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Divide Monomials

Simplify expressions using the Quotient Property for Exponents

Simplify Expressions Using the Quotient Property for Exponents

Earlier in this chapter, we developed the properties of exponents for multiplication. We summarize these properties below.

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.

As before, we’ll try to discover a property by looking at some examples.

Consider	\(\frac{{x}^{5}}{{x}^{2}}\)	and	\(\frac{{x}^{2}}{{x}^{3}}\)
What do they mean?	\(\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\)		\(\frac{x\cdot x}{x\cdot x\cdot x}\)
Use the Equivalent Fractions Property.	\(\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\)		\(\frac{x\cdot x\cdot 1}{x\cdot x\cdot x}\)
Simplify.	\({x}^{3}\)		\(\frac{1}{x}\)

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

We write:

\[\begin{array}{llll}\frac{{x}^{5}}{{x}^{2}} & & & \frac{{x}^{2}}{{x}^{3}} \\ {x}^{5-2} & & & \frac{1}{{x}^{3-2}} \\ {x}^{3} & & & \frac{1}{x}\end{array}\]\[\begin{array}{llllllll}\frac{{3}^{4}}{{3}^{2}} & = & {3}^{4-2} & & & \ \frac{{5}^{2}}{{5}^{3}} & = & \frac{1}{{5}^{3-2}} \\ \frac{81}{9} & = & {3}^{2} & & & \ \frac{25}{125} & = & \frac{1}{{5}^{1}} \\ 9 & = & 9✓ & & & \ \frac{1}{5} & = & \frac{1}{5}✓\end{array}\]

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions with an Exponent of Zero

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like \(\frac{{a}^{m}}{{a}^{m}}\). From your earlier work with fractions, you know that:

\[\frac{2}{2}=1\ \frac{17}{17}=1\ \frac{-43}{-43}=1\]

In words, a number divided by itself is 1. So, \(\frac{x}{x}=1\), for any \(x\ (x\ne 0)\), since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify \(\frac{{a}^{m}}{{a}^{n}}\) when \(m>n\) and when \(n

Consider \(\frac{8}{8}\), which we know is 1.

	\(\begin{array}{l}\frac{8}{8}=1\end{array}\)
Write \(8\) as \({2}^{3}\).	\(\begin{array}{l}\frac{{2}^{3}}{{2}^{3}}=1\end{array}\)
Subtract exponents.	\(\begin{array}{l}{2}^{3-3}=1\end{array}\)
Simplify.	\(\begin{array}{l}{2}^{0}=1\end{array}\)

Now we will simplify \(\frac{{a}^{m}}{{a}^{m}}\) in two ways to lead us to the definition of the zero exponent. In general, for \(a\ne 0\):

We see \(\frac{{a}^{m}}{{a}^{m}}\) simplifies to \({a}^{0}\) and to 1. So \({a}^{0}=1\).

In this text, we assume any variable that we raise to the zero power is not zero.

Example

Try it.

Simplify: ⓐ \({9}^{0}\) ⓑ \({n}^{0}.\)

Solution

The definition says any non-zero number raised to the zero power is 1.

ⓐ Use the definition of the zero exponent.	\(\begin{array}{l}{9}^{0} \\ 1\end{array}\)
ⓑ Use the definition of the zero exponent.	\(\begin{array}{l}{n}^{0} \\ 1\end{array}\)

	\({(2x)}^{0}\)
Use the product to a power rule.	\({2}^{0}{x}^{0}\)
Use the zero exponent property.	\(1\cdot 1\)
Simplify.	\(1\)

Example

Try it.

Simplify: ⓐ \({(5b)}^{0}\) ⓑ \({(-4{a}^{2}b)}^{0}.\)

Solution

ⓐ	\({(5b)}^{0}\)
Use the definition of the zero exponent.	\(1\)
ⓑ	\({(-4{a}^{2}b)}^{0}\)
Use the definition of the zero exponent.	\(1\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

	\({(\frac{x}{y})}^{3}\)
This means:	\(\frac{x}{y}\cdot \frac{x}{y}\cdot \frac{x}{y}\)
Multiply the fractions.	\(\frac{x\cdot x\cdot x}{y\cdot y\cdot y}\)
Write with exponents.	\(\frac{{x}^{3}}{{y}^{3}}\)

Notice that the exponent applies to both the numerator and the denominator.

