# 6.5 Divide monomials  (Page 4/4)

 Page 4 / 4

Simplify: $\frac{{\left(2{x}^{4}\right)}^{5}}{{\left(4{x}^{3}\right)}^{2}{\left({x}^{3}\right)}^{5}}.$

$\frac{2}{x}$

## Divide monomials

You have now been introduced to all the properties of exponents and used them to simplify expressions. Next, you’ll see how to use these properties to divide monomials. Later, you’ll use them to divide polynomials.

Find the quotient: $56{x}^{7}÷8{x}^{3}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}56{x}^{7}÷8{x}^{3}\hfill \\ \text{Rewrite as a fraction.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{56{x}^{7}}{8{x}^{3}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{56}{8}\cdot \frac{{x}^{7}}{{x}^{3}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}7{x}^{4}\hfill \end{array}$

Find the quotient: $42{y}^{9}÷6{y}^{3}.$

$7{y}^{6}$

Find the quotient: $48{z}^{8}÷8{z}^{2}.$

$6{z}^{6}$

Find the quotient: $\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}.$

## Solution

When we divide monomials with more than one variable, we write one fraction for each variable.

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{45}{-5}·\frac{{a}^{2}}{a}·\frac{{b}^{3}}{{b}^{5}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}-9·a·\frac{1}{{b}^{2}}\hfill \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}-\frac{9a}{{b}^{2}}\hfill \end{array}$

Find the quotient: $\frac{-72{a}^{7}{b}^{3}}{8{a}^{12}{b}^{4}}.$

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

Find the quotient: $\frac{-63{c}^{8}{d}^{3}}{7{c}^{12}{d}^{2}}.$

$\frac{-9d}{{c}^{4}}$

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

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{24}{48}·\frac{{a}^{5}}{a}·\frac{{b}^{3}}{{b}^{4}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{2}·{a}^{4}·\frac{1}{b}\hfill \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{a}^{4}}{2b}\hfill \end{array}$

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

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

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{2{y}^{6}}{3{x}^{4}}\hfill \end{array}$

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. This follows the order of operations. Remember, a fraction bar is a grouping symbol.

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

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}\hfill \\ \text{Simplify the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{30{x}^{5}{y}^{5}}{3{x}^{4}{y}^{5}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}10x\hfill \end{array}$

Find the quotient: $\frac{\left(6{a}^{4}{b}^{5}\right)\left(4{a}^{2}{b}^{5}\right)}{12{a}^{5}{b}^{8}}.$

$2a{b}^{2}$

Find the quotient: $\frac{\left(-12{x}^{6}{y}^{9}\right)\left(-4{x}^{5}{y}^{8}\right)}{-12{x}^{10}{y}^{12}}.$

$-4x{y}^{5}$

Access these online resources for additional instruction and practice with dividing monomials:

## Key concepts

• Quotient Property for Exponents:
• If $a$ is a real number, $a\ne 0$ , and $m,n$ are whole numbers, then:
$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\phantom{\rule{0.5em}{0ex}}\text{and}\phantom{\rule{0.5em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m$
• Zero Exponent
• If $a$ is a non-zero number, then ${a}^{0}=1$ .

• Quotient to a Power Property for Exponents :
• If $a$ and $b$ are real numbers, $b\ne 0,$ and $m$ is a counting number, then:
${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$
• To raise a fraction to a power, raise the numerator and denominator to that power.

• Summary of Exponent Properties
• If $a,b$ are real numbers and $m,n$ are whole numbers, then
$\begin{array}{ccccc}\mathbf{\text{Product Property}}\hfill & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & \hfill \frac{{a}^{m}}{{b}^{m}}& =\hfill & {a}^{m-n},a\ne 0,m>n\hfill \\ & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & \frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & \hfill {a}^{o}& =\hfill & 1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & \hfill {\left(\frac{a}{b}\right)}^{m}& =\hfill & \frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \end{array}$

