Lesson Explainer: Multiplying and Dividing Rational Expressions | Nagwa Lesson Explainer: Multiplying and Dividing Rational Expressions | Nagwa

Lesson Explainer: Multiplying and Dividing Rational Expressions Mathematics

Join Nagwa Classes

Attend live Mathematics sessions on Nagwa Classes to learn more about this topic from an expert teacher!

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 23Γ—154 by multiplying the numerators and denominators separately to get 23Γ—154=2Γ—153Γ—4.

We can then evaluate and simplify; however, in this case, it is easier to factor and cancel. We note that 4=2Γ—2 and 15=3Γ—5, so 2Γ—153Γ—4=2Γ—3Γ—53Γ—2Γ—2.

We can then see that the numerator and denominator share factors of 2 and 3. We can cancel these shared factors to obtain 2Γ—3Γ—53Γ—2Γ—2=2Γ—3Γ—53Γ—2Γ—2=52.

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: π‘₯3Γ—15π‘₯.

We know that π‘₯ 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: π‘₯3Γ—15π‘₯=π‘₯Γ—153Γ—π‘₯.

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 15=3Γ—5,π‘₯=π‘₯Γ—π‘₯.

Therefore, π‘₯Γ—153Γ—π‘₯=π‘₯Γ—π‘₯Γ—3Γ—53Γ—π‘₯.

We now cancel the shared factors of 3 and π‘₯ in the numerator and denominator, where we can note that π‘₯π‘₯=1 only when π‘₯β‰ 0. This means that our expression is only valid when π‘₯β‰ 0. We obtain π‘₯Γ—π‘₯Γ—3Γ—53Γ—π‘₯=π‘₯Γ—π‘₯Γ—3Γ—53Γ—π‘₯=5π‘₯.

We can cancel any shared factors in this way, even more complicated factors such as 2(π‘₯+3)(π‘₯βˆ’1)3(π‘₯βˆ’1)(π‘₯βˆ’2)=2(π‘₯+3)(π‘₯βˆ’1)3(π‘₯βˆ’1)(π‘₯βˆ’2)=2(π‘₯+3)3(π‘₯βˆ’2).

In this case, we can only cancel the factors of π‘₯βˆ’1 if π‘₯β‰ 1. 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 2π‘ŽΓ—π‘Ž6.

Answer

To multiply fractions, we multiply the numerators and denominators separately. We have 2π‘ŽΓ—π‘Ž6=2Γ—π‘Žπ‘ŽΓ—6=2π‘Ž6π‘Ž.

We can then note that there are shared factors in the numerator and denominator; since 2 and 6 share a factor of 2 and π‘Ž and π‘ŽοŠ¨ share a factor of π‘Ž, we can write these products out in full and cancel the shared factors to get 2π‘Ž6π‘Ž=2Γ—π‘ŽΓ—π‘Ž2Γ—3Γ—π‘Ž=π‘Ž3.

Hence, 2π‘ŽΓ—π‘Ž6=π‘Ž3.

In our next example, we will simplify the product of algebraic fractions involving canceling a shared linear factor.

Example 2: Multiplying Algebraic Fractions

Simplify π‘₯+2π‘₯+3Γ—2π‘₯+6π‘₯+1.

Answer

To multiply any fractions, we multiply the numerators and denominators separately. We have π‘₯+2π‘₯+3Γ—2π‘₯+6π‘₯+1=(π‘₯+2)Γ—(2π‘₯+6)(π‘₯+3)Γ—(π‘₯+1).

In order to simplify algebraic fractions, we need to fully factorize the numerator and denominator. We note that 2π‘₯+6=2(π‘₯+3). This gives us (π‘₯+2)Γ—(2π‘₯+6)(π‘₯+3)Γ—(π‘₯+1)=(π‘₯+2)Γ—2(π‘₯+3)(π‘₯+3)Γ—(π‘₯+1)=2(π‘₯+2)(π‘₯+3)(π‘₯+3)(π‘₯+1).