We write:	\({(\frac{x}{y})}^{3}\)
	\(\frac{{x}^{3}}{{y}^{3}}\)

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

An example with numbers may help you understand this property:

\[\begin{array}{lll}{(\frac{2}{3})}^{3} & = & \frac{{2}^{3}}{{3}^{3}} \\ \frac{2}{3}\cdot \frac{2}{3}\cdot \frac{2}{3} & = & \frac{8}{27} \\ \frac{8}{27} & = & \frac{8}{27}✓\end{array}\]

Example

Try it.

Simplify: ⓐ \({(\frac{3}{7})}^{2}\) ⓑ \({(\frac{b}{3})}^{4}\) ⓒ \({(\frac{k}{j})}^{3}.\)

Solution

ⓐ


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

ⓑ


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

ⓒ


Raise the numerator and denominator to the third power.

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.

Example

Try it.

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

Solution

	\(\frac{{({y}^{4})}^{2}}{{y}^{6}}\)
Multiply the exponents in the numerator.	\(\frac{{y}^{8}}{{y}^{6}}\)
Subtract the exponents.	\({y}^{2}\)

Example

Try it.

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

Solution

	\(\frac{{b}^{12}}{{({b}^{2})}^{6}}\)
Multiply the exponents in the numerator.	\(\frac{{b}^{12}}{{b}^{12}}\)
Subtract the exponents.	\({b}^{0}\)
Simplify.	\(1\)

Example

Try it.

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

Solution

	\({(\frac{{y}^{9}}{{y}^{4}})}^{2}\)
Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents.	\({({y}^{5})}^{2}\)
Multiply the exponents.	\({y}^{10}\)

Example

Try it.

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

Solution

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

	\({(\frac{{j}^{2}}{{k}^{3}})}^{4}\)
Raise the numerator and denominator to the fourth power using the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\).	\(\)
Use the Power Property and simplify.	\(\frac{{j}^{8}}{{k}^{12}}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

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.

Example

Try it.

Find the quotient: \(56{x}^{7}\div 8{x}^{3}.\)

Solution

	\(56{x}^{7}\div 8{x}^{3}\)
Rewrite as a fraction.	\(\frac{56{x}^{7}}{8{x}^{3}}\)
Use fraction multiplication.	\(\frac{56}{8}⋅\frac{{x}^{7}}{{x}^{3}}\)
Simplify and use the Quotient Property.	\(7{x}^{4}\)

Example

Try it.

Find the quotient: \(\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}.\)

Solution

	\(\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}\)
Use fraction multiplication.	\(\frac{45}{-5}\cdot \frac{{a}^{2}}{a}\cdot \frac{{b}^{3}}{{b}^{5}}\)
Simplify and use the Quotient Property.	\(-9\cdot a\cdot \frac{1}{{b}^{2}}\)
Multiply.	\(-\frac{9a}{{b}^{2}}\)

Example

Try it.

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.	\(\frac{{a}^{4}}{2b}\)

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.

Example

Try it.

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.

	\(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\)
Simplify and use the Quotient Property.	\(\frac{2{y}^{6}}{3{x}^{4}}\)

Example

Try it.

Find the quotient: \(\frac{(6{x}^{2}{y}^{3})(5{x}^{3}{y}^{2})}{(3{x}^{4}{y}^{5})}.\)

Solution

	\(\frac{(6{x}^{2}{y}^{3})(5{x}^{3}{y}^{2})}{(3{x}^{4}{y}^{5})}\)
Simplify the numerator.	\(\frac{30{x}^{5}{y}^{5}}{3{x}^{4}{y}^{5}}\)
Simplify.	\(10x\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Quotient Property for Exponents:
- If \(a\) is a real number, \(a\ne 0\), and \(m,n\) are whole numbers, then:
  \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\ \text{and}\ \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m\)
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:
  \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\)
- 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
  \(\begin{array}{lllll}\text{Product Property} & & {a}^{m}\cdot {a}^{n} & = & {a}^{m+n} \\ \text{Power Property} & & {({a}^{m})}^{n} & = & {a}^{m\cdot n} \\ \text{Product to a Power} & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \text{Quotient Property} & & \frac{{a}^{m}}{{a}^{n}} & = & {a}^{m-n},a\ne 0,m>n \\ & & \frac{{a}^{m}}{{a}^{n}} & = & \frac{1}{{a}^{n-m}},a\ne 0,n>m \\ \text{Zero Exponent Definition} & & {a}^{o} & = & 1,a\ne 0 \\ \text{Quotient to a Power Property} & & {(\frac{a}{b})}^{m} & = & \frac{{a}^{m}}{{b}^{m}},b\ne 0\end{array}\)

