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Divide Monomials

Simplify expressions using the Quotient Property of Exponents

Simplify Expressions Using the Quotient Property of Exponents

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

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

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

\(\begin{array}{llllllllll}\text{Consider} & & & \frac{{x}^{5}}{{x}^{2}} & & & \text{and} & & & \frac{{x}^{2}}{{x}^{3}} \\ \text{What do they mean?} & & & \frac{x⋅x⋅x⋅x⋅x}{x⋅x} & & & & & & \frac{x⋅x}{x⋅x⋅x} \\ \text{Use the Equivalent Fractions Property.} & & & \frac{x⋅x⋅x⋅x⋅x}{x⋅x⋅1} & & & & & & \frac{x⋅x⋅1}{x⋅x⋅x} \\ \text{Simplify.} & & & {x}^{3} & & & & & & \frac{1}{x}\end{array}\)

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

When the larger exponent was in the numerator, we were left with factors in the numerator and \(1\) in the denominator, which we simplified.
When the larger exponent was in the denominator, we were left with factors in the denominator, and \(1\) in the numerator, which could not be simplified.

We write:

\[\begin{array}{lllll}\frac{{x}^{5}}{{x}^{2}} & \ & & & \frac{{x}^{2}}{{x}^{3}} \\ {x}^{5-2} & \ & & & \frac{1}{{x}^{3-2}} \\ {x}^{3} & \ & & & \frac{1}{x}\end{array}\]

A couple of examples with numbers may help to verify this property.

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

Condensed — the full section is in OpenStax Prealgebra 2e.

Simplify Expressions with Zero Exponents

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 earlier work with fractions, we 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 of Exponents shows us how to simplify \(\frac{{a}^{m}}{{a}^{n}}\) when \(m>n\) and when \(n

Now we will simplify \(\frac{{a}^{m}}{{a}^{m}}\) in two ways to lead us to the definition of the zero exponent.

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

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

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

Example

Try it.

Simplify:

ⓐ \(\ {12}^{0}\)
ⓑ \(\ {y}^{0}\)

Solution

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

ⓐ
	\({12}^{0}\)
Use the definition of the zero exponent.	1

ⓑ
	\({y}^{0}\)
Use the definition of the zero exponent.	1

Example

Try it.

Simplify: \({(7z)}^{0}.\)

Solution

	\({(7z)}^{0}\)
Use the definition of the zero exponent.	1

Example

Try it.

Simplify:

ⓐ \(\ {(-3{x}^{2}y)}^{0}\)
ⓑ \(\ -3{x}^{2}{y}^{0}\)

Solution

ⓐ
The product is raised to the zero power.	\({(-3{x}^{2}y)}^{0}\)
Use the definition of the zero exponent.	\(1\)

ⓑ
Notice that only the variable \(y\) is being raised to the zero power.	\({-3{x}^{2}y}^{0}\)
Use the definition of the zero exponent.	\(-3{x}^{2}⋅1\)
Simplify.	\(-3{x}^{2}\)

Condensed — the full section is in OpenStax Prealgebra 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	\(\frac{x}{y}⋅\frac{x}{y}⋅\frac{x}{y}\)
Multiply the fractions.	\(\frac{x⋅x⋅x}{y⋅y⋅y}\)
Write with exponents.	\(\frac{{x}^{3}}{{y}^{3}}\)

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

We see that \({(\frac{x}{y})}^{3}\) is \(\frac{{x}^{3}}{{y}^{3}}.\)

\(\begin{array}{lllll}\text{We write:} & & & & {(\frac{x}{y})}^{3} \\ & & & & \frac{{x}^{3}}{{y}^{3}}\end{array}\)

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

An example with numbers may help you understand this property:

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

Example

Try it.

Simplify:

ⓐ \(\ {(\frac{5}{8})}^{2}\)
ⓑ \(\ {(\frac{x}{3})}^{4}\)
ⓒ \(\ {(\frac{y}{m})}^{3}\)

Solution

ⓐ

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

ⓑ

Use the Quotient to a Power 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: \(\frac{{({x}^{2})}^{3}}{{x}^{5}}.\)

Solution

	\(\frac{{({x}^{2})}^{3}}{{x}^{5}}\)
Multiply the exponents in the numerator, using the Power Property.	\(\frac{{x}^{6}}{{x}^{5}}\)
Subtract the exponents.	\(x\)

Example

Try it.

