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Simplify: $\frac{{\left(2{x}^{4}\right)}^{5}}{{\left(4{x}^{3}\right)}^{2}{\left({x}^{3}\right)}^{5}}.$
$\frac{2}{x}$
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: $56{x}^{7}\xf78{x}^{3}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}56{x}^{7}\xf78{x}^{3}\hfill \\ \text{Rewrite as a fraction.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{56{x}^{7}}{8{x}^{3}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{56}{8}\cdot \frac{{x}^{7}}{{x}^{3}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}7{x}^{4}\hfill \end{array}$
Find the quotient: $\frac{45{a}^{2}{b}^{3}}{\mathrm{-5}a{b}^{5}}.$
When we divide monomials with more than one variable, we write one fraction for each variable.
$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{45{a}^{2}{b}^{3}}{\mathrm{-5}a{b}^{5}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{45}{\mathrm{-5}}\xb7\frac{{a}^{2}}{a}\xb7\frac{{b}^{3}}{{b}^{5}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\mathrm{-9}\xb7a\xb7\frac{1}{{b}^{2}}\hfill \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}-\frac{9a}{{b}^{2}}\hfill \end{array}$
Find the quotient: $\frac{\mathrm{-72}{a}^{7}{b}^{3}}{8{a}^{12}{b}^{4}}.$
$-\frac{9}{{a}^{5}b}$
Find the quotient: $\frac{\mathrm{-63}{c}^{8}{d}^{3}}{7{c}^{12}{d}^{2}}.$
$\frac{\mathrm{-9}d}{{c}^{4}}$
Find the quotient: $\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{24}{48}\xb7\frac{{a}^{5}}{a}\xb7\frac{{b}^{3}}{{b}^{4}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{2}\xb7{a}^{4}\xb7\frac{1}{b}\hfill \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{a}^{4}}{2b}\hfill \end{array}$
Find the quotient: $\frac{16{a}^{7}{b}^{6}}{24a{b}^{8}}.$
$\frac{2{a}^{6}}{3{b}^{2}}$
Find the quotient: $\frac{27{p}^{4}{q}^{7}}{\mathrm{-45}{p}^{12}q}.$
$-\frac{3{q}^{6}}{5{p}^{8}}$
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: $\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.$
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: $\frac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}.$
$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}\hfill \\ \text{Simplify the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{30{x}^{5}{y}^{5}}{3{x}^{4}{y}^{5}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}10x\hfill \end{array}$
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: $\frac{\left(\mathrm{-12}{x}^{6}{y}^{9}\right)\left(\mathrm{-4}{x}^{5}{y}^{8}\right)}{\mathrm{-12}{x}^{10}{y}^{12}}.$
$\mathrm{-4}x{y}^{5}$
Access these online resources for additional instruction and practice with dividing monomials:
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.
ⓐ
${20}^{0}$
ⓑ
${b}^{0}$
ⓐ
$\text{\u2212}{27}^{0}$
ⓑ
$\text{\u2212}\left({27}^{0}\right)$
ⓐ
$\text{\u2212}{15}^{0}$