Divide Monomials

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

Try it.

ⓐ \(\frac{{x}^{18}}{{x}^{3}}\) ⓑ \(\frac{{5}^{12}}{{5}^{3}}\)

Try it.

ⓐ \(\frac{{y}^{20}}{{y}^{10}}\) ⓑ \(\frac{{7}^{16}}{{7}^{2}}\)

Solution

ⓐ \({y}^{10}\) ⓑ \({7}^{14}\)

Try it.

ⓐ \(\frac{{p}^{21}}{{p}^{7}}\) ⓑ \(\frac{{4}^{16}}{{4}^{4}}\)

Try it.

ⓐ \(\frac{{u}^{24}}{{u}^{3}}\) ⓑ \(\frac{{9}^{15}}{{9}^{5}}\)

Solution

ⓐ \({u}^{21}\) ⓑ \({9}^{10}\)

Try it.

ⓐ \(\frac{{q}^{18}}{{q}^{36}}\) ⓑ \(\frac{{10}^{2}}{{10}^{3}}\)

Try it.

ⓐ \(\frac{{t}^{10}}{{t}^{40}}\) ⓑ \(\frac{{8}^{3}}{{8}^{5}}\)

Solution

ⓐ \(\frac{1}{{t}^{30}}\) ⓑ \(\frac{1}{64}\)

Try it.

ⓐ \(\frac{b}{{b}^{9}}\) ⓑ \(\frac{4}{{4}^{6}}\)

Try it.

ⓐ \(\frac{x}{{x}^{7}}\) ⓑ \(\frac{10}{{10}^{3}}\)

Solution

ⓐ \(\frac{1}{{x}^{6}}\) ⓑ \(\frac{1}{100}\)

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Try it.

ⓐ \({20}^{0}\) ⓑ \({b}^{0}\)

Try it.

ⓐ \({13}^{0}\) ⓑ \({k}^{0}\)

Solution

ⓐ 1 ⓑ 1

Try it.

ⓐ \(\text{-}{27}^{0}\) ⓑ \(\text{-}({27}^{0})\)

Try it.

ⓐ \(\text{-}{15}^{0}\) ⓑ \(\text{-}({15}^{0})\)

Solution

ⓐ \(-1\) ⓑ \(-1\)

Try it.

ⓐ \({(25x)}^{0}\) ⓑ \(25{x}^{0}\)

Try it.

ⓐ \({(6y)}^{0}\) ⓑ \(6{y}^{0}\)

Solution

ⓐ 1 ⓑ 6

Try it.

ⓐ \({(12x)}^{0}\) ⓑ \({(-56{p}^{4}{q}^{3})}^{0}\)

Try it.

ⓐ \(7{y}^{0}\)\({(17y)}^{0}\) ⓑ \({(-93{c}^{7}{d}^{15})}^{0}\)

Solution

ⓐ 7 ⓑ 1

Try it.

ⓐ \(12{n}^{0}-18{m}^{0}\) ⓑ \({(12n)}^{0}-{(18m)}^{0}\)

Try it.

ⓐ \(15{r}^{0}-22{s}^{0}\) ⓑ \({(15r)}^{0}-{(22s)}^{0}\)

Solution

ⓐ \(-7\) ⓑ 0

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Try it.

ⓐ \({(\frac{3}{4})}^{3}\) ⓑ \({(\frac{p}{2})}^{5}\) ⓒ \({(\frac{x}{y})}^{6}\)

Try it.