Simplify: \(\frac{{m}^{8}}{{({m}^{2})}^{4}}.\)

Solution

	\(\frac{{m}^{8}}{{({m}^{2})}^{4}}\)
Multiply the exponents in the numerator, using the Power Property.	\(\frac{{m}^{8}}{{m}^{8}}\)
Subtract the exponents.	\({m}^{0}\)
Zero power property	\(1\)

Example

Try it.

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

Solution

	\({(\frac{{x}^{7}}{{x}^{3}})}^{2}\)
Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents.	\({({x}^{7-3})}^{2}\)
Simplify.	\({({x}^{4})}^{2}\)
Multiply the exponents.	\({x}^{8}\)

Example

Try it.

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

Solution

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

	\({(\frac{{p}^{2}}{{q}^{5}})}^{3}\)
Raise the numerator and denominator to the third power using the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\)	\(\frac{{({p}^{2})}^{3}}{{({q}^{5})}^{3}}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m⋅n}.\)	\(\frac{{p}^{6}}{{q}^{15}}\)

Condensed — the full section is in OpenStax Prealgebra 2e.

Divide Monomials

We have now seen all the properties of exponents. We'll use them to divide monomials. Later, you'll use them to divide polynomials.

Example

Try it.

Find the quotient: \(56{x}^{5}\div 7{x}^{2}.\)

Solution

	\(56{x}^{5}\div 7{x}^{2}\)
Rewrite as a fraction.	\(\frac{56{x}^{5}}{7{x}^{2}}\)
Use fraction multiplication to separate the number part from the variable part.	\(\frac{56}{7}⋅\frac{{x}^{5}}{{x}^{2}}\)
Use the Quotient Property.	\(8{x}^{3}\)

When we divide monomials with more than one variable, we write one fraction for each variable.

Example

Try it.

Find the quotient: \(\frac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}.\)

Solution

	\(\frac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}\)
Use fraction multiplication.	\(\frac{42}{-7}⋅\frac{{x}^{2}}{x}⋅\frac{{y}^{3}}{{y}^{5}}\)
Simplify and use the Quotient Property.	\(-6⋅x⋅\frac{1}{{y}^{2}}\)
Multiply.	\(-\frac{6x}{{y}^{2}}\)

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.	\(\frac{24}{48}⋅\frac{{a}^{5}}{a}⋅\frac{{b}^{3}}{{b}^{4}}\)
Simplify and use the Quotient Property.	\(\frac{1}{2}⋅{a}^{4}⋅\frac{1}{b}\)
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

	\(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\)
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.

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.

Condensed — the full section is in OpenStax Prealgebra 2e.

Key Concepts

Equivalent Fractions Property
- If \(a,\ b,\ c\) are whole numbers where \(b\ne 0,\ c\ne 0,\) then \[\frac{a}{b}=\frac{a\ \cdot \ c}{b\ \cdot \ c}\ \text{and}\ \frac{a\ \cdot \ c}{b\ \cdot \ c}=\frac{a}{b}\]
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 \(m,\ n\) are whole numbers, then \[\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\ m>n\ \text{and}\ \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\ n>m\]
Quotient to a Power Property for Exponents
- If \(a\) and \(b\) are real numbers, \(b\ne 0,\) and \(m\) is a counting number, then \[{(\frac{a}{b})}^{m}=\ \frac{{a}^{m}}{{b}^{m}}\]
- To raise a fraction to a power, raise the numerator and denominator to that power.

Divide Monomials

Simplify Expressions Using the Quotient Property of Exponents

In the following exercises, simplify.

Try it.

\(\frac{{4}^{8}}{{4}^{2}}\)

Solution

4⁶

Try it.

\(\frac{{3}^{12}}{{3}^{4}}\)

Try it.

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

Solution

x⁹

Try it.

\(\frac{{u}^{9}}{{u}^{3}}\)

Try it.

\(\frac{{r}^{5}}{r}\)

Solution

r⁴

Try it.

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

Try it.

\(\frac{{y}^{4}}{{y}^{20}}\)

Solution

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

Try it.

\(\frac{{x}^{10}}{{x}^{30}}\)

Try it.

\(\frac{{10}^{3}}{{10}^{15}}\)

Solution

\(\frac{1}{{10}^{12}}\)

Try it.

\(\frac{{r}^{2}}{{r}^{8}}\)

Try it.

