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Lesson Explainer: Multiplying and Dividing Rational Expressions Mathematics

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

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

We can then see that the numerator and denominator share factors of 2 and 3. We can cancel these shared factors to obtain 2×3×53×2×2=2×3×53×2×2=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: 𝑥3×15𝑥.

We know that 𝑥 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: 𝑥3×15𝑥=𝑥×153×𝑥.

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=3×5,𝑥=𝑥×𝑥.

Therefore, 𝑥×153×𝑥=𝑥×𝑥×3×53×𝑥.

We now cancel the shared factors of 3 and 𝑥 in the numerator and denominator, where we can note that 𝑥𝑥=1 only when 𝑥0. This means that our expression is only valid when 𝑥0. We obtain 𝑥×𝑥×3×53×𝑥=𝑥×𝑥×3×53×𝑥=5𝑥.

We can cancel any shared factors in this way, even more complicated factors such as 2(𝑥+3)(𝑥1)3(𝑥1)(𝑥2)=2(𝑥+3)(𝑥1)3(𝑥1)(𝑥2)=2(𝑥+3)3(𝑥2).

In this case, we can only cancel the factors of 𝑥1 if 𝑥1. 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 2𝑎×𝑎6.

Answer

To multiply fractions, we multiply the numerators and denominators separately. We have 2𝑎×𝑎6=2×𝑎𝑎×6=2𝑎6𝑎.

We can then note that there are shared factors in the numerator and denominator; since 2 and 6 share a factor of 2 and 𝑎 and 𝑎 share a factor of 𝑎, we can write these products out in full and cancel the shared factors to get 2𝑎6𝑎=2×𝑎×𝑎2×3×𝑎=𝑎3.

Hence, 2𝑎×𝑎6=𝑎3.

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

Example 2: Multiplying Algebraic Fractions

Simplify 𝑥+2𝑥+3×2𝑥+6𝑥+1.

Answer

To multiply any fractions, we multiply the numerators and denominators separately. We have 𝑥+2𝑥+3×2𝑥+6𝑥+1=(𝑥+2)×(2𝑥+6)(𝑥+3)×(𝑥+1).

In order to simplify algebraic fractions, we need to fully factorize the numerator and denominator. We note that 2𝑥+6=2(𝑥+3). This gives us (𝑥+2)×(2𝑥+6)(𝑥+3)×(𝑥+1)=(𝑥+2)×2(𝑥+3)(𝑥+3)×(𝑥+1)=2(𝑥+2)(𝑥+3)(𝑥+3)(𝑥+1).

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

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

Hence, 𝑥+2𝑥+3×2𝑥+6𝑥+1=2𝑥+4𝑥+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 45𝑥.

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=45𝑥.

We can write the cube out in full to obtain 𝑣=45𝑥×45𝑥×45𝑥.

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 45𝑥×45𝑥×45𝑥=4×4×45𝑥×5𝑥×5𝑥.

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

Hence, the volume of the cube 𝑣 is given by the equation 𝑣=64125𝑥.

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 45𝑥=4×4×45𝑥×5𝑥×5𝑥=4(5𝑥).

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 2𝑥5÷4𝑥3. We can start by changing the division into a multiplication by instead multiplying by the reciprocal of the divisor: 2𝑥5÷4𝑥3=2𝑥5×34𝑥.

We can now follow the same process we do for multiplying algebraic fractions. We start by multiplying the numerators and denominators separately to obtain 2𝑥5×34𝑥=2𝑥×35×4𝑥.

We can now factor and cancel the shared factors in the numerator and denominator to get 2𝑥×35×4𝑥=2×𝑥×35×2×𝑥=2×𝑥×35×2×𝑥=35.

Of course, since we cancel the shared factor of 𝑥 in the numerator and denominator, this is only valid when 𝑥0.

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

Example 4: Dividing Algebraic Fractions

Simplify 2𝑎3÷2𝑎4.

Answer

To divide two fractions, we can instead multiply by the reciprocal of the second fraction: 2𝑎3÷2𝑎4=2𝑎3×42𝑎.

To multiply any fractions, we multiply the numerators and denominators separately. We have 2𝑎3×42𝑎=2𝑎×43×2𝑎.

We can then note that there are shared factors in the numerator and denominator: 2 and 𝑎. We can cancel these shared factors to get 2𝑎×43×2𝑎=2𝑎×43×2𝑎=43𝑎.

Hence, 2𝑎3÷2𝑎4=43𝑎.

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 5𝑥𝑥+1÷3𝑥2𝑥+2.

Answer

To divide two fractions, we can instead multiply by the reciprocal of the second fraction: 5𝑥𝑥+1÷3𝑥2𝑥+2=5𝑥𝑥+1×2𝑥+23𝑥.

We then need to multiply two algebraic fractions; we do this by multiplying the numerators and denominators separately. We have 5𝑥𝑥+1×2𝑥+22𝑥=5𝑥×(2𝑥+2)(𝑥+1)×3𝑥.

Instead of leaving our answer like this, we should try to simplify by factorizing and canceling any shared factors. We can note that 2𝑥+2=2(𝑥+1), so we have 5𝑥×(2𝑥+2)(𝑥+1)×3𝑥=5𝑥×2(𝑥+1)(𝑥+1)×3𝑥.

We can then cancel the shared factors of 𝑥 and 𝑥+1 in the numerator and denominator, where we note 𝑥𝑥=𝑥=𝑥 to get 5𝑥×2(𝑥+1)(𝑥+1)×3𝑥=5𝑥×2(𝑥+1)(𝑥+1)×3𝑥=5𝑥×23=10𝑥3.

Hence, 5𝑥𝑥+1÷3𝑥2𝑥+2=10𝑥3.

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 14𝑥21𝑥4𝑥20÷4𝑥62𝑥1.

Answer

To divide two algebraic fractions, we can instead multiply by the reciprocal of the second fraction. This gives us 14𝑥21𝑥4𝑥20÷4𝑥62𝑥1=14𝑥21𝑥4𝑥20×2𝑥14𝑥6.

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 14𝑥21𝑥4𝑥20×2𝑥14𝑥6=14𝑥21𝑥×(2𝑥1)(4𝑥20)×(4𝑥6).

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 14𝑥21𝑥=7𝑥(2𝑥3),4𝑥20=4(𝑥5),4𝑥6=2(2𝑥3).

Substituting the factorized expressions into the algebraic fraction yields 14𝑥21𝑥×(2𝑥1)(4𝑥20)×(4𝑥6)=7𝑥(2𝑥3)×(2𝑥1)4(𝑥5)×2(2𝑥3).

We can then cancel the shared factor of 2𝑥3 in the numerator and denominator to obtain 7𝑥(2𝑥3)×(2𝑥1)4(𝑥5)×2(2𝑥3)=7𝑥(2𝑥3)×(2𝑥1)4(𝑥5)×2(2𝑥3)=7𝑥(2𝑥1)4(𝑥5)×2=7𝑥(2𝑥1)8(𝑥5).

Hence, 14𝑥21𝑥4𝑥20÷4𝑥62𝑥1=7𝑥(2𝑥1)8(𝑥5).

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.

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