# 10.4 Divide monomials  (Page 3/5)

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Find the quotient: $\frac{-72{a}^{4}{b}^{5}}{-8{a}^{9}{b}^{5}}.$

$\frac{9}{{a}^{5}}$

Find the quotient: $\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}.$

## Solution

 $\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}$ Use fraction multiplication. $\frac{24}{48}\cdot \frac{{a}^{5}}{a}\cdot \frac{{b}^{3}}{{b}^{4}}$ Simplify and use the Quotient Property. $\frac{1}{2}\cdot {a}^{4}\cdot \frac{1}{b}$ Multiply. $\frac{{a}^{4}}{2b}$

Find the quotient: $\frac{16{a}^{7}{b}^{6}}{24a{b}^{8}}.$

$\frac{2{a}^{6}}{3{b}^{2}}$

Find the quotient: $\frac{27{p}^{4}{q}^{7}}{-45{p}^{12}{q}^{}}.$

$-\frac{3{q}^{6}}{5{p}^{8}}$

Once you become familiar with the process and have practiced it step by step several times, you may be able to simplify a fraction in one step.

Find the quotient: $\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.$

## Solution

 $\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}$ Simplify and use the Quotient Property. $\frac{2{y}^{6}}{3{x}^{4}}$

Be very careful to simplify $\frac{14}{21}$ by dividing out a common factor, and to simplify the variables by subtracting their exponents.

Find the quotient: $\frac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}.$

$\frac{4{y}^{2}}{7{x}^{4}}$

Find the quotient: $\frac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}.$

$\frac{5}{8{m}^{5}{n}^{3}}$

In all examples so far, there was no work to do in the numerator or denominator before simplifying the fraction. In the next example, we'll first find the product of two monomials in the numerator before we simplify the fraction.

Find the quotient: $\frac{\left(3{x}^{3}{y}^{2}\right)\left(10{x}^{2}{y}^{3}\right)}{6{x}^{4}{y}^{5}}.$

## Solution

Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.

 $\frac{\left(3{x}^{3}{y}^{2}\right)\left(10{x}^{2}{y}^{3}\right)}{6{x}^{4}{y}^{5}}$ Simplify the numerator. $\frac{30{x}^{5}{y}^{5}}{6{x}^{4}{y}^{5}}$ Simplify, using the Quotient Rule. $5x$

Find the quotient: $\frac{\left(3{x}^{4}{y}^{5}\right)\left(8{x}^{2}{y}^{5}\right)}{12{x}^{5}{y}^{8}}.$

2 xy 2

Find the quotient: $\frac{\left(-6{a}^{6}{b}^{9}\right)\left(-8{a}^{5}{b}^{8}\right)}{-12{a}^{10}{b}^{12}}.$

−4 ab 5

## Key concepts

• Equivalent Fractions Property
• If $a,\phantom{\rule{0.2em}{0ex}}b,\phantom{\rule{0.2em}{0ex}}c$ are whole numbers where $b\ne 0,\phantom{\rule{0.2em}{0ex}}c\ne 0,$ then
$\frac{a}{b}=\frac{a\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}{b\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{a\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}{b\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}c}=\frac{a}{b}$
• Zero Exponent
• If $a$ is a non-zero number, then ${a}^{0}=1.$
• Any nonzero number raised to the zero power is $1.$
• Quotient Property for Exponents
• If $a$ is a real number, $a\ne 0,$ and $m,\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then
$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\phantom{\rule{2em}{0ex}}m>n\phantom{\rule{2em}{0ex}}\text{and}\phantom{\rule{2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\phantom{\rule{2em}{0ex}}n>m$
• Quotient to a Power Property for Exponents
• If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then
${\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}$
• To raise a fraction to a power, raise the numerator and denominator to that power.

## Practice makes perfect

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

$\frac{{4}^{8}}{{4}^{2}}$

4 6

$\frac{{3}^{12}}{{3}^{4}}$

$\frac{{x}^{12}}{{x}^{3}}$

x 9

$\frac{{u}^{9}}{{u}^{3}}$

$\frac{{r}^{5}}{r}$

r 4

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

$\frac{{y}^{4}}{{y}^{20}}$

$\frac{1}{{y}^{16}}$

$\frac{{x}^{10}}{{x}^{30}}$

$\frac{{10}^{3}}{{10}^{15}}$

$\frac{1}{{10}^{12}}$

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

$\frac{a}{{a}^{9}}$

$\frac{1}{{a}^{8}}$

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

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

${5}^{0}$

1

${10}^{0}$

${a}^{0}$

1

${x}^{0}$

$-{7}^{0}$

−1

$-{4}^{0}$

1. $\phantom{\rule{0.2em}{0ex}}{\left(10p\right)}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}10{p}^{0}$

1. 1
2. 10

1. $\phantom{\rule{0.2em}{0ex}}{\left(3a\right)}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}3{a}^{0}$

1. $\phantom{\rule{0.2em}{0ex}}{\left(-27{x}^{5}y\right)}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}-27{x}^{5}{y}^{0}$