We can then cancel the shared factor of π‘₯+3 in the numerator and denominator to obtain 2(π‘₯+2)(π‘₯+3)(π‘₯+3)(π‘₯+1)=2(π‘₯+2)π‘₯+1.

We could leave our answer like this; however, we can also expand the brackets in the numerator to get 2(π‘₯+2)π‘₯+1=2π‘₯+4π‘₯+1.

Hence, π‘₯+2π‘₯+3Γ—2π‘₯+6π‘₯+1=2π‘₯+4π‘₯+1.

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 45π‘₯.

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 𝑣, then we have 𝑣=ο€Ό45π‘₯.

We can write the cube out in full to obtain 𝑣=45π‘₯Γ—45π‘₯Γ—45π‘₯.

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 45π‘₯Γ—45π‘₯Γ—45π‘₯=4Γ—4Γ—45π‘₯Γ—5π‘₯Γ—5π‘₯.

We can evaluate the numerator by calculating 4Γ—4Γ—4=64. In the denominator, we rearrange the product to be 5π‘₯Γ—5π‘₯Γ—5π‘₯=(5Γ—5Γ—5)(π‘₯Γ—π‘₯Γ—π‘₯). We then calculate 5Γ—5Γ—5=125 and rewrite π‘₯Γ—π‘₯Γ—π‘₯=π‘₯. This is equivalent to multiplying the coefficients and variables separately. We obtain 4Γ—4Γ—45π‘₯Γ—5π‘₯Γ—5π‘₯=64125π‘₯.

Hence, the volume of the cube 𝑣 is given by the equation 𝑣=64125π‘₯.

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 ο€Ό45π‘₯=4Γ—4Γ—45π‘₯Γ—5π‘₯Γ—5π‘₯=4(5π‘₯).

In general, if we have a positive integer 𝑛, 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 2π‘₯5Γ·4π‘₯3. We can start by changing the division into a multiplication by instead multiplying by the reciprocal of the divisor: 2π‘₯5Γ·4π‘₯3=2π‘₯5Γ—34π‘₯.

We can now follow the same process we do for multiplying algebraic fractions. We start by multiplying the numerators and denominators separately to obtain 2π‘₯5Γ—34π‘₯=2π‘₯Γ—35Γ—4π‘₯.

We can now factor and cancel the shared factors in the numerator and denominator to get 2π‘₯Γ—35Γ—4π‘₯=2Γ—π‘₯Γ—35Γ—2Γ—π‘₯=2Γ—π‘₯Γ—35Γ—2Γ—π‘₯=35.

Of course, since we cancel the shared factor of π‘₯ in the numerator and denominator, this is only valid when π‘₯β‰ 0.

In our next example, we will use this process to simplify the division of two algebraic fractions.

Example 4: Dividing Algebraic Fractions

Simplify 2π‘Ž3Γ·2π‘Ž4.

Answer

To divide two fractions, we can instead multiply by the reciprocal of the second fraction: 2π‘Ž3Γ·2π‘Ž4=2π‘Ž3Γ—42π‘Ž.

To multiply any fractions, we multiply the numerators and denominators separately. We have 2π‘Ž3Γ—42π‘Ž=2π‘ŽΓ—43Γ—2π‘Ž.

We can then note that there are shared factors in the numerator and denominator: 2 and π‘Ž. We can cancel these shared factors to get 2π‘ŽΓ—43Γ—2π‘Ž=2π‘ŽΓ—43Γ—2π‘Ž=43π‘Ž.

Hence, 2π‘Ž3Γ·2π‘Ž4=43π‘Ž.

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 5π‘₯π‘₯+1Γ·3π‘₯2π‘₯+2.

Answer

To divide two fractions, we can instead multiply by the reciprocal of the second fraction: 5π‘₯π‘₯+1Γ·3π‘₯2π‘₯+2=5π‘₯π‘₯+1Γ—2π‘₯+23π‘₯.