ⓑ
$\text{\u2212}\left({15}^{0}\right)$
ⓐ $\mathrm{-1}$ ⓑ $\mathrm{-1}$
ⓐ
${\left(25x\right)}^{0}$
ⓑ
$25{x}^{0}$
ⓐ
${\left(12x\right)}^{0}$
ⓑ
${\left(\mathrm{-56}{p}^{4}{q}^{3}\right)}^{0}$
ⓐ
$7{y}^{0}$
${\left(17y\right)}^{0}$
ⓑ
${\left(\mathrm{-93}{c}^{7}{d}^{15}\right)}^{0}$
ⓐ 7 ⓑ 1
ⓐ
$12{n}^{0}-18{m}^{0}$
ⓑ
${\left(12n\right)}^{0}-{\left(18m\right)}^{0}$
ⓐ
$15{r}^{0}-22{s}^{0}$
ⓑ
${\left(15r\right)}^{0}-{\left(22s\right)}^{0}$
ⓐ $\mathrm{-7}$ ⓑ 0
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
ⓐ ${\left(\frac{3}{4}\right)}^{3}$ ⓑ ${\left(\frac{p}{2}\right)}^{5}$ ⓒ ${\left(\frac{x}{y}\right)}^{6}$
ⓐ ${\left(\frac{2}{5}\right)}^{2}$ ⓑ ${\left(\frac{x}{3}\right)}^{4}$ ⓒ ${\left(\frac{a}{b}\right)}^{5}$
ⓐ $\frac{4}{25}$ ⓑ $\frac{{x}^{4}}{81}$ ⓒ $\frac{{a}^{5}}{{b}^{5}}$
ⓐ ${\left(\frac{a}{3b}\right)}^{4}$ ⓑ ${\left(\frac{5}{4m}\right)}^{2}$
ⓐ ${\left(\frac{x}{2y}\right)}^{3}$ ⓑ ${\left(\frac{10}{3q}\right)}^{4}$
ⓐ $\frac{{x}^{3}}{8{y}^{3}}$ ⓑ $\frac{\mathrm{10,000}}{81{q}^{4}}$
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
$\frac{{\left({a}^{2}\right)}^{3}}{{a}^{4}}$
$\frac{{\left({y}^{3}\right)}^{4}}{{y}^{10}}$
$\frac{{u}^{6}}{{\left({u}^{3}\right)}^{2}}$
$\frac{{m}^{12}}{{\left({m}^{8}\right)}^{3}}$
$\frac{{n}^{8}}{{\left({n}^{6}\right)}^{4}}$
$\frac{1}{{n}^{16}}$
${\left(\frac{{p}^{9}}{{p}^{3}}\right)}^{5}$
${\left(\frac{{r}^{2}}{{r}^{6}}\right)}^{3}$
${\left(\frac{{m}^{4}}{{m}^{7}}\right)}^{4}$
$\frac{1}{{m}^{12}}$
${\left(\frac{p}{{r}^{11}}\right)}^{2}$
${\left(\frac{a}{{b}^{6}}\right)}^{3}$
$\frac{{a}^{3}}{{b}^{18}}$
${\left(\frac{{w}^{5}}{{x}^{3}}\right)}^{8}$
${\left(\frac{{y}^{4}}{{z}^{10}}\right)}^{5}$
$\frac{{y}^{20}}{{z}^{50}}$
${\left(\frac{2{j}^{3}}{3k}\right)}^{4}$
${\left(\frac{3{m}^{5}}{5n}\right)}^{3}$
$\frac{27{m}^{15}}{125{n}^{3}}$
${\left(\frac{3{c}^{2}}{4{d}^{6}}\right)}^{3}$
${\left(\frac{5{u}^{7}}{2{v}^{3}}\right)}^{4}$
$\frac{625{u}^{28}}{16{v}^{{}^{12}}}$
${\left(\frac{{k}^{2}{k}^{8}}{{k}^{3}}\right)}^{2}$
${\left(\frac{{j}^{2}{j}^{5}}{{j}^{4}}\right)}^{3}$
${j}^{9}$
$\frac{{\left({t}^{2}\right)}^{5}{\left({t}^{4}\right)}^{2}}{{\left({t}^{3}\right)}^{7}}$
$\frac{{\left({q}^{3}\right)}^{6}{\left({q}^{2}\right)}^{3}}{{\left({q}^{4}\right)}^{8}}$
$\frac{1}{{q}^{8}}$
$\frac{{\left(\mathrm{-2}{p}^{2}\right)}^{4}{\left(3{p}^{4}\right)}^{2}}{{\left(\mathrm{-6}{p}^{3}\right)}^{2}}$
$\frac{{\left(\mathrm{-2}{k}^{3}\right)}^{2}{\left(6{k}^{2}\right)}^{4}}{{\left(9{k}^{4}\right)}^{2}}$
$64{k}^{6}$
$\frac{{\left(\mathrm{-4}{m}^{3}\right)}^{2}{\left(5{m}^{4}\right)}^{3}}{{\left(\mathrm{-10}{m}^{6}\right)}^{3}}$
$\frac{{\left(\mathrm{-10}{n}^{2}\right)}^{3}{\left(4{n}^{5}\right)}^{2}}{{\left(2{n}^{8}\right)}^{2}}$
$\mathrm{-4,000}$
Divide Monomials
In the following exercises, divide the monomials.
