# 6.5 Divide monomials  (Page 3/4)

 Page 3 / 4

This leads to the Quotient to a Power Property for Exponents .

## 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.

$\begin{array}{ccc}\hfill {\left(\frac{2}{3}\right)}^{3}& =\hfill & \frac{{2}^{3}}{{3}^{3}}\hfill \\ \hfill \frac{2}{3}·\frac{2}{3}·\frac{2}{3}& =\hfill & \frac{8}{27}\hfill \\ \hfill \frac{8}{27}& =\hfill & \frac{8}{27}✓\hfill \end{array}$

Simplify: ${\left(\frac{3}{7}\right)}^{2}$ ${\left(\frac{b}{3}\right)}^{4}$ ${\left(\frac{k}{j}\right)}^{3}.$

## Solution

 Use the Quotient Property, ${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$ . Simplify.

 Use the Quotient Property, ${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$ . Simplify.

 Raise the numerator and denominator to the third power.

Simplify: ${\left(\frac{5}{8}\right)}^{2}$ ${\left(\frac{p}{10}\right)}^{4}$ ${\left(\frac{m}{n}\right)}^{7}.$

$\frac{25}{64}$ $\frac{{p}^{4}}{10,000}$ $\frac{{m}^{7}}{{n}^{7}}$

Simplify: ${\left(\frac{1}{3}\right)}^{3}$ ${\left(\frac{-2}{q}\right)}^{3}$ ${\left(\frac{w}{x}\right)}^{4}.$

$\frac{1}{27}$ $\frac{-8}{{q}^{3}}$ $\frac{{w}^{4}}{{x}^{4}}$

## Simplify expressions by applying several properties

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

## Summary of exponent properties

If $a\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}b$ are real numbers, and $m\phantom{\rule{0.2em}{0ex}}\text{and}\phantom{\rule{0.2em}{0ex}}n$ are whole numbers, then

$\begin{array}{ccccc}\mathbf{\text{Product Property}}\hfill & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \\ \mathbf{\text{Quotient Property}}\hfill & & \hfill \frac{{a}^{m}}{{b}^{m}}& =\hfill & {a}^{m-n},a\ne 0,m>n\hfill \\ & & \hfill \frac{{a}^{m}}{{a}^{n}}& =\hfill & \frac{1}{{a}^{n-m}},a\ne 0,n>m\hfill \\ \mathbf{\text{Zero Exponent Definition}}\hfill & & \hfill {a}^{o}& =\hfill & 1,a\ne 0\hfill \\ \mathbf{\text{Quotient to a Power Property}}\hfill & & \hfill {\left(\frac{a}{b}\right)}^{m}& =\hfill & \frac{{a}^{m}}{{b}^{m}},b\ne 0\hfill \end{array}$

Simplify: $\frac{{\left({y}^{4}\right)}^{2}}{{y}^{6}}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left({y}^{4}\right)}^{2}}{{y}^{6}}\hfill \\ \text{Multiply the exponents in the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{y}^{8}}{{y}^{6}}\hfill \\ \text{Subtract the exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{y}^{2}\hfill \end{array}$

Simplify: $\frac{{\left({m}^{5}\right)}^{4}}{{m}^{7}}.$

${m}^{13}$

Simplify: $\frac{{\left({k}^{2}\right)}^{6}}{{k}^{7}}.$

${k}^{5}$

Simplify: $\frac{{b}^{12}}{{\left({b}^{2}\right)}^{6}}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{b}^{12}}{{\left({b}^{2}\right)}^{6}}\hfill \\ \text{Multiply the exponents in the denominator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{b}^{12}}{{b}^{12}}\hfill \\ \text{Subtract the exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{b}^{0}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}1\hfill \end{array}$

Notice that after we simplified the denominator in the first step, the numerator and the denominator were equal. So the final value is equal to 1.

Simplify: $\frac{{n}^{12}}{{\left({n}^{3}\right)}^{4}}.$

1

Simplify: $\frac{{x}^{15}}{{\left({x}^{3}\right)}^{5}}.$

1

Simplify: ${\left(\frac{{y}^{9}}{{y}^{4}}\right)}^{2}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{{y}^{9}}{{y}^{4}}\right)}^{2}\hfill \\ \begin{array}{c}\text{Remember parentheses come before exponents.}\hfill \\ \text{Notice the bases are the same, so we can simplify}\hfill \\ \text{inside the parentheses. Subtract the exponents.}\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{\left({y}^{5}\right)}^{2}\hfill \\ \text{Multiply the exponents.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}{y}^{10}\hfill \end{array}$

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

${r}^{8}$

Simplify: ${\left(\frac{{v}^{6}}{{v}^{4}}\right)}^{3}.$

${v}^{6}$

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

## Solution

Here we cannot simplify inside the parentheses first, since the bases are not the same.