## Practice makes perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

$\frac{{x}^{18}}{{x}^{3}}$ $\frac{{5}^{12}}{{5}^{3}}$

$\frac{{y}^{20}}{{y}^{10}}$ $\frac{{7}^{16}}{{7}^{2}}$

${y}^{10}$ ${7}^{14}$

$\frac{{p}^{21}}{{p}^{7}}$ $\frac{{4}^{16}}{{4}^{4}}$

$\frac{{u}^{24}}{{u}^{3}}$ $\frac{{9}^{15}}{{9}^{5}}$

${u}^{21}$ ${9}^{10}$

$\frac{{q}^{18}}{{q}^{36}}$ $\frac{{10}^{2}}{{10}^{3}}$

$\frac{{t}^{10}}{{t}^{40}}$ $\frac{{8}^{3}}{{8}^{5}}$

$\frac{1}{{t}^{30}}$ $\frac{1}{64}$

$\frac{b}{{b}^{9}}$ $\frac{4}{{4}^{6}}$

$\frac{x}{{x}^{7}}$ $\frac{10}{{10}^{3}}$

$\frac{1}{{x}^{6}}$ $\frac{1}{100}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

${20}^{0}$
${b}^{0}$

${13}^{0}$
${k}^{0}$

1 1

$\text{−}{27}^{0}$
$\text{−}\left({27}^{0}\right)$

$\text{−}{15}^{0}$
$\text{−}\left({15}^{0}\right)$

$-1$ $-1$

${\left(25x\right)}^{0}$
$25{x}^{0}$

${\left(6y\right)}^{0}$
$6{y}^{0}$

1 6

${\left(12x\right)}^{0}$
${\left(-56{p}^{4}{q}^{3}\right)}^{0}$

$7{y}^{0}$ ${\left(17y\right)}^{0}$
${\left(-93{c}^{7}{d}^{15}\right)}^{0}$

7 1

$12{n}^{0}-18{m}^{0}$
${\left(12n\right)}^{0}-{\left(18m\right)}^{0}$

$15{r}^{0}-22{s}^{0}$
${\left(15r\right)}^{0}-{\left(22s\right)}^{0}$

$-7$ 0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

${\left(\frac{3}{4}\right)}^{3}$ ${\left(\frac{p}{2}\right)}^{5}$ ${\left(\frac{x}{y}\right)}^{6}$

${\left(\frac{2}{5}\right)}^{2}$ ${\left(\frac{x}{3}\right)}^{4}$ ${\left(\frac{a}{b}\right)}^{5}$

$\frac{4}{25}$ $\frac{{x}^{4}}{81}$ $\frac{{a}^{5}}{{b}^{5}}$

${\left(\frac{a}{3b}\right)}^{4}$ ${\left(\frac{5}{4m}\right)}^{2}$

${\left(\frac{x}{2y}\right)}^{3}$ ${\left(\frac{10}{3q}\right)}^{4}$

$\frac{{x}^{3}}{8{y}^{3}}$ $\frac{10,000}{81{q}^{4}}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

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

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

${p}^{7}$

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

$\frac{{\left({x}^{4}\right)}^{5}}{{x}^{15}}$

${x}^{5}$

$\frac{{u}^{6}}{{\left({u}^{3}\right)}^{2}}$

$\frac{{v}^{20}}{{\left({v}^{4}\right)}^{5}}$

1

$\frac{{m}^{12}}{{\left({m}^{8}\right)}^{3}}$

$\frac{{n}^{8}}{{\left({n}^{6}\right)}^{4}}$

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

${\left(\frac{{p}^{9}}{{p}^{3}}\right)}^{5}$

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

${q}^{18}$

${\left(\frac{{r}^{2}}{{r}^{6}}\right)}^{3}$

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

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

${\left(\frac{p}{{r}^{11}}\right)}^{2}$

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

$\frac{{a}^{3}}{{b}^{18}}$

${\left(\frac{{w}^{5}}{{x}^{3}}\right)}^{8}$

${\left(\frac{{y}^{4}}{{z}^{10}}\right)}^{5}$

$\frac{{y}^{20}}{{z}^{50}}$

${\left(\frac{2{j}^{3}}{3k}\right)}^{4}$

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

$\frac{27{m}^{15}}{125{n}^{3}}$

${\left(\frac{3{c}^{2}}{4{d}^{6}}\right)}^{3}$

${\left(\frac{5{u}^{7}}{2{v}^{3}}\right)}^{4}$

$\frac{625{u}^{28}}{16{v}^{{}^{12}}}$

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

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

${j}^{9}$

$\frac{{\left({t}^{2}\right)}^{5}{\left({t}^{4}\right)}^{2}}{{\left({t}^{3}\right)}^{7}}$

$\frac{{\left({q}^{3}\right)}^{6}{\left({q}^{2}\right)}^{3}}{{\left({q}^{4}\right)}^{8}}$

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

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

$\frac{{\left(-2{k}^{3}\right)}^{2}{\left(6{k}^{2}\right)}^{4}}{{\left(9{k}^{4}\right)}^{2}}$

$64{k}^{6}$

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

$\frac{{\left(-10{n}^{2}\right)}^{3}{\left(4{n}^{5}\right)}^{2}}{{\left(2{n}^{8}\right)}^{2}}$

$-4,000$

Divide Monomials

In the following exercises, divide the monomials.

