# 5.03 Multiplying and dividing rational expressions

Lesson

## Multiply and divide algebraic fractions

### Multiplying

When it comes to working with algebraic fractions and applying the four operations, the process is exactly the same as when we worked with numeric fractions.

Let's have a look at a simple example of multiplying two algebraic fractions.

#### Worked example

##### question 1

Simplify $\frac{y}{5}\times\frac{3}{m}$y5×3m

 $\frac{y}{5}\times\frac{3}{m}$y5​×3m​ $=$= $\frac{y\times3}{5\times m}$y×35×m​    Multiplying numerator and denominators $=$= $\frac{3y}{5m}$3y5m​     Simplifying the numerator

Again, since the numerator $3y$3y and the denominator $5m$5m don't have any common factors, $\frac{3y}{5m}$3y5m is the simplest form of our answer.

### Dividing

Again, the process for dividing is the same as when we divided numeric fractions. We need to multiply by the reciprocal of the second fraction.

#### Worked example

##### question 2

Simplify $\frac{m}{3}\div\frac{5}{x}$m3÷​5x

 $\frac{m}{3}\div\frac{5}{x}$m3​÷​5x​ $=$= $\frac{m}{3}\times\frac{x}{5}$m3​×x5​ Dividing by a fraction is the same as multiplying by its reciprocal. $=$= $\frac{m\times x}{3\times5}$m×x3×5​ Multiply numerators and denominators respectively. $=$= $\frac{mx}{15}$mx15​

Again, since the numerator $mx$mx and the denominator $15$15 don't have any common factors, $\frac{mx}{15}$mx15 is the simplest form of our answer.

#### Practice questions

##### Question 3

Simplify the expression:

$\frac{m}{8}\div\frac{3}{n}$m8÷​3n

##### Question 4

Simplify the following: $\frac{-2x}{11}\div\frac{7y}{5}$2x11÷​7y5

##### Question 5

Simplify $\frac{-2x}{11}\div\frac{2x}{3}$2x11÷​2x3.

## Using factoring

Sometimes the fractions involved in these multiplication and division problems are too complicated to see their factors immediately, and that's where factoring comes in. We can use the various factoring techniques we have learned previously.

#### Worked examples

##### Question 6

Factor and simplify the following: $\frac{9x^2}{3xy-6x}\div\frac{3y+9}{y^2+y-6}$9x23xy6x÷​3y+9y2+y6

Think about how division is just multiplication with the second fraction inverted

Do: So our problem can be rewritten as:

$\frac{9x^2}{3xy-6x}\times\frac{y^2+y-6}{3y+9}$9x23xy6x×y2+y63y+9

The denominator of the first fraction can be factored using GCFs:

 $\frac{9x^2}{3xy-6x}$9x23xy−6x​ $=$= $\frac{9x^2}{3x\left(y-2\right)}$9x23x(y−2)​ $=$= $\frac{3x}{y-2}$3xy−2​ by canceling out $3x$3x from top and bottom

The second fraction can be factored using the cross method on top and GCFs on the bottom:

 $\frac{y^2+y-6}{3y+9}$y2+y−63y+9​ $=$= $\frac{\left(y-2\right)\left(y+3\right)}{3\left(y+3\right)}$(y−2)(y+3)3(y+3)​ $=$= $\frac{y-2}{3}$y−23​ by canceling out $y+3$y+3 on top and bottom

So our problem is now:

 $\frac{3x}{y-2}\times\frac{y-2}{3}$3xy−2​×y−23​ $=$= $\frac{3x}{1}\times\frac{1}{3}$3x1​×13​ by diagonally canceling out $y-2$y−2 $=$= $\frac{x}{1}\times\frac{1}{1}$x1​×11​ by diagonally canceling out $3$3 $=$= $x$x

##### Question 7

Factor and simplify

$\frac{5q}{50pq^2-8p}\times\frac{4pq+24p^2}{q^2+12pq+36p^2}$5q50pq28p×4pq+24p2q2+12pq+36p2