ⓐ \({(\frac{2}{5})}^{2}\) ⓑ \({(\frac{x}{3})}^{4}\) ⓒ \({(\frac{a}{b})}^{5}\)

Solution

ⓐ \(\frac{4}{25}\) ⓑ \(\frac{{x}^{4}}{81}\) ⓒ \(\frac{{a}^{5}}{{b}^{5}}\)

Try it.

ⓐ \({(\frac{a}{3b})}^{4}\) ⓑ \({(\frac{5}{4m})}^{2}\)

Try it.

ⓐ \({(\frac{x}{2y})}^{3}\) ⓑ \({(\frac{10}{3q})}^{4}\)

Solution

ⓐ \(\frac{{x}^{3}}{8{y}^{3}}\) ⓑ \(\frac{10,000}{81{q}^{4}}\)

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Try it.

\(\frac{{({a}^{2})}^{3}}{{a}^{4}}\)

Try it.

\(\frac{{({p}^{3})}^{4}}{{p}^{5}}\)

Solution

\({p}^{7}\)

Try it.

\(\frac{{({y}^{3})}^{4}}{{y}^{10}}\)

Try it.

\(\frac{{({x}^{4})}^{5}}{{x}^{15}}\)

Solution

\({x}^{5}\)

Try it.

\(\frac{{u}^{6}}{{({u}^{3})}^{2}}\)

Try it.

\(\frac{{v}^{20}}{{({v}^{4})}^{5}}\)

Solution

Try it.

\(\frac{{m}^{12}}{{({m}^{8})}^{3}}\)

Try it.

\(\frac{{n}^{8}}{{({n}^{6})}^{4}}\)

Solution

\(\frac{1}{{n}^{16}}\)

Try it.

\({(\frac{{p}^{9}}{{p}^{3}})}^{5}\)

Try it.

\({(\frac{{q}^{8}}{{q}^{2}})}^{3}\)

Solution

\({q}^{18}\)

Try it.

\({(\frac{{r}^{2}}{{r}^{6}})}^{3}\)

Try it.

\({(\frac{{m}^{4}}{{m}^{7}})}^{4}\)

Solution

\(\frac{1}{{m}^{12}}\)

Try it.

\({(\frac{p}{{r}^{11}})}^{2}\)

Try it.

\({(\frac{a}{{b}^{6}})}^{3}\)

Solution

\(\frac{{a}^{3}}{{b}^{18}}\)

Try it.

\({(\frac{{w}^{5}}{{x}^{3}})}^{8}\)

Try it.

\({(\frac{{y}^{4}}{{z}^{10}})}^{5}\)

Solution

\(\frac{{y}^{20}}{{z}^{50}}\)

Try it.

\({(\frac{2{j}^{3}}{3k})}^{4}\)

Try it.

\({(\frac{3{m}^{5}}{5n})}^{3}\)

Solution

\(\frac{27{m}^{15}}{125{n}^{3}}\)

Try it.

\({(\frac{3{c}^{2}}{4{d}^{6}})}^{3}\)

Try it.

\({(\frac{5{u}^{7}}{2{v}^{3}})}^{4}\)

Solution

\(\frac{625{u}^{28}}{16{v}^{{}^{12}}}\)

Try it.

\({(\frac{{k}^{2}{k}^{8}}{{k}^{3}})}^{2}\)

Try it.

\({(\frac{{j}^{2}{j}^{5}}{{j}^{4}})}^{3}\)

Solution

\({j}^{9}\)

Try it.

\(\frac{{({t}^{2})}^{5}{({t}^{4})}^{2}}{{({t}^{3})}^{7}}\)

Try it.

\(\frac{{({q}^{3})}^{6}{({q}^{2})}^{3}}{{({q}^{4})}^{8}}\)

Solution

\(\frac{1}{{q}^{8}}\)

Try it.

\(\frac{{(-2{p}^{2})}^{4}{(3{p}^{4})}^{2}}{{(-6{p}^{3})}^{2}}\)

Try it.

\(\frac{{(-2{k}^{3})}^{2}{(6{k}^{2})}^{4}}{{(9{k}^{4})}^{2}}\)

Solution

\(64{k}^{6}\)

Try it.

