# 10.4 Divide monomials  (Page 3/5)

 Page 4 / 5

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?

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