Think about how some quadratics don't need to be factored using the cross method

Do

 $\frac{5q}{50pq^2-8p}\times\frac{4pq+24p^2}{q^2+12pq+36p^2}$5q50pq2−8p​×4pq+24p2q2+12pq+36p2​ $=$= $\frac{5q}{2p\left(25q^2-4\right)}\times\frac{4p\left(q+6p\right)}{q^2+12pq+36p^2}$5q2p(25q2−4)​×4p(q+6p)q2+12pq+36p2​ $=$= $\frac{5q}{2p\left(5q+2\right)\left(5q-2\right)}\times\frac{4p\left(q+6p\right)}{q^2+12pq+36p^2}$5q2p(5q+2)(5q−2)​×4p(q+6p)q2+12pq+36p2​ $=$= $\frac{5q}{2p\left(5q+2\right)\left(5q-2\right)}\times\frac{4p\left(q+6p\right)}{\left(q+6p\right)^2}$5q2p(5q+2)(5q−2)​×4p(q+6p)(q+6p)2​ the denominator is a perfect square as $q^2$q2 and $36p^2$36p2 are both squares and $12pq=2\times q\times6p$12pq=2×q×6p $=$= $\frac{5q}{2p\left(5q+2\right)\left(5q-2\right)}\times\frac{4p}{q+6p}$5q2p(5q+2)(5q−2)​×4pq+6p​ $=$= $\frac{5q}{\left(5q+2\right)\left(5q-2\right)}\times\frac{2}{q+6p}$5q(5q+2)(5q−2)​×2q+6p​ $=$= $\frac{10q}{\left(5q+2\right)\left(5q-2\right)\left(q+6p\right)}$10q(5q+2)(5q−2)(q+6p)​

#### Practice questions

##### Question 8

Simplify the following: $\frac{5x+8}{8xy^2}\times\frac{9xy}{25x+40}$5x+88xy2×9xy25x+40

##### Question 9

Simplify the following expression:

$\frac{p+7}{5}\times\frac{5p-2}{p^2+14p+49}$p+75×5p2p2+14p+49

##### Question 10

Simplify the following expression:

$\frac{a^2-16}{a\left(a+4\right)}\times\frac{7a+28}{28\left(a-4\right)}$a216a(a+4)×7a+2828(a4)

## Further simplifying algebraic fractions

We want to look at fractions in which either the numerator, denominator, or both contain fractions themselves.  There are two techniques for simplifying these more complex fractions.  The first is to simplify by rewriting as division, and then multiplying by the reciprocal of the second fraction.

#### Worked example

##### Question 11

Simplify the expression $\frac{\frac{1}{6}}{\frac{-9}{8}}$1698.

Firstly, when we have one fraction divided by another, we can multiply the first by the reciprocal of the second.

$=\frac{1}{6}\times\frac{8}{-9}$=16×89

Now we want to simplify. We look vertically and diagonally for numbers that have greatest common factors. Doing this we see that $6$6 and $8$8 can be divided by $2$2.

$=\frac{1}{3}\times\frac{4}{-9}$=13×49

Now that we've simplified everything we can see, we multiply across horizontally.

$=\frac{4}{-27}$=427

#### Practice questions

##### QUESTION 12

Simplify the expression $\frac{-\frac{3}{8}}{-\frac{9}{4}}$3894.

##### QUESTION 13

Fill in the empty boxes to simplify the expression.

1. $\frac{\frac{4}{n}}{\frac{5}{n^2}}$4n5n2$=$=$\frac{\editable{}\left(\frac{4}{n}\right)}{\editable{}\left(\frac{5}{n^2}\right)}=\frac{\editable{}}{\editable{}}$(4n)(5n2)=

##### QUESTION 14

Simplify the expression $\frac{\frac{28uv^2}{25}}{\frac{49u^2v}{15}}$28uv22549u2v15.

### Outcomes

#### A.SSE.A.1^

Interpret expressions that represent a quantity in terms of its context. ^Limited to polynomial and rational functions.

#### A.SSE.A.1.A^

Interpret parts of an expression, such as terms, factors, and coefficients. ^Limited to polynomial and rational functions.

#### A.SSE.A.1.B^

Interpret complicated expressions by viewing one or more of their parts as a single entity. ^Limited to polynomial and rational functions.

#### A.SSE.A.2^

Use the structure of an expression to identify ways to rewrite it. ^Limited to polynomial and rational functions.

#### A.APR.D.7

(+) Understand that rational expressions form a system analogous to the rational numbers, closed under addition, subtraction, multiplication, and division by a nonzero rational expression; add, subtract, multiply, and divide rational expressions.