\(\frac{{(-4{m}^{3})}^{2}{(5{m}^{4})}^{3}}{{(-10{m}^{6})}^{3}}\)

Try it.

\(\frac{{(-10{n}^{2})}^{3}{(4{n}^{5})}^{2}}{{(2{n}^{8})}^{2}}\)

Solution

\(-4,000\)

Divide Monomials

In the following exercises, divide the monomials.

Try it.

\(56{b}^{8}\div 7{b}^{2}\)

Try it.

\(63{v}^{10}\div 9{v}^{2}\)

Solution

\(7{v}^{8}\)

Try it.

\(-88{y}^{15}\div 8{y}^{3}\)

Try it.

\(-72{u}^{12}\div 12{u}^{4}\)

Solution

\(-6{u}^{8}\)

Try it.

\(\frac{45{a}^{6}{b}^{8}}{-15{a}^{10}{b}^{2}}\)

Try it.

\(\frac{54{x}^{9}{y}^{3}}{-18{x}^{6}{y}^{15}}\)

Solution

\(-\frac{3{x}^{3}}{{y}^{12}}\)

Try it.

\(\frac{15{r}^{4}{s}^{9}}{18{r}^{9}{s}^{2}}\)

Try it.

\(\frac{20{m}^{8}{n}^{4}}{30{m}^{5}{n}^{9}}\)

Solution

\(\frac{2{m}^{3}}{3{n}^{5}}\)

Try it.

\(\frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}}\)

Try it.

\(\frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}\)

Solution

\(\frac{-3{y}^{3}}{4{x}^{3}}\)

Try it.

\(\frac{64{q}^{11}{r}^{9}{s}^{3}}{48{q}^{6}{r}^{8}{s}^{5}}\)

Try it.

\(\frac{65{a}^{10}{b}^{8}{c}^{5}}{42{a}^{7}{b}^{6}{c}^{8}}\)

Solution

\(\frac{65{a}^{3}{b}^{2}}{42{c}^{3}}\)

Try it.

\(\frac{(10{m}^{5}{n}^{4})(5{m}^{3}{n}^{6})}{25{m}^{7}{n}^{5}}\)

Try it.

\(\frac{(-18{p}^{4}{q}^{7})(-6{p}^{3}{q}^{8})}{-36{p}^{12}{q}^{10}}\)

Solution

\(\frac{-3{q}^{5}}{{p}^{5}}\)

Try it.

\(\frac{(6{a}^{4}{b}^{3})(4a{b}^{5})}{(12{a}^{2}b)({a}^{3}b)}\)

Try it.

\(\frac{(4{u}^{2}{v}^{5})(15{u}^{3}v)}{(12{u}^{3}v)({u}^{4}v)}\)

Solution

\(\frac{5{v}^{4}}{{u}^{2}}\)

Mixed Practice

Try it.

Solution

Try it.

ⓐ \({p}^{4}\cdot {p}^{6}\) ⓑ \({({p}^{4})}^{6}\)

Try it.

ⓐ \({q}^{5}\cdot {q}^{3}\) ⓑ \({({q}^{5})}^{3}\)

Solution

ⓐ \({q}^{8}\) ⓑ \({q}^{15}\)

Try it.

ⓐ \(\frac{{y}^{3}}{y}\) ⓑ \(\frac{y}{{y}^{3}}\)

Try it.

ⓐ \(\frac{{z}^{6}}{{z}^{5}}\) ⓑ \(\frac{{z}^{5}}{{z}^{6}}\)

Solution

ⓐ \(z\) ⓑ \(\frac{1}{z}\)

Try it.

\((8{x}^{5})(9x)\div 6{x}^{3}\)

Try it.

\((4y)(12{y}^{7})\div 8{y}^{2}\)

Solution

\(6{y}^{6}\)

Try it.

\(\frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}}\)

Try it.

\(\frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}\)

Solution

\(15{c}^{6}\)

Try it.

\(\frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}}\)

Try it.

\(\frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}\)

Solution

\(3{x}^{2}\)

Try it.

\(\frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s}\)

Try it.

\(\frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}\)

Solution

\(\text{-}y{z}^{2}\)

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Simplify: \(\frac{8}{24}.\)
If you missed this problem, review .

