# 8.2 Reducing rational expressions  (Page 2/2)

 Page 2 / 2

Consider the fraction $\frac{6}{24}$ . Multiply this fraction by 1. This is written $\frac{6}{24}·1$ . But 1 can be rewritten as $\frac{\frac{1}{6}}{\frac{1}{6}}$ .

$\frac{6}{24}\cdot \frac{\frac{1}{6}}{\frac{1}{6}}=\frac{6\cdot \frac{1}{6}}{24\cdot \frac{1}{6}}=\frac{1}{4}$

The answer, $\frac{1}{4}$ , is the reduced form. Notice that in $\frac{1}{4}$ there is no factor common to both the numerator and denominator. This reasoning provides justification for the following rule.

## Cancelling

Multiplying or dividing the numerator and denominator by the same nonzero number does not change the value of a fraction.

## The process

We can now state a process for reducing a rational expression.

## Reducing a rational expression

1. Factor the numerator and denominator completely.
2. Divide the numerator and denominator by all factors they have in common, that is, remove all factors of 1.

## Reduced to lowest terms

1. A rational expression is said to be reduced to lowest terms when the numerator and denominator have no factors in common.

## Sample set a

Reduce the following rational expressions.

$\begin{array}{l}\begin{array}{lll}\frac{15x}{20x}.\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{15x}{20x}=\frac{5·3·x}{5·2·2·x}\hfill & \hfill & \begin{array}{l}\text{The factors that are common to both the numerator and}\\ \text{denominator are 5 and \hspace{0.17em}}x\text{. Divide each by\hspace{0.17em}}5x\text{.}\end{array}\hfill \end{array}\\ \frac{\overline{)5}·3·\overline{)x}}{\overline{)5}·2·2·\overline{)x}}=\frac{3}{4},\text{\hspace{0.17em}}x\ne 0\\ \text{\hspace{0.17em}}\\ \text{It is helpful to draw a line through the divided-out factors}.\end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{{x}^{2}-4}{{x}^{2}-6x+8}.\hfill & \hfill & \text{Factor.}\hfill \\ \frac{\left(x+2\right)\left(x-2\right)}{\left(x-2\right)\left(x-4\right)}\hfill & \hfill & \begin{array}{l}\text{The factor that is common to both the numerator}\\ \text{and denominator is\hspace{0.17em}}x-2.\text{\hspace{0.17em}Divide each by\hspace{0.17em}}x-2.\end{array}\hfill \end{array}\\ \frac{\left(x+2\right)\overline{)\left(x-2\right)}}{\overline{)\left(x-2\right)}\left(x-4\right)}=\frac{x+2}{x-4},\text{\hspace{0.17em}}x\ne 2,\text{\hspace{0.17em}}4\end{array}$

The expression $\frac{x-2}{x-4}$ is the reduced form since there are no factors common to both the numerator and denominator. Although there is an $x$ in both, it is a common term , not a common factor , and therefore cannot be divided out.

CAUTION — This is a common error: $\frac{x-2}{x-4}=\frac{\overline{)x}-2}{\overline{)x}-4}=\frac{2}{4}$ is incorrect!

$\begin{array}{l}\begin{array}{ll}\frac{a+2b}{6a+12b}.\hfill & \text{Factor}\text{.}\hfill \end{array}\\ \frac{a+2b}{6\left(a+2b\right)}=\frac{\overline{)a+2b}}{6\overline{)\left(a+2b\right)}}=\frac{1}{6},\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\ne \text{\hspace{0.17em}}-2b\end{array}$
Since $a+2b$ is a common factor to both the numerator and denominator, we divide both by $a+2b$ . Since $\frac{\left(a+2b\right)}{\left(a+2b\right)}=1$ , we get 1 in the numerator.

Sometimes we may reduce a rational expression by using the division rule of exponents.

