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Find the quotient: $\frac{\mathrm{-72}{a}^{4}{b}^{5}}{\mathrm{-8}{a}^{9}{b}^{5}}.$
$\frac{9}{{a}^{5}}$
Find the quotient: $\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}.$
$\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}$ | |
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: $\frac{27{p}^{4}{q}^{7}}{\mathrm{-45}{p}^{12}{q}^{}}.$
$-\frac{3{q}^{6}}{5{p}^{8}}$
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: $\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.$
$\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}$ | |
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: $\frac{(3{x}^{3}{y}^{2})(10{x}^{2}{y}^{3})}{6{x}^{4}{y}^{5}}.$
Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.
$\frac{(3{x}^{3}{y}^{2})(10{x}^{2}{y}^{3})}{6{x}^{4}{y}^{5}}$ | |
Simplify the numerator. | $\frac{30{x}^{5}{y}^{5}}{6{x}^{4}{y}^{5}}$ |
Simplify, using the Quotient Rule. | $5x$ |
Find the quotient: $\frac{(3{x}^{4}{y}^{5})(8{x}^{2}{y}^{5})}{12{x}^{5}{y}^{8}}.$
2 xy ^{2}
Find the quotient: $\frac{(\mathrm{-6}{a}^{6}{b}^{9})(\mathrm{-8}{a}^{5}{b}^{8})}{\mathrm{-12}{a}^{10}{b}^{12}}.$
−4 ab ^{5}
Simplify Expressions Using the Quotient Property of Exponents
In the following exercises, simplify.
$\frac{{3}^{12}}{{3}^{4}}$
$\frac{{u}^{9}}{{u}^{3}}$
$\frac{{y}^{4}}{y}$
$\frac{{x}^{10}}{{x}^{30}}$
$\frac{{r}^{2}}{{r}^{8}}$
$\frac{2}{{2}^{5}}$
Simplify Expressions with Zero Exponents
In the following exercises, simplify.
${10}^{0}$
${x}^{0}$
$-{4}^{0}$
$2\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{x}^{0}+5\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{y}^{0}$
7
$8\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{m}^{0}-4\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{n}^{0}$
Simplify Expressions Using the Quotient to a Power Property
In the following exercises, simplify.
${\left(\frac{4}{5}\right)}^{3}$
${\left(\frac{p}{2}\right)}^{5}$
${\left(\frac{x}{y}\right)}^{10}$
$\frac{{x}^{10}}{{y}^{10}}$
${\left(\frac{a}{b}\right)}^{8}$
${\left(\frac{a}{3b}\right)}^{2}$
$\frac{{a}^{2}}{9{b}^{2}}$
${\left(\frac{2x}{y}\right)}^{4}$
Simplify Expressions by Applying Several Properties
In the following exercises, simplify.
$\frac{{\left({y}^{4}\right)}^{3}}{{y}^{7}}$
$\frac{{\left({y}^{2}\right)}^{5}}{{y}^{6}}$
$\frac{{y}^{8}}{{\left({y}^{5}\right)}^{2}}$
$\frac{1}{{y}^{2}}$
$\frac{{p}^{11}}{{\left({p}^{5}\right)}^{3}}$
$\frac{{r}^{5}}{{r}^{4}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}r}$
1
$\frac{{a}^{3}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{a}^{4}}{{a}^{7}}$
${\left(\frac{{x}^{2}}{{x}^{8}}\right)}^{3}$
$\frac{1}{{x}^{18}}$
${\left(\frac{u}{{u}^{10}}\right)}^{2}$
${\left(\frac{{a}^{4}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{a}^{6}}{{a}^{3}}\right)}^{2}$
a ^{14}
${\left(\frac{{x}^{3}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{x}^{8}}{{x}^{4}}\right)}^{3}$
$\frac{{\left({y}^{3}\right)}^{5}}{{\left({y}^{4}\right)}^{3}}$
y ^{3}
$\frac{{\left({z}^{6}\right)}^{2}}{{\left({z}^{2}\right)}^{4}}$
$\frac{{\left({x}^{3}\right)}^{6}}{{\left({x}^{4}\right)}^{7}}$
$\frac{1}{{x}^{10}}$