Avslöja svaret

\(\frac{1}{3}\)
Simplify: \({(2{m}^{3})}^{5}.\)
If you missed this problem, review .

Avslöja svaret

\(32{m}^{15}\)
Simplify: \(\frac{12x}{12y}.\)
If you missed this problem, review .

Avslöja svaret

\(\frac{x}{y}\)
Simplify: ⓐ \(\frac{{x}^{9}}{{x}^{7}}\) ⓑ \(\frac{{3}^{10}}{{3}^{2}}.\)
Avslöja svaret
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
1. ⓐ
  
  Since 9 > 7, there are more factors of x in the numerator.
  Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).
  Simplify.
2. ⓑ
  
  Since 10 > 2, there are more factors of x in the numerator.
  Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).
  Simplify.
  
  Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Simplify: ⓐ \(\frac{{x}^{15}}{{x}^{10}}\) ⓑ \(\frac{{6}^{14}}{{6}^{5}}.\)

Avslöja svaret

ⓐ \({x}^{5}\) ⓑ \({6}^{9}\)
Simplify: ⓐ \(\frac{{y}^{43}}{{y}^{37}}\) ⓑ \(\frac{{10}^{15}}{{10}^{7}}.\)

Avslöja svaret

ⓐ \({y}^{6}\) ⓑ \({10}^{8}\)
Simplify: ⓐ \(\frac{{b}^{8}}{{b}^{12}}\) ⓑ \(\frac{{7}^{3}}{{7}^{5}}.\)
Avslöja svaret
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
1. ⓐ
  
  Since 12 > 8, there are more factors of b in the denominator.
  Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).
  Simplify.
2. ⓑ
  
  Since 5 > 3, there are more factors of 3 in the denominator.
  Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).
  Simplify.
  Simplify.
  
  Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
Simplify: ⓐ \(\frac{{x}^{18}}{{x}^{22}}\) ⓑ \(\frac{{12}^{15}}{{12}^{30}}.\)

Avslöja svaret

ⓐ \(\frac{1}{{x}^{4}}\) ⓑ \(\frac{1}{{12}^{15}}\)
Simplify: ⓐ \(\frac{{m}^{7}}{{m}^{15}}\) ⓑ \(\frac{{9}^{8}}{{9}^{19}}.\)

Avslöja svaret

ⓐ \(\frac{1}{{m}^{8}}\) ⓑ \(\frac{1}{{9}^{11}}\)
Simplify: ⓐ \(\frac{{a}^{5}}{{a}^{9}}\) ⓑ \(\frac{{x}^{11}}{{x}^{7}}.\)
Avslöja svaret
1. ⓐ Is the exponent of \(a\) larger in the numerator or denominator? Since 9 > 5, there are more \(a'\text{s}\) in the denominator and so we will end up with factors in the denominator.
  
  Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).
  Simplify.
2. ⓑ Notice there are more factors of \(x\) in the numerator, since 11 > 7. So we will end up with factors in the numerator.
  
  Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).
  Simplify.
Simplify: ⓐ \(\frac{{b}^{19}}{{b}^{11}}\) ⓑ \(\frac{{z}^{5}}{{z}^{11}}.\)

Avslöja svaret

ⓐ \({b}^{8}\) ⓑ \(\frac{1}{{z}^{6}}\)
Simplify: ⓐ \(\frac{{p}^{9}}{{p}^{17}}\) ⓑ \(\frac{{w}^{13}}{{w}^{9}}.\)

Avslöja svaret

ⓐ \(\frac{1}{{p}^{8}}\) ⓑ \({w}^{4}\)
Simplify: ⓐ \({9}^{0}\) ⓑ \({n}^{0}.\)

Avslöja svaret

The definition says any non-zero number raised to the zero power is 1.
ⓐ
Use the definition of the zero exponent. \(\begin{array}{l}{9}^{0} \\ 1\end{array}\)
ⓑ
Use the definition of the zero exponent. \(\begin{array}{l}{n}^{0} \\ 1\end{array}\)
Simplify: ⓐ \({15}^{0}\) ⓑ \({m}^{0}.\)

Avslöja svaret

ⓐ 1 ⓑ 1
Simplify: ⓐ \({k}^{0}\) ⓑ \({29}^{0}.\)