$\begin{array}{lll}\frac{8{x}^{2}{y}^{5}}{4x{y}^{2}}.\hfill & \hfill & \text{Factor and use the rule\hspace{0.17em}}\frac{{a}^{n}}{{a}^{m}}={a}^{n-m}.\hfill \\ \frac{8{x}^{2}{y}^{5}}{4x{y}^{2}}\hfill & =\hfill & \frac{2\cdot 2\cdot 2}{2\cdot 2}{x}^{2-1}{y}^{5-2}\hfill \\ \hfill & =\hfill & 2x{y}^{3},\text{\hspace{0.17em}}x\ne 0,\text{\hspace{0.17em}}y\ne 0\hfill \end{array}$

$\begin{array}{lll}\frac{-10{x}^{3}a\left({x}^{2}-36\right)}{2{x}^{3}-10{x}^{2}-12x}.\hfill & \hfill & \text{Factor}\text{.}\hfill \\ \frac{-10{x}^{3}a\left({x}^{2}-36\right)}{2{x}^{3}-10{x}^{2}-12x}\hfill & =\hfill & \frac{-5\cdot 2{x}^{3}a\left(x+6\right)\left(x-6\right)}{2x\left({x}^{2}-5x-6\right)}\hfill \\ \hfill & =\hfill & \frac{-5\cdot 2{x}^{3}a\left(x+6\right)\left(x-6\right)}{2x\left(x-6\right)\left(x+1\right)}\hfill \\ \hfill & =\hfill & \frac{-5\cdot \overline{)2}{x}^{\begin{array}{l}2\\ \overline{)3}\end{array}}a\left(x+6\right)\overline{)\left(x-6\right)}}{\overline{)2}\overline{)x}\overline{)\left(x-6\right)}\left(x+1\right)}\hfill \\ \hfill & =\hfill & \frac{-5{x}^{2}a\left(x+6\right)}{x-1},\text{\hspace{0.17em}}x\ne -1,\text{\hspace{0.17em}}6\hfill \end{array}$

$\begin{array}{l}\begin{array}{lll}\frac{{x}^{2}-x-12}{-{x}^{2}+2x+8}.\hfill & \hfill & \begin{array}{l}\text{Since it is most convenient to have the leading terms of a}\\ \text{polynomial positive, factor out}-\text{1 from the denominator}\text{.}\end{array}\hfill \\ \frac{{x}^{2}-x-12}{-\left({x}^{2}-2x-8\right)}\hfill & \hfill & \text{Rewrite this}\text{.}\hfill \\ -\frac{{x}^{2}-x-12}{{x}^{2}-2x-8}\hfill & \hfill & \text{Factor}\text{.}\hfill \\ -\frac{\overline{)\left(x-4\right)}\left(x+3\right)}{\overline{)\left(x-4\right)}\left(x+2\right)}\hfill & \hfill & \hfill \end{array}\\ -\frac{x+3}{x+2}=\frac{-\left(x+3\right)}{x+2}=\frac{-x-3}{x+2},\text{\hspace{0.17em}}x\ne -2,\text{\hspace{0.17em}}4\end{array}$

$\begin{array}{l}\begin{array}{ll}\frac{a-b}{b-a}.\hfill & \begin{array}{l}\text{The numerator and denominator have the same terms but they}\\ \text{occur with opposite signs}\text{. Factor}-\text{1 from the denominator}\text{.}\end{array}\hfill \end{array}\\ \frac{a-b}{-\left(-b+a\right)}=\frac{a-b}{-\left(a-b\right)}=-\frac{\overline{)a-b}}{\overline{)a-b}}=-1,\text{\hspace{0.17em}}a\ne b\end{array}$

## Practice set a

Reduce each of the following fractions to lowest terms.

$\frac{30y}{35y}$

$\frac{6}{7}$

$\frac{{x}^{2}-9}{{x}^{2}+5x+6}$

$\frac{x-3}{x+2}$

$\frac{x+2b}{4x+8b}$

$\frac{1}{4}$

$\frac{18{a}^{3}{b}^{5}{c}^{7}}{3a{b}^{3}{c}^{5}}$

$6{a}^{2}{b}^{2}{c}^{2}$

$\frac{-3{a}^{4}+75{a}^{2}}{2{a}^{3}-16{a}^{2}+30a}$

$\frac{-3a\left(a+5\right)}{2\left(a-3\right)}$

$\frac{{x}^{2}-5x+4}{-{x}^{2}+12x-32}$

$\frac{-x+1}{x-8}$

$\frac{2x-y}{y-2x}$

−1

## Excercises

For the following problems, reduce each rational expression to lowest terms.