1. 1
2. −27 x 5

1. $\phantom{\rule{0.2em}{0ex}}{\left(-92{y}^{8}z\right)}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}-92{y}^{8}{z}^{0}$

1. $\phantom{\rule{0.2em}{0ex}}{15}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}{15}^{1}$

1. 1
2. 15

1. $\phantom{\rule{0.2em}{0ex}}-{6}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}-{6}^{1}$

$2\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{x}^{0}+5\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{y}^{0}$

7

$8\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{m}^{0}-4\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{n}^{0}$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

${\left(\frac{3}{2}\right)}^{5}$

$\frac{243}{32}$

${\left(\frac{4}{5}\right)}^{3}$

${\left(\frac{m}{6}\right)}^{3}$

$\frac{{m}^{3}}{216}$

${\left(\frac{p}{2}\right)}^{5}$

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

$\frac{{x}^{10}}{{y}^{10}}$

${\left(\frac{a}{b}\right)}^{8}$

${\left(\frac{a}{3b}\right)}^{2}$

$\frac{{a}^{2}}{9{b}^{2}}$

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

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

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

x 3

$\frac{{\left({y}^{4}\right)}^{3}}{{y}^{7}}$

$\frac{{\left({u}^{3}\right)}^{4}}{{u}^{10}}$

u 2

$\frac{{\left({y}^{2}\right)}^{5}}{{y}^{6}}$

$\frac{{y}^{8}}{{\left({y}^{5}\right)}^{2}}$

$\frac{1}{{y}^{2}}$

$\frac{{p}^{11}}{{\left({p}^{5}\right)}^{3}}$

$\frac{{r}^{5}}{{r}^{4}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}r}$

1

$\frac{{a}^{3}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{a}^{4}}{{a}^{7}}$

${\left(\frac{{x}^{2}}{{x}^{8}}\right)}^{3}$

$\frac{1}{{x}^{18}}$

${\left(\frac{u}{{u}^{10}}\right)}^{2}$

${\left(\frac{{a}^{4}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{a}^{6}}{{a}^{3}}\right)}^{2}$

a 14

${\left(\frac{{x}^{3}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{x}^{8}}{{x}^{4}}\right)}^{3}$

$\frac{{\left({y}^{3}\right)}^{5}}{{\left({y}^{4}\right)}^{3}}$

y 3

$\frac{{\left({z}^{6}\right)}^{2}}{{\left({z}^{2}\right)}^{4}}$

$\frac{{\left({x}^{3}\right)}^{6}}{{\left({x}^{4}\right)}^{7}}$

$\frac{1}{{x}^{10}}$

$\frac{{\left({x}^{4}\right)}^{8}}{{\left({x}^{5}\right)}^{7}}$

${\left(\frac{2{r}^{3}}{5s}\right)}^{4}$

$\frac{16{r}^{12}}{625{s}^{4}}$

${\left(\frac{3{m}^{2}}{4n}\right)}^{3}$

${\left(\frac{3{y}^{2}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{y}^{5}}{{y}^{15}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{y}^{8}}\right)}^{0}$

1

${\left(\frac{15{z}^{4}\phantom{\rule{0.2em}{0ex}}·\phantom{\rule{0.2em}{0ex}}{z}^{9}}{0.3{z}^{2}}\right)}^{0}$

$\frac{{\left({r}^{2}\right)}^{5}\phantom{\rule{0.2em}{0ex}}{\left({r}^{4}\right)}^{2}}{{\left({r}^{3}\right)}^{7}}$

$\frac{1}{{r}^{3}}$

$\frac{{\left({p}^{4}\right)}^{2}\phantom{\rule{0.2em}{0ex}}{\left({p}^{3}\right)}^{5}}{{\left({p}^{2}\right)}^{9}}$

$\frac{{\left(3{x}^{4}\right)}^{3}\phantom{\rule{0.2em}{0ex}}{\left(2{x}^{3}\right)}^{2}}{{\left(6{x}^{5}\right)}^{2}}$

3 x 8

$\frac{{\left(-2{y}^{3}\right)}^{4}\phantom{\rule{0.2em}{0ex}}{\left(3{y}^{4}\right)}^{2}}{{\left(-6{y}^{3}\right)}^{2}}$

Divide Monomials

In the following exercises, divide the monomials.