$56{b}^{8}\xf77{b}^{2}$
$\mathrm{-88}{y}^{15}\xf78{y}^{3}$
$\frac{45{a}^{6}{b}^{8}}{\mathrm{-15}{a}^{10}{b}^{2}}$
$\frac{54{x}^{9}{y}^{3}}{\mathrm{-18}{x}^{6}{y}^{15}}$
$-\frac{3{x}^{3}}{{y}^{12}}$
$\frac{15{r}^{4}{s}^{9}}{18{r}^{9}{s}^{2}}$
$\frac{20{m}^{8}{n}^{4}}{30{m}^{5}{n}^{9}}$
$\frac{\mathrm{-2}{m}^{3}}{3{n}^{5}}$
$\frac{18{a}^{4}{b}^{8}}{\mathrm{-27}{a}^{9}{b}^{5}}$
$\frac{45{x}^{5}{y}^{9}}{\mathrm{-60}{x}^{8}{y}^{6}}$
$\frac{\mathrm{-3}{y}^{3}}{4{x}^{3}}$
$\frac{64{q}^{11}{r}^{9}{s}^{3}}{48{q}^{6}{r}^{8}{s}^{5}}$
$\frac{65{a}^{10}{b}^{8}{c}^{5}}{42{a}^{7}{b}^{6}{c}^{8}}$
$\frac{65{a}^{3}{b}^{2}}{42{c}^{3}}$
$\frac{\left(10{m}^{5}{n}^{4}\right)\left(5{m}^{3}{n}^{6}\right)}{25{m}^{7}{n}^{5}}$
$\frac{\left(\mathrm{-18}{p}^{4}{q}^{7}\right)\left(\mathrm{-6}{p}^{3}{q}^{8}\right)}{\mathrm{-36}{p}^{12}{q}^{10}}$
$\frac{\mathrm{-3}{q}^{5}}{{p}^{5}}$
$\frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{2}b\right)\left({a}^{3}b\right)}$
$\frac{\left(4{u}^{2}{v}^{5}\right)\left(15{u}^{3}v\right)}{\left(12{u}^{3}v\right)\left({u}^{4}v\right)}$
$\frac{5{v}^{4}}{{u}^{2}}$
Mixed Practice
ⓐ
$24{a}^{5}+2{a}^{5}$
ⓑ
$24{a}^{5}-2{a}^{5}$
ⓒ
$24{a}^{5}\xb72{a}^{5}$
ⓓ
$24{a}^{5}\xf72{a}^{5}$
ⓐ
$15{n}^{10}+3{n}^{10}$
ⓑ
$15{n}^{10}-3{n}^{10}$
ⓒ
$15{n}^{10}\xb73{n}^{10}$
ⓓ
$15{n}^{10}\xf73{n}^{10}$
ⓐ
$18{n}^{10}$
ⓑ
$12{n}^{10}$
ⓒ
$45{n}^{20}$
ⓓ 5
ⓐ
${p}^{4}\xb7{p}^{6}$
ⓑ
${\left({p}^{4}\right)}^{6}$
ⓐ
${q}^{5}\xb7{q}^{3}$
ⓑ
${\left({q}^{5}\right)}^{3}$
ⓐ
${q}^{8}$
ⓑ
${q}^{15}$
ⓐ
$\frac{{y}^{3}}{y}$
ⓑ
$\frac{y}{{y}^{3}}$
ⓐ
$\frac{{z}^{6}}{{z}^{5}}$
ⓑ
$\frac{{z}^{5}}{{z}^{6}}$
ⓐ $z$ ⓑ $\frac{1}{z}$
$\left(8{x}^{5}\right)\left(9x\right)\xf76{x}^{3}$
$\left(4y\right)\left(12{y}^{7}\right)\xf78{y}^{2}$
$6{y}^{6}$
$\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}}$
$\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}$
$15{c}^{6}$
$\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}}$
$\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}$
$3{x}^{2}$
$\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s}$
$\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}$
$\text{\u2212}y{z}^{2}$
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?
${10}^{3}$
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?
Answers will vary.
When Drake simplified $\text{\u2212}{3}^{0}$ and ${\left(\mathrm{-3}\right)}^{0}$ 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?
Answers will vary.
ⓐ 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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