Avslöja svaret

ⓐ 1 ⓑ 1
Simplify: ⓐ \({(5b)}^{0}\) ⓑ \({(-4{a}^{2}b)}^{0}.\)

Avslöja svaret

ⓐ \({(5b)}^{0}\)
Use the definition of the zero exponent. \(1\)
ⓑ \({(-4{a}^{2}b)}^{0}\)
Use the definition of the zero exponent. \(1\)
Simplify: ⓐ \({(11z)}^{0}\) ⓑ \({(-11p{q}^{3})}^{0}.\)

Avslöja svaret

ⓐ \(1\) ⓑ \(1\)
Simplify: ⓐ \({(-6d)}^{0}\) ⓑ \({(-8{m}^{2}{n}^{3})}^{0}.\)

Avslöja svaret

ⓐ \(1\) ⓑ \(1\)
Simplify: ⓐ \({(\frac{3}{7})}^{2}\) ⓑ \({(\frac{b}{3})}^{4}\) ⓒ \({(\frac{k}{j})}^{3}.\)

Avslöja svaret

ⓐ

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

ⓑ

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

ⓒ

Raise the numerator and denominator to the third power.
Simplify: ⓐ \({(\frac{5}{8})}^{2}\) ⓑ \({(\frac{p}{10})}^{4}\) ⓒ \({(\frac{m}{n})}^{7}.\)

Avslöja svaret

ⓐ \(\frac{25}{64}\) ⓑ \(\frac{{p}^{4}}{10,000}\) ⓒ \(\frac{{m}^{7}}{{n}^{7}}\)
Simplify: ⓐ \({(\frac{1}{3})}^{3}\) ⓑ \({(\frac{-2}{q})}^{3}\) ⓒ \({(\frac{w}{x})}^{4}.\)

Avslöja svaret

ⓐ \(\frac{1}{27}\) ⓑ \(\frac{-8}{{q}^{3}}\) ⓒ \(\frac{{w}^{4}}{{x}^{4}}\)
Simplify: \(\frac{{({y}^{4})}^{2}}{{y}^{6}}.\)

Avslöja svaret

\(\frac{{({y}^{4})}^{2}}{{y}^{6}}\)
Multiply the exponents in the numerator. \(\frac{{y}^{8}}{{y}^{6}}\)
Subtract the exponents. \({y}^{2}\)
Simplify: \(\frac{{({m}^{5})}^{4}}{{m}^{7}}.\)

Avslöja svaret

\({m}^{13}\)
Simplify: \(\frac{{({k}^{2})}^{6}}{{k}^{7}}.\)

Avslöja svaret

\({k}^{5}\)
Simplify: \(\frac{{b}^{12}}{{({b}^{2})}^{6}}.\)

Avslöja svaret

\(\frac{{b}^{12}}{{({b}^{2})}^{6}}\)
Multiply the exponents in the numerator. \(\frac{{b}^{12}}{{b}^{12}}\)
Subtract the exponents. \({b}^{0}\)
Simplify. \(1\)
Simplify: \(\frac{{n}^{12}}{{({n}^{3})}^{4}}.\)

Avslöja svaret

1
Simplify: \(\frac{{x}^{15}}{{({x}^{3})}^{5}}.\)

Avslöja svaret

1

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

Avslöja svaret

	\({(\frac{{y}^{9}}{{y}^{4}})}^{2}\)
Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents.	\({({y}^{5})}^{2}\)
Multiply the exponents.	\({y}^{10}\)

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

Avslöja svaret

\({r}^{8}\)
Simplify: \({(\frac{{v}^{6}}{{v}^{4}})}^{3}.\)

Avslöja svaret

\({v}^{6}\)

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

Avslöja svaret

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

	\({(\frac{{j}^{2}}{{k}^{3}})}^{4}\)
Raise the numerator and denominator to the fourth power using the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\).	\(\)
Use the Power Property and simplify.	\(\frac{{j}^{8}}{{k}^{12}}\)

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

Avslöja svaret

\(\frac{{a}^{12}}{{b}^{8}}\)
Simplify: \({(\frac{{q}^{7}}{{r}^{5}})}^{3}.\)