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{{j}^{2}}{{k}^{3}}\right)}^{4}\hfill \\ \begin{array}{c}\text{Raise the numerator and denominator to the third power}\hfill \\ \text{using the Quotient to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}.\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left({j}^{2}\right)}^{4}}{{\left({k}^{3}\right)}^{4}}\hfill \\ \text{Use the Power Property and simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{j}^{8}}{{k}^{12}}\hfill \end{array}$

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

$\frac{{a}^{12}}{{b}^{8}}$

Simplify: ${\left(\frac{{q}^{7}}{{r}^{5}}\right)}^{3}.$

$\frac{{q}^{21}}{{r}^{15}}$

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

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}{\left(\frac{2{m}^{2}}{5n}\right)}^{4}\hfill \\ \begin{array}{c}\text{Raise the numerator and denominator to the fourth}\hfill \\ \text{power, using the Quotient to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}.\hfill \end{array}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(2{m}^{2}\right)}^{4}}{{\left(5n\right)}^{4}}\hfill \\ \text{Raise each factor to the fourth power.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{2}^{4}{\left({m}^{2}\right)}^{4}}{{5}^{4}{n}^{4}}\hfill \\ \text{Use the Power Property and simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{16{m}^{8}}{625{n}^{4}}\hfill \end{array}$

Simplify: ${\left(\frac{7{x}^{3}}{9y}\right)}^{2}.$

$\frac{49{x}^{6}}{81{y}^{2}}$

Simplify: ${\left(\frac{3{x}^{4}}{7y}\right)}^{2}.$

$\frac{9{x}^{8}}{49{y}^{2}}$

Simplify: $\frac{{\left({x}^{3}\right)}^{4}{\left({x}^{2}\right)}^{5}}{{\left({x}^{6}\right)}^{5}}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left({x}^{3}\right)}^{4}{\left({x}^{2}\right)}^{5}}{{\left({x}^{6}\right)}^{5}}\hfill \\ \text{Use the Power Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m·n}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{\left({x}^{12}\right)\left({x}^{10}\right)}{\left({x}^{30}\right)}\hfill \\ \text{Add the exponents in the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{x}^{22}}{{x}^{30}}\hfill \\ \text{Use the Quotient Property,}\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{{x}^{8}}\hfill \end{array}$

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

$\frac{1}{{a}^{6}}$

Simplify: $\frac{{\left({p}^{3}\right)}^{4}{\left({p}^{5}\right)}^{3}}{{\left({p}^{7}\right)}^{6}}.$

$\frac{1}{{p}^{15}}$

Simplify: $\frac{{\left(10{p}^{3}\right)}^{2}}{{\left(5p\right)}^{3}{\left(2{p}^{5}\right)}^{4}}.$

## Solution

$\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(10{p}^{3}\right)}^{2}}{{\left(5p\right)}^{3}{\left(2{p}^{5}\right)}^{4}}\hfill \\ \text{Use the Product to a Power Property,}\phantom{\rule{0.2em}{0ex}}{\left(ab\right)}^{m}={a}^{m}{b}^{m}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{\left(10\right)}^{2}{\left({p}^{3}\right)}^{2}}{{\left(5\right)}^{3}{\left(p\right)}^{3}{\left(2\right)}^{4}{\left({p}^{5}\right)}^{4}}\hfill \\ \text{Use the Power Property,}\phantom{\rule{0.2em}{0ex}}{\left({a}^{m}\right)}^{n}={a}^{m·n}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100{p}^{6}}{125{p}^{3}·16{p}^{20}}\hfill \\ \text{Add the exponents in the denominator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100{p}^{6}}{125·16{p}^{23}}\hfill \\ \text{Use the Quotient Property,}\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{100}{125·16{p}^{17}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{20{p}^{17}}\hfill \end{array}$

Simplify: $\frac{{\left(3{r}^{3}\right)}^{2}{\left({r}^{3}\right)}^{7}}{{\left({r}^{3}\right)}^{3}}.$

$9{r}^{18}$

how did you get the value of 2000N.What calculations are needed to arrive at it
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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
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Tremayne
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