Simplify Products or Quotients of Single-Variable Expressions

Recall that when you add and subtract terms in an expression, you can only combine like terms.

However, you can multiply or divide terms whether they are like terms or not.

For example, and  are like terms because both terms include the variable . We can multiply them to simplify an expression like this. .

However, even though  and 3 are not like terms, we can still multiply them, like this.

The Commutative and Associative Properties of Multiplication may help you understand how to multiply expressions with variables. Remember, the Commutative property states that factors can be multiplied in any order. The Associative property states that the grouping of factors does not matter.

Example 1

We can take these two terms and multiply them together.

First, we multiply the number parts.

Next, we multiply the variables.

Example 2

Even though these two terms are different, we can still multiply them together.

First, we multiply the number parts.

Next, we multiply the variables.

Example 3

Find the product . and  are not like terms, however, you can multiply terms even if they are not like terms.

Use the commutative and associative properties to rearrange the factors to make it easier to see how they can be multiplied.

According to the commutative property, the order of the factors does not matter.

So, .

According to the associative property, the grouping of the factors does not matter. Group the factors so that the numbers are multiplied first.

So, .

Now, multiply.

The product is .

Remember that the word PRODUCT means multiplication and the word QUOTIENT means division.

Example 4

Find the quotient .

It may help you to rewrite the problem like this. Then separate out the numbers and variables like this.

Now, divide 42 by 7 to find the quotient.

The quotient is.

Example 5

Find the quotient .

It may help you to rewrite the problem like this. Then separate out the numbers and variables like this.

Now, divide 50 by 10 and divide by to find the quotient. Since any number over itself is equal to 1, you know that.

The quotient is 5.