make the product clearer, we can rewrite this product by including the multiplication signs that are implicit within the monomials, that is, between the coefficient and the variables:
Now, we can use the commutativity and associativity of multiplication to rearrange our right-hand side expression:
Now, we multiply the coefficients together:
Since the bases in the final two factors are both and the powers are nonnegative integers, the product rule for exponents tells us that we can evaluate their product by adding the exponents. We have
The product of monomials is not limited to positive integer coefficients, as we will see in our next example.
Example 3: Multiplying Monomials with Fractional Coefficients
Simplify .
Answer
We first rewrite the product, using the commutativity and associativity of multiplication, as
Next, we can evaluate each product inside the parentheses separately. To multiply two fractions, we multiply the numerators and denominators. We have
To evaluate , we need to use the product rule for exponents, which tells us that we can evaluate this product by adding the exponents. This gives
Substituting these back into the expression gives
In our next example, we will use this process for multiplying monomials to simplify an expression for the volume of a given rectangular prism.
Example 4: Writing an Algebraic Expression for the Volume of a Rectangular Prism by Multiplying Monomials
Find an expression for the volume of the rectangular prism shown.
Answer
We first recall that the volume of a rectangular prism is given by the product of its length, width, and height. If we call this volume , we have
We can simplify this expression by noting that this is the product of three monomials. This means we can multiply the coefficients and variables separately by using the commutativity and associativity of multiplication. We have
We then note that the repeated multiplication can be written as . Thus,
We can use the product rule for exponents to simplify the product of any monomials even if they have multiple variables. For example, let’s say we want to find the product of and . We can start by using commutativity and associativity of multiplication to rewrite the product as
Now, we can evaluate each product inside the parentheses separately. We calculate that and we can evaluate the other products by using the power rule:
It is also worth noting that this process shows us that the product of any monomials is always a monomial, since we add the integer exponents to get new integer exponents for the variables.
Example 5: Multiplying Monomials Involving More Than One Variable
Simplify .
Answer
We first rewrite the product, using the associativity and commutativity of multiplication, as
Now, we evaluate the product of the coefficients, and the product of the variables using the product rule :
We get
In our final example, we will simplify the product of monomials in multiple variables with fractional coefficients.
Example 6: Multiplying Monomials Involving More Than One Variable
Simplify .
Answer
In this question, we want to multiply two monomials. To make the multiplication clearer, we can start by writing each monomial as a product of its coefficients and variables:
Now, we can use the commutativity and associativity of multiplication to rearrange our right-hand side expression:
We can evaluate the product of the coefficients by multiplying the numerators and denominators:
We can evaluate the products of the variables by using the product rule :
We get
Let’s finish by recapping some of the important points from this explainer.
Key Points
- A monomial is a single algebraic term where every variable is raised to a nonnegative integer power.
- The product rule for exponents tells us that for any rational number and nonnegative integers and , we have .
- The product of monomials is a monomial.