# 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 many typos can we find...?
5
Joseph
In the LCM Prime Factors exercises, the LCM of 28 and 40 is 280. Not 420!
4x+7y=29,x+3y=11 substitute method of linear equation
substitute method of linear equation
Srinu
Solve one equation for one variable. Using the 2nd equation, x=11-3y. Substitute that for x in first equation. this will find y. then use the value for y to find the value for x.
bruce
I want to learn
Elizebeth
help
Elizebeth
I want to learn. Please teach me?
Wayne
1) Use any equation, and solve for any of the variables. Since the coefficient of x (the number in front of the x) in the second equation is 1 (it actually isn't shown, but 1 * x = x), use that equation. Subtract 3y from both sides (this isolates the x on the left side of the equal sign).
bruce
2) This results in x=11-3y. x is note in terms of y. Use that as the value of x and substitute for all x in the first equation. The first equation becomes 4(11-3y)+7y =29. Note that the only variable left in the first equation is the y. If you have multiple variable, then something is wrong.
bruce
3) Distribute (multiply) the 4 across 11-3y to get 44-12y. Add this to the 7y. So, the equation is now 44-5y=29.
bruce
4) Solve 44-5y=29 for y. Isolate the y by subtracting 44 from birth sides, resulting in -5y=-15. Now, divide birth sides by -5 (since you have -5y). This results in y=3. You now have the value of one variable.
bruce
5) The last step is to take the value of y from Step 4) and substitute into the 2nd equation. Therefore: x+3y=11 becomes x+3(3)=11. Then multiplying, x+9=11. Finally, solve for x by subtracting 9 from both sides. Therefore, x=2.
bruce
6) The ordered pair of (2, 3) is the proposed solution. To check, substitute those values into either equation. If the result is true, then the solution is correct. 4(2)+7(3)=8+21=29. TRUE! Finished.
bruce
At 1:30 Marlon left his house to go to the beach, a distance of 5.625 miles. He rose his skateboard until 2:15, and then walked the rest of the way. He arrived at the beach at 3:00. Marlon's speed on his skateboard is 1.5 times his walking speed. Find his speed when skateboarding and when walking.
divide 3x⁴-4x³-3x-1 by x-3
how to multiply the monomial
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!
how do u solve that question
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
Brandon has a cup of quarters and dimes with a total of 5.55\$. The number of quarters is five less than three times the number of dimes
app is wrong how can 350 be divisible by 3.
June needs 48 gallons of punch for a party and has two different coolers to carry it in. The bigger cooler is five times as large as the smaller cooler. How many gallons can each cooler hold?
Susanna if the first cooler holds five times the gallons then the other cooler. The big cooler holda 40 gallons and the 2nd will hold 8 gallons is that correct?
Georgie
@Susanna that person is correct if you divide 40 by 8 you can see it's 5 it's simple
Ashley
@Geogie my bad that was meant for u
Ashley
Hi everyone, I'm glad to be connected with you all. from France.
I'm getting "math processing error" on math problems. Anyone know why?
Can you all help me I don't get any of this
4^×=9