$48{b}^{8}÷6{b}^{2}$

8 b 6

$42{a}^{14}÷6{a}^{2}$

$36{x}^{3}÷\left(-2{x}^{9}\right)$

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

$20{u}^{8}÷\left(-4{u}^{6}\right)$

$\frac{18{x}^{3}}{9{x}^{2}}$

2 x

$\frac{36{y}^{9}}{4{y}^{7}}$

$\frac{-35{x}^{7}}{-42{x}^{13}}$

$\frac{5}{6{x}^{6}}$

$\frac{18{x}^{5}}{-27{x}^{9}}$

$\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}}$

$\frac{6{r}^{2}}{{s}^{8}}$

$\frac{24{p}^{7}q}{6{p}^{2}{q}^{5}}$

$\frac{8m{n}^{10}}{64m{n}^{4}}$

$\frac{{n}^{6}}{8}$

$\frac{10{a}^{4}b}{50{a}^{2}{b}^{6}}$

$\frac{-12{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}}$

$-\frac{4{y}^{6}}{5{x}^{2}}$

$\frac{48{x}^{11}{y}^{9}{z}^{3}}{36{x}^{6}{y}^{8}{z}^{5}}$

$\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}}$

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

$\frac{\left(10{u}^{2}v\right)\left(4{u}^{3}{v}^{6}\right)}{5{u}^{9}{v}^{2}}$

$\frac{\left(6{m}^{2}n\right)\left(5{m}^{4}{n}^{3}\right)}{3{m}^{10}{n}^{2}}$

$\frac{10{n}^{2}}{{m}^{4}}$

$\frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{8}b\right)\left({a}^{3}b\right)}$

$\frac{\left(4{u}^{5}{v}^{4}\right)\left(15{u}^{8}v\right)}{\left(12{u}^{3}v\right)\left({u}^{6}v\right)}$

5 u 4 v 3

## Mixed practice

1. $\phantom{\rule{0.2em}{0ex}}24{a}^{5}+2{a}^{5}$
2. $\phantom{\rule{0.2em}{0ex}}24{a}^{5}-2{a}^{5}$
3. $\phantom{\rule{0.2em}{0ex}}24{a}^{5}\cdot 2{a}^{5}$
4. $\phantom{\rule{0.2em}{0ex}}24{a}^{5}÷2{a}^{5}$

1. $\phantom{\rule{0.2em}{0ex}}15{n}^{10}+3{n}^{10}$
2. $\phantom{\rule{0.2em}{0ex}}15{n}^{10}-3{n}^{10}$
3. $\phantom{\rule{0.2em}{0ex}}15{n}^{10}\cdot 3{n}^{10}$
4. $\phantom{\rule{0.2em}{0ex}}15{n}^{10}÷3{n}^{10}$

1. $\phantom{\rule{0.2em}{0ex}}18{n}^{10}$
2. $\phantom{\rule{0.2em}{0ex}}12{n}^{10}$
3. $\phantom{\rule{0.2em}{0ex}}45{n}^{20}$
4. $\phantom{\rule{0.2em}{0ex}}5$

1. $\phantom{\rule{0.2em}{0ex}}{p}^{4}\cdot {p}^{6}$
2. $\phantom{\rule{0.2em}{0ex}}{\left({p}^{4}\right)}^{6}$

1. $\phantom{\rule{0.2em}{0ex}}{q}^{5}\cdot {q}^{3}$
2. $\phantom{\rule{0.2em}{0ex}}{\left({q}^{5}\right)}^{3}$

1. $\phantom{\rule{0.2em}{0ex}}{q}^{8}$
2. $\phantom{\rule{0.2em}{0ex}}{q}^{15}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{{y}^{3}}{y}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{y}{{y}^{3}}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{{z}^{6}}{{z}^{5}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{{z}^{5}}{{z}^{6}}$

1. $\phantom{\rule{0.2em}{0ex}}z$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{z}$

$\left(8{x}^{5}\right)\left(9x\right)÷6{x}^{3}$

$\left(4{y}^{5}\right)\left(12{y}^{7}\right)÷8{y}^{2}$

$6{y}^{6}$

$\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}}$

$\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}$

$15{c}^{6}$

$\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}}$

$\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}$

$3{x}^{2}$

$\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s}$

$\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}$

$y{z}^{2}$

## Everyday math

Memory One megabyte is approximately ${10}^{6}$ bytes. One gigabyte is approximately ${10}^{9}$ bytes. How many megabytes are in one gigabyte?

Memory One megabyte is approximately ${10}^{6}$ bytes. One terabyte is approximately ${10}^{12}$ bytes. How many megabytes are in one terabyte?

1,000,000

## Writing exercises

Vic thinks the quotient $\frac{{x}^{20}}{{x}^{4}}$ simplifies to ${x}^{5}.$ What is wrong with his reasoning?

Mai simplifies the quotient $\frac{{y}^{3}}{y}$ by writing $\frac{{\overline{)y}}^{3}}{\overline{)y}}=3.$ What is wrong with her reasoning?

When Dimple simplified $-{3}^{0}$ and ${\left(-3\right)}^{0}$ she got the same answer. Explain how using the Order of Operations correctly gives different answers.

Roxie thinks ${n}^{0}$ simplifies to $0.$ What would you say to convince Roxie she is wrong?

## Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

where we get a research paper on Nano chemistry....?
nanopartical of organic/inorganic / physical chemistry , pdf / thesis / review
Ali
what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
hey
Giriraj
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
ya I also want to know the raman spectra
Bhagvanji
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
Nasa has use it in the 60's, copper as water purification in the moon travel.
Alexandre
nanocopper obvius
Alexandre
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
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?