In this explainer, we will learn how to multiply monomials involving single and multiple variables.
A monomial is an algebraic expression consisting of one term. So, all the following expressions are monomials: , , , .
A monomial may contain one or more variables raised to different powers. Multiplying monomials therefore may involve multiplying powers of different variables.
First, we need to remember what a power means. We write a power in the form , where is called the base and the exponent. And this means that , which can be a number or a variable, is multiplied by itself times:
Now, when we multiply two powers of the same base, we get:
This is known as the Product Rule.
The Product Rule
The product of two powers that have the same base is a power of this same base with an exponent equal to the sum of the exponents:
It is important to understand here why the product rule is true only if the two powers have the same base. Let us look at a simple 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).
So far, we have assumed that the exponent is positive. To understand the meaning of a negative exponent, let us look at what happens when we divide the power of a number by this number; for instance, divided by 6. Given that , we see that .
If we perform the same division again on , we get .
The number is of course simply 6, but it is useful to write the exponent here, as you will see in the following.
So, we start seeing a pattern here: every time we divide by 6, the exponent decreases by 1.
If we carry out the division by 6 once more, we get
And once more,
Let us summarize these findings in a table.
We have just found out something very interesting. The exponent, when positive, tells us how many times the base is used in a multiplication with 1. When it is negative, it tells us how many times the based is used in a division of 1. From this, it follows that any power of a number is the reciprocal of the power with the same base but with an opposite exponent. This is given by the “negative exponent rule.”
Negative Exponent Rule
Any power can be expressed as the reciprocal (or multiplicative inverse) of a power having the same base but an opposite exponent:
Example 1: Applying the Product Rule
We want to simplify this product of two powers. We see that the base is the same ().
Therefore, we can apply the product rule, saying that 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.
Example 2: Multiplying Monomials with Coefficients of Different Signs
In this question, we want to simplify the product of two monomials. To make it clearer, we can rewrite this product adding the multiplication signs that are implicit within the monomials, that is, between the coefficient and the variables:
Now, we can use the commutativity of multiplication to rearrange our right-hand side expression: and multiply the coefficients together: and the powers of together. For this, we apply the product rule:
Putting these two calculations together, we find
Example 3: Multiplying Monomials with Two Variables and Coefficients of Different Signs
Here, we can rewrite this product as
Using the commutativity of multiplication, we can pair like terms together in our expression: and then first evaluate .
Then, we evaluate the product of the powers of by applying the product rule: We do the same for the product of the powers of : The final answer is then
Example 4: Multiplying Monomials with Exponents of Different Signs
Simplify , where .
We have here the product of three powers with the same base, . We can therefore apply the product rule:
We can also use the negative exponent rule to rewrite our expression and better understand what is happening. We know that . Hence,