$\frac{{\left({x}^{4}\right)}^{8}}{{\left({x}^{5}\right)}^{7}}$
${\left(\frac{2{r}^{3}}{5s}\right)}^{4}$
$\frac{16{r}^{12}}{625{s}^{4}}$
${\left(\frac{3{m}^{2}}{4n}\right)}^{3}$
${\left(\frac{3{y}^{2}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{y}^{5}}{{y}^{15}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{y}^{8}}\right)}^{0}$
1
${\left(\frac{15{z}^{4}\phantom{\rule{0.2em}{0ex}}\xb7\phantom{\rule{0.2em}{0ex}}{z}^{9}}{0.3{z}^{2}}\right)}^{0}$
$\frac{{\left({r}^{2}\right)}^{5}\phantom{\rule{0.2em}{0ex}}{\left({r}^{4}\right)}^{2}}{{\left({r}^{3}\right)}^{7}}$
$\frac{1}{{r}^{3}}$
$\frac{{\left({p}^{4}\right)}^{2}\phantom{\rule{0.2em}{0ex}}{\left({p}^{3}\right)}^{5}}{{\left({p}^{2}\right)}^{9}}$
$\frac{{\left(3{x}^{4}\right)}^{3}\phantom{\rule{0.2em}{0ex}}{\left(2{x}^{3}\right)}^{2}}{{\left(6{x}^{5}\right)}^{2}}$
3 x ^{8}
$\frac{{\left(\mathrm{-2}{y}^{3}\right)}^{4}\phantom{\rule{0.2em}{0ex}}{\left(3{y}^{4}\right)}^{2}}{{\left(\mathrm{-6}{y}^{3}\right)}^{2}}$
Divide Monomials
In the following exercises, divide the monomials.
$42{a}^{14}\xf76{a}^{2}$
$36{x}^{3}\xf7\left(\mathrm{-2}{x}^{9}\right)$
$\frac{\mathrm{-18}}{{x}^{6}}$
$20{u}^{8}\xf7\left(\mathrm{-4}{u}^{6}\right)$
$\frac{36{y}^{9}}{4{y}^{7}}$
$\frac{\mathrm{-35}{x}^{7}}{\mathrm{-42}{x}^{13}}$
$\frac{5}{6{x}^{6}}$
$\frac{18{x}^{5}}{\mathrm{-27}{x}^{9}}$
$\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}}$
$\frac{6{r}^{2}}{{s}^{8}}$
$\frac{24{p}^{7}q}{6{p}^{2}{q}^{5}}$
$\frac{10{a}^{4}b}{50{a}^{2}{b}^{6}}$
$\frac{\mathrm{-12}{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}}$
$-\frac{4{y}^{6}}{5{x}^{2}}$
$\frac{48{x}^{11}{y}^{9}{z}^{3}}{36{x}^{6}{y}^{8}{z}^{5}}$
$\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}}$
$\frac{4z}{3{x}^{2}{y}^{3}}$
$\frac{(10{u}^{2}v)(4{u}^{3}{v}^{6})}{5{u}^{9}{v}^{2}}$
$\frac{(6{m}^{2}n)(5{m}^{4}{n}^{3})}{3{m}^{10}{n}^{2}}$
$\frac{10{n}^{2}}{{m}^{4}}$
$\frac{(6{a}^{4}{b}^{3})(4a{b}^{5})}{(12{a}^{8}b)({a}^{3}b)}$
$\frac{(4{u}^{5}{v}^{4})(15{u}^{8}v)}{(12{u}^{3}v)({u}^{6}v)}$
5 u ^{4} v ^{3}
$\left(8{x}^{5}\right)\left(9x\right)\xf76{x}^{3}$
$\left(4{y}^{5}\right)\left(12{y}^{7}\right)\xf78{y}^{2}$
$6{y}^{6}$
$\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}}$
$\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}$
$15{c}^{6}$
$\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}}$
$\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}$
$3{x}^{2}$
$\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s}$
$\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}$
$y{z}^{2}$
Memory One megabyte is approximately ${10}^{6}$ bytes. One gigabyte is approximately ${10}^{9}$ bytes. How many megabytes are in one gigabyte?
Memory One megabyte is approximately ${10}^{6}$ bytes. One terabyte is approximately ${10}^{12}$ bytes. How many megabytes are in one terabyte?
1,000,000
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?
Answers will vary.
When Dimple simplified $-{3}^{0}$ and ${\left(\mathrm{-3}\right)}^{0}$ 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?
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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