We then need to multiply two algebraic fractions; we do this by multiplying the numerators and denominators separately. We have 5π‘₯π‘₯+1Γ—2π‘₯+22π‘₯=5π‘₯Γ—(2π‘₯+2)(π‘₯+1)Γ—3π‘₯.

Instead of leaving our answer like this, we should try to simplify by factorizing and canceling any shared factors. We can note that 2π‘₯+2=2(π‘₯+1), so we have 5π‘₯Γ—(2π‘₯+2)(π‘₯+1)Γ—3π‘₯=5π‘₯Γ—2(π‘₯+1)(π‘₯+1)Γ—3π‘₯.

We can then cancel the shared factors of π‘₯ and π‘₯+1 in the numerator and denominator, where we note π‘₯π‘₯=π‘₯=π‘₯ to get 5π‘₯Γ—2(π‘₯+1)(π‘₯+1)Γ—3π‘₯=5π‘₯Γ—2(π‘₯+1)(π‘₯+1)Γ—3π‘₯=5π‘₯Γ—23=10π‘₯3.

Hence, 5π‘₯π‘₯+1Γ·3π‘₯2π‘₯+2=10π‘₯3.

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 14π‘₯βˆ’21π‘₯4π‘₯βˆ’20Γ·4π‘₯βˆ’62π‘₯βˆ’1.

Answer

To divide two algebraic fractions, we can instead multiply by the reciprocal of the second fraction. This gives us 14π‘₯βˆ’21π‘₯4π‘₯βˆ’20Γ·4π‘₯βˆ’62π‘₯βˆ’1=14π‘₯βˆ’21π‘₯4π‘₯βˆ’20Γ—2π‘₯βˆ’14π‘₯βˆ’6.

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 14π‘₯βˆ’21π‘₯4π‘₯βˆ’20Γ—2π‘₯βˆ’14π‘₯βˆ’6=ο€Ή14π‘₯βˆ’21π‘₯×(2π‘₯βˆ’1)(4π‘₯βˆ’20)Γ—(4π‘₯βˆ’6).

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 14π‘₯βˆ’21π‘₯=7π‘₯(2π‘₯βˆ’3),4π‘₯βˆ’20=4(π‘₯βˆ’5),4π‘₯βˆ’6=2(2π‘₯βˆ’3).

Substituting the factorized expressions into the algebraic fraction yields ο€Ή14π‘₯βˆ’21π‘₯×(2π‘₯βˆ’1)(4π‘₯βˆ’20)Γ—(4π‘₯βˆ’6)=7π‘₯(2π‘₯βˆ’3)Γ—(2π‘₯βˆ’1)4(π‘₯βˆ’5)Γ—2(2π‘₯βˆ’3).

We can then cancel the shared factor of 2π‘₯βˆ’3 in the numerator and denominator to obtain 7π‘₯(2π‘₯βˆ’3)Γ—(2π‘₯βˆ’1)4(π‘₯βˆ’5)Γ—2(2π‘₯βˆ’3)=7π‘₯(2π‘₯βˆ’3)Γ—(2π‘₯βˆ’1)4(π‘₯βˆ’5)Γ—2(2π‘₯βˆ’3)=7π‘₯(2π‘₯βˆ’1)4(π‘₯βˆ’5)Γ—2=7π‘₯(2π‘₯βˆ’1)8(π‘₯βˆ’5).

Hence, 14π‘₯βˆ’21π‘₯4π‘₯βˆ’20Γ·4π‘₯βˆ’62π‘₯βˆ’1=7π‘₯(2π‘₯βˆ’1)8(π‘₯βˆ’5).

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.

Join Nagwa Classes

Attend live sessions on Nagwa Classes to boost your learning with guidance and advice from an expert teacher!

  • Interactive Sessions
  • Chat & Messaging
  • Realistic Exam Questions

Nagwa uses cookies to ensure you get the best experience on our website. Learn more about our Privacy Policy