Avslöja svaret

\(\frac{{q}^{21}}{{r}^{15}}\)

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

Avslöja svaret

	\({(\frac{2{m}^{2}}{5n})}^{4}\)
Raise the numberator and denominator to the fourth power, using the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\).	\(\frac{{(2{m}^{2})}^{4}}{{(5n)}^{4}}\)
Raise each factor to the fourth power.	\(\frac{{(2{m}^{2})}^{4}}{{(5n)}^{4}}\)
Use the Power Property and simplify.	\(\frac{16{m}^{8}}{625{n}^{4}}\)

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

Avslöja svaret

\(\frac{49{x}^{6}}{81{y}^{2}}\)
Simplify: \({(\frac{3{x}^{4}}{7y})}^{2}.\)

Avslöja svaret

\(\frac{9{x}^{8}}{49{y}^{2}}\)

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

Avslöja svaret

	\(\frac{{({x}^{3})}^{4}{({x}^{2})}^{5}}{{({x}^{6})}^{5}}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}\).	\(\frac{({x}^{12})({x}^{10})}{({x}^{30})}\)
Add the exponents in the numerator.	\(\frac{{x}^{22}}{{x}^{30}}\)
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).	\(\frac{1}{{x}^{8}}\)

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

Avslöja svaret

\(\frac{1}{{a}^{6}}\)
Simplify: \(\frac{{({p}^{3})}^{4}{({p}^{5})}^{3}}{{({p}^{7})}^{6}}.\)

Avslöja svaret

\(\frac{1}{{p}^{15}}\)

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

Avslöja svaret

	\(\frac{{(10{p}^{3})}^{2}}{{(5p)}^{3}{(2{p}^{5})}^{4}}\)
Use the Product to a Power Property, \({(ab)}^{m}={a}^{m}{b}^{m}\).	\(\frac{{(10)}^{2}{({p}^{3})}^{2}}{{(5)}^{3}{(p)}^{3}{(2)}^{4}{({p}^{5})}^{4}}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m\cdot n}\).	\(\frac{100{p}^{6}}{125{p}^{3}\cdot 16{p}^{20}}\)
Add the exponents in the denominator.	\(\frac{100{p}^{6}}{125\cdot 16{p}^{23}}\)
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).	\(\frac{100}{125\cdot 16{p}^{17}}\)
Simplify.	\(\frac{1}{20{p}^{17}}\)

Symbols used here

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Divide Monomials

Simplify expressions using the Quotient Property for Exponents
Simplify expressions with zero exponents
Simplify expressions using the quotient to a Power Property
Simplify expressions by applying several properties
Divide monomials
If we start with more factors in the numerator, we will end up with factors in the numerator.
If we start with more factors in the denominator, we will end up with factors in the denominator.
If

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Prova själv

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

Mer information Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value

Product Property	\({a}^{m}\cdot {a}^{n}={a}^{m+n}\)
Power Property	\({({a}^{m})}^{n}={a}^{m\cdot n}\)
Product to a Power	\({(ab)}^{m}={a}^{m}{b}^{m}\)

Product Property	\({a}^{m}\cdot {a}^{n}={a}^{m+n}\)
Power Property	\({({a}^{m})}^{n}={a}^{m\cdot n}\)
Product to a Power	\({(ab)}^{m}={a}^{m}{b}^{m}\)
Quotient Property	\(\begin{array}{lll}\frac{{a}^{m}}{{a}^{n}} & = & {a}^{m-n},a\ne 0,m>n \\ \frac{{a}^{m}}{{a}^{n}} & = & \frac{1}{{a}^{n-m}},a\ne 0,n>m\end{array}\)
Zero Exponent Definition	\({a}^{o}=1,a\ne 0\)
Quotient to a Power Property	\({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},b\ne 0\)

Since 9 > 7, there are more factors of x in the numerator.
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).
Simplify.

Since 10 > 2, there are more factors of x in the numerator.
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).
Simplify.

Since 12 > 8, there are more factors of b in the denominator.
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).
Simplify.

Since 5 > 3, there are more factors of 3 in the denominator.
Use the Quotient Property, \(\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}\).
Simplify.
Simplify.