In this explainer, we will learn how to multiply and divide algebraic fractions.
We multiply two fractions by multiplying their numerators and denominators separately. For example, we can calculate by multiplying the numerators and denominators separately to get
We can then evaluate and simplify; however, in this case, it is easier to factor and cancel. We note that and , so
We can then see that the numerator and denominator share factors of 2 and 3. We can cancel these shared factors to obtain
We can follow this same process if we are dealing with any types of fractions; however, we do need to be a little bit careful. For example, let’s say we were asked to multiply two algebraic fractions: .
We know that is a variable; it represents an unknown number, so we can treat it like a number. This means that we can multiply these algebraic fractions in the same way, by multiplying their numerators and denominators separately:
We now want to simplify the fraction by canceling any shared factors in the numerator and denominator. To do this, we can first factor the numerators. We have
Therefore,
We now cancel the shared factors of 3 and in the numerator and denominator, where we can note that only when . This means that our expression is only valid when . We obtain
We can cancel any shared factors in this way, even more complicated factors such as
In this case, we can only cancel the factors of if . Otherwise, we would be dividing by 0, which is not defined.
In our first example, we will simplify the product of two algebraic fractions.
Example 1: Multiplying Algebraic Fractions
Simplify .
Answer
To multiply fractions, we multiply the numerators and denominators separately. We have
We can then note that there are shared factors in the numerator and denominator; since 2 and 6 share a factor of 2 and and share a factor of , we can write these products out in full and cancel the shared factors to get
Hence,
In our next example, we will simplify the product of algebraic fractions involving canceling a shared linear factor.
Example 2: Multiplying Algebraic Fractions
Simplify .
Answer
To multiply any fractions, we multiply the numerators and denominators separately. We have
In order to simplify algebraic fractions, we need to fully factorize the numerator and denominator. We note that . This gives us
We can then cancel the shared factor of in the numerator and denominator to obtain
We could leave our answer like this; however, we can also expand the brackets in the numerator to get
Hence,
In our next example, we will apply this method for simplifying the product of algebraic fractions to simplify an expression for the volume of a cube given an algebraic fraction representing its side length.
Example 3: Applications of Multiplying Algebraic Fractions
Find the volume of a cube whose side length is .
Answer
We first recall that the volume of a cube is the cube of its side length. So, if we call the volume of the cube , then we have
We can write the cube out in full to obtain
We could now evaluate the product in pairs; however, it is not necessary. Instead, we can multiply any number of algebraic fractions by multiplying their numerators and denominators separately. We find
We can evaluate the numerator by calculating . In the denominator, we rearrange the product to be . We then calculate and rewrite . This is equivalent to multiplying the coefficients and variables separately. We obtain
Hence, the volume of the cube is given by the equation .
In the previous example, we noted that we could cube an algebraic fraction by writing the product in full and then multiplying the numerators and denominators separately. In particular, we saw
In general, if we have a positive integer , then we can follow this same process to obtain
We can also apply this process of multiplying algebraic fractions to dividing two algebraic fractions by recalling that division by a fraction is equivalent to multiplying by its reciprocal. For example, consider . We can start by changing the division into a multiplication by instead multiplying by the reciprocal of the divisor:
We can now follow the same process we do for multiplying algebraic fractions. We start by multiplying the numerators and denominators separately to obtain
We can now factor and cancel the shared factors in the numerator and denominator to get
Of course, since we cancel the shared factor of in the numerator and denominator, this is only valid when .
In our next example, we will use this process to simplify the division of two algebraic fractions.
Example 4: Dividing Algebraic Fractions
Simplify .
Answer
To divide two fractions, we can instead multiply by the reciprocal of the second fraction:
To multiply any fractions, we multiply the numerators and denominators separately. We have
We can then note that there are shared factors in the numerator and denominator: 2 and . We can cancel these shared factors to get
Hence,
In our next example, we will simplify the division of two algebraic fractions by canceling their shared linear factors.
Example 5: Dividing Algebraic Fractions
Simplify .
Answer
To divide two fractions, we can instead multiply by the reciprocal of the second fraction:
We then need to multiply two algebraic fractions; we do this by multiplying the numerators and denominators separately. We have
Instead of leaving our answer like this, we should try to simplify by factorizing and canceling any shared factors. We can note that , so we have
We can then cancel the shared factors of and in the numerator and denominator, where we note to get
Hence,
In our final example, we will simplify the division of two algebraic fractions by canceling their shared linear factors.
Example 6: Dividing Algebraic Fractions
Simplify .
Answer
To divide two algebraic fractions, we can instead multiply by the reciprocal of the second fraction. This gives us
We then need to multiply these two algebraic fractions; we recall that we can do this by multiplying the numerators and denominators separately. We have
Instead of leaving our answer like this, we should try to simplify by factorizing and canceling any shared factors. We can factorize each polynomial separately. We have
Substituting the factorized expressions into the algebraic fraction yields
We can then cancel the shared factor of in the numerator and denominator to obtain
Hence,
Let’s finish by recapping some of the important points from this explainer.
Key Points
- To multiply two or more algebraic fractions, we multiply the numerators and denominators separately. We can then factorize and cancel shared factors in the numerator and denominator.
- To divide two algebraic fractions, we multiply the first algebraic fraction by the reciprocal of the second algebraic fraction. We can then multiply the numerators and denominators separately and factorize and cancel shared factors in the numerator and denominator.