$\frac{6}{3x-12}$

$\frac{2}{\left(x-4\right)}$

$\frac{8}{4a-16}$

$\frac{9}{3y-21}$

$\frac{3}{\left(y-7\right)}$

$\frac{10}{5x-5}$

$\frac{7}{7x-14}$

$\frac{1}{\left(x-2\right)}$

$\frac{6}{6x-18}$

$\frac{2{y}^{2}}{8y}$

$\frac{1}{4}y$

$\frac{4{x}^{3}}{2x}$

$\frac{16{a}^{2}{b}^{3}}{2a{b}^{2}}$

$8ab$

$\frac{20{a}^{4}{b}^{4}}{4a{b}^{2}}$

$\frac{\left(x+3\right)\left(x-2\right)}{\left(x+3\right)\left(x+5\right)}$

$\frac{x-2}{x+5}$

$\frac{\left(y-1\right)\left(y-7\right)}{\left(y-1\right)\left(y+6\right)}$

$\frac{\left(a+6\right)\left(a-5\right)}{\left(a-5\right)\left(a+2\right)}$

$\frac{a+6}{a+2}$

$\frac{\left(m-3\right)\left(m-1\right)}{\left(m-1\right)\left(m+4\right)}$

$\frac{\left(y-2\right)\left(y-3\right)}{\left(y-3\right)\left(y-2\right)}$

1

$\frac{\left(x+7\right)\left(x+8\right)}{\left(x+8\right)\left(x+7\right)}$

$\frac{-12{x}^{2}\left(x+4\right)}{4x}$

$-3x\left(x+4\right)$

$\frac{-3{a}^{4}\left(a-1\right)\left(a+5\right)}{-2{a}^{3}\left(a-1\right)\left(a+9\right)}$

$\frac{6{x}^{2}{y}^{5}\left(x-1\right)\left(x+4\right)}{-2xy\left(x+4\right)}$

$-3x{y}^{4}\left(x-1\right)$

$\frac{22{a}^{4}{b}^{6}{c}^{7}\left(a+2\right)\left(a-7\right)}{4c\left(a+2\right)\left(a-5\right)}$

$\frac{{\left(x+10\right)}^{3}}{x+10}$

${\left(x+10\right)}^{2}$

$\frac{{\left(y-6\right)}^{7}}{y-6}$

$\frac{{\left(x-8\right)}^{2}{\left(x+6\right)}^{4}}{\left(x-8\right)\left(x+6\right)}$

$\left(x-8\right){\left(x+6\right)}^{3}$

$\frac{{\left(a+1\right)}^{5}{\left(a-1\right)}^{7}}{{\left(a+1\right)}^{3}{\left(a-1\right)}^{4}}$

$\frac{{\left(y-2\right)}^{6}{\left(y-1\right)}^{4}}{{\left(y-2\right)}^{3}{\left(y-1\right)}^{2}}$

${\left(y-2\right)}^{3}{\left(y-1\right)}^{2}$

$\frac{{\left(x+10\right)}^{5}{\left(x-6\right)}^{3}}{\left(x-6\right){\left(x+10\right)}^{2}}$

$\frac{{\left(a+6\right)}^{2}{\left(a-7\right)}^{6}}{{\left(a+6\right)}^{5}{\left(a-7\right)}^{2}}$

$\frac{{\left(a-7\right)}^{4}}{{\left(a+6\right)}^{3}}$

$\frac{{\left(m+7\right)}^{4}{\left(m-8\right)}^{5}}{{\left(m+7\right)}^{7}{\left(m-8\right)}^{2}}$

$\frac{\left(a+2\right){\left(a-1\right)}^{3}}{\left(a+1\right)\left(a-1\right)}$