$56{b}^{8}÷7{b}^{2}$

$63{v}^{10}÷9{v}^{2}$

$7{v}^{8}$

$-88{y}^{15}÷8{y}^{3}$

$-72{u}^{12}÷12{u}^{4}$

$-6{u}^{8}$

$\frac{45{a}^{6}{b}^{8}}{-15{a}^{10}{b}^{2}}$

$\frac{54{x}^{9}{y}^{3}}{-18{x}^{6}{y}^{15}}$

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

$\frac{15{r}^{4}{s}^{9}}{18{r}^{9}{s}^{2}}$

$\frac{20{m}^{8}{n}^{4}}{30{m}^{5}{n}^{9}}$

$\frac{-2{m}^{3}}{3{n}^{5}}$

$\frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}}$

$\frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}$

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

$\frac{64{q}^{11}{r}^{9}{s}^{3}}{48{q}^{6}{r}^{8}{s}^{5}}$

$\frac{65{a}^{10}{b}^{8}{c}^{5}}{42{a}^{7}{b}^{6}{c}^{8}}$

$\frac{65{a}^{3}{b}^{2}}{42{c}^{3}}$

$\frac{\left(10{m}^{5}{n}^{4}\right)\left(5{m}^{3}{n}^{6}\right)}{25{m}^{7}{n}^{5}}$

$\frac{\left(-18{p}^{4}{q}^{7}\right)\left(-6{p}^{3}{q}^{8}\right)}{-36{p}^{12}{q}^{10}}$

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

$\frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{2}b\right)\left({a}^{3}b\right)}$

$\frac{\left(4{u}^{2}{v}^{5}\right)\left(15{u}^{3}v\right)}{\left(12{u}^{3}v\right)\left({u}^{4}v\right)}$

$\frac{5{v}^{4}}{{u}^{2}}$

Mixed Practice

$24{a}^{5}+2{a}^{5}$
$24{a}^{5}-2{a}^{5}$
$24{a}^{5}·2{a}^{5}$
$24{a}^{5}÷2{a}^{5}$

$15{n}^{10}+3{n}^{10}$
$15{n}^{10}-3{n}^{10}$
$15{n}^{10}·3{n}^{10}$
$15{n}^{10}÷3{n}^{10}$

$18{n}^{10}$
$12{n}^{10}$
$45{n}^{20}$
5

${p}^{4}·{p}^{6}$
${\left({p}^{4}\right)}^{6}$

${q}^{5}·{q}^{3}$
${\left({q}^{5}\right)}^{3}$

${q}^{8}$
${q}^{15}$

$\frac{{y}^{3}}{y}$
$\frac{y}{{y}^{3}}$

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

$z$ $\frac{1}{z}$

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

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

$6{y}^{6}$

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

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

$15{c}^{6}$

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

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

$3{x}^{2}$

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

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

$\text{−}y{z}^{2}$

## 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 gigabyte is approximately ${10}^{9}$ bytes. One terabyte is approximately ${10}^{12}$ bytes. How many gigabytes are in one terabyte?

${10}^{3}$

## Writing exercises

Jennifer thinks the quotient $\frac{{a}^{24}}{{a}^{6}}$ simplifies to ${a}^{4}$ . What is wrong with her reasoning?

Maurice simplifies the quotient $\frac{{d}^{7}}{d}$ by writing $\frac{{\overline{)d}}^{7}}{\overline{)d}}=7$ . What is wrong with his reasoning?

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

Robert thinks ${x}^{0}$ simplifies to 0. What would you say to convince Robert he 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?

how did you get the value of 2000N.What calculations are needed to arrive at it
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Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike? Got questions? Get instant answers now!
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Seera
Two sisters like to compete on their bike rides. Tamara can go 4 mph faster than her sister, Samantha. If it takes Samantha 1 hours longer than Tamara to go 80 miles, how fast can Samantha ride her bike?
Seera
Speed=distance ÷ time
Tremayne
x-3y =1; 3x-2y+4=0 graph
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