$\frac{\left(a+2\right){\left(a-1\right)}^{2}}{\left(a+1\right)}$

$\frac{\left(b+6\right){\left(b-2\right)}^{4}}{\left(b-1\right)\left(b-2\right)}$

$\frac{8{\left(x+2\right)}^{3}{\left(x-5\right)}^{6}}{2\left(x+2\right){\left(x-5\right)}^{2}}$

$4{\left(x+2\right)}^{2}{\left(x-5\right)}^{4}$

$\frac{14{\left(x-4\right)}^{3}{\left(x-10\right)}^{6}}{-7{\left(x-4\right)}^{2}{\left(x-10\right)}^{2}}$

$\frac{{x}^{2}+x-12}{{x}^{2}-4x+3}$

$\frac{\left(x+4\right)}{\left(x-1\right)}$

$\frac{{x}^{2}+3x-10}{{x}^{2}+2x-15}$

$\frac{{x}^{2}-10x+21}{{x}^{2}-6x-7}$

$\frac{\left(x-3\right)}{\left(x+1\right)}$

$\frac{{x}^{2}+10x+24}{{x}^{2}+6x}$

$\frac{{x}^{2}+9x+14}{{x}^{2}+7x}$

$\frac{\left(x+2\right)}{x}$

$\frac{6{b}^{2}-b}{6{b}^{2}+11b-2}$

$\frac{3{b}^{2}+10b+3}{3{b}^{2}+7b+2}$

$\frac{b+3}{b+2}$

$\frac{4{b}^{2}-1}{2{b}^{2}+5b-3}$

$\frac{16{a}^{2}-9}{4{a}^{2}-a-3}$

$\frac{\left(4a-3\right)}{\left(a-1\right)}$

$\frac{20{x}^{2}+28xy+9{y}^{2}}{4{x}^{2}+4xy+{y}^{2}}$

For the following problems, reduce each rational expression if possible. If not possible, state the answer in lowest terms.

$\frac{x+3}{x+4}$

$\frac{\left(x+3\right)}{\left(x+4\right)}$

$\frac{a+7}{a-1}$

$\frac{3a+6}{3}$

$a+2$

$\frac{4x+12}{4}$

$\frac{5a-5}{-5}$

$\begin{array}{lllll}-\left(a-1\right)\hfill & \hfill & \text{or}\hfill & \hfill & -a+1\hfill \end{array}$

$\frac{6b-6}{-3}$

$\frac{8x-16}{-4}$

$-2\left(x-2\right)$

$\frac{4x-7}{-7}$

$\frac{-3x+10}{10}$

$\frac{-3x+10}{10}$

$\frac{x-2}{2-x}$

$\frac{a-3}{3-a}$

$-1$

$\frac{{x}^{3}-x}{x}$

$\frac{{y}^{4}-y}{y}$

${y}^{3}-1$

$\frac{{a}^{5}-{a}^{2}}{a}$

$\frac{{a}^{6}-{a}^{4}}{{a}^{3}}$

$a\left(a+1\right)\left(a-1\right)$

$\frac{4{b}^{2}+3b}{b}$

$\frac{2{a}^{3}+5a}{a}$

$2{a}^{2}+5$

$\frac{a}{{a}^{3}+a}$

$\frac{{x}^{4}}{{x}^{5}-3x}$

$\frac{{x}^{3}}{{x}^{4}-3}$

$\frac{-a}{-{a}^{2}-a}$

## Excercises for review

( [link] ) Write ${\left(\frac{{4}^{4}{a}^{8}{b}^{10}}{{4}^{2}{a}^{6}{b}^{2}}\right)}^{-1}$ so that only positive exponents appear.

$\frac{1}{16{a}^{2}{b}^{8}}$

( [link] ) Factor ${y}^{4}-16$ .

( [link] ) Factor $10{x}^{2}-17x+3$ .

$\left(5x-1\right)\left(2x-3\right)$

( [link] ) Supply the missing word. An equation expressed in the form $ax+by=c$ is said to be expressed in form.

( [link] ) Find the domain of the rational expression $\frac{2}{{x}^{2}-3x-18}$ .

$x\ne -3,\text{\hspace{0.17em}}6$

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.