\(\frac{a}{{a}^{9}}\)

Solution

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

Try it.

\(\frac{2}{{2}^{5}}\)

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

Try it.

\({5}^{0}\)

Solution

Try it.

\({10}^{0}\)

Try it.

\({a}^{0}\)

Solution

Try it.

\({x}^{0}\)

Try it.

\(-{7}^{0}\)

Solution

−1

Try it.

\(-{4}^{0}\)

Try it.

ⓐ \(\ {(10p)}^{0}\)
ⓑ \(\ 10{p}^{0}\)

Solution

ⓐ 1
ⓑ 10

Try it.

ⓐ \(\ {(3a)}^{0}\)
ⓑ \(\ 3{a}^{0}\)

Try it.

ⓐ \(\ {(-27{x}^{5}y)}^{0}\)
ⓑ \(\ -27{x}^{5}{y}^{0}\)

Solution

ⓐ 1
ⓑ −27x⁵

Try it.

ⓐ \(\ {(-92{y}^{8}z)}^{0}\)
ⓑ \(\ -92{y}^{8}{z}^{0}\)

Try it.

ⓐ \(\ {15}^{0}\)
ⓑ \(\ {15}^{1}\)

Solution

ⓐ 1
ⓑ 15

Try it.

ⓐ \(\ -{6}^{0}\)
ⓑ \(\ -{6}^{1}\)

Try it.

\(2\ \cdot \ {x}^{0}+5\ \cdot \ {y}^{0}\)

Solution

Try it.

\(8\ \cdot \ {m}^{0}-4\ \cdot \ {n}^{0}\)

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

Try it.

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

Solution

\(\frac{243}{32}\)

Try it.

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

Try it.

\({(\frac{m}{6})}^{3}\)

Solution

\(\frac{{m}^{3}}{216}\)

Try it.

\({(\frac{p}{2})}^{5}\)

Try it.

\({(\frac{x}{y})}^{10}\)

Solution

\(\frac{{x}^{10}}{{y}^{10}}\)

Try it.

\({(\frac{a}{b})}^{8}\)

Try it.

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

Solution

\(\frac{{a}^{2}}{9{b}^{2}}\)

Try it.

\({(\frac{2x}{y})}^{4}\)

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

Try it.

\(\frac{{({x}^{2})}^{4}}{{x}^{5}}\)

Solution

x³

Try it.

\(\frac{{({y}^{4})}^{3}}{{y}^{7}}\)

Try it.

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

Solution

u²

Try it.

\(\frac{{({y}^{2})}^{5}}{{y}^{6}}\)

Try it.

\(\frac{{y}^{8}}{{({y}^{5})}^{2}}\)

Solution

\(\frac{1}{{y}^{2}}\)

Try it.

\(\frac{{p}^{11}}{{({p}^{5})}^{3}}\)

Try it.

\(\frac{{r}^{5}}{{r}^{4}\ \cdot \ r}\)

Solution

Try it.

\(\frac{{a}^{3}\ \cdot \ {a}^{4}}{{a}^{7}}\)

Try it.

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

Solution

\(\frac{1}{{x}^{18}}\)

Try it.

\({(\frac{u}{{u}^{10}})}^{2}\)

Try it.

\({(\frac{{a}^{4}\ \cdot \ {a}^{6}}{{a}^{3}})}^{2}\)

Solution

a¹⁴

Try it.

\({(\frac{{x}^{3}\ \cdot \ {x}^{8}}{{x}^{4}})}^{3}\)

Try it.

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

Solution

y³

Try it.

\(\frac{{({z}^{6})}^{2}}{{({z}^{2})}^{4}}\)

Try it.

\(\frac{{({x}^{3})}^{6}}{{({x}^{4})}^{7}}\)

Solution

\(\frac{1}{{x}^{10}}\)

Try it.

\(\frac{{({x}^{4})}^{8}}{{({x}^{5})}^{7}}\)

Try it.

\({(\frac{2{r}^{3}}{5s})}^{4}\)

Solution

\(\frac{16{r}^{12}}{625{s}^{4}}\)

Try it.

\({(\frac{3{m}^{2}}{4n})}^{3}\)

Try it.

\({(\frac{3{y}^{2}\ \cdot \ {y}^{5}}{{y}^{15}\ \cdot \ {y}^{8}})}^{0}\)

Solution

Try it.

\({(\frac{15{z}^{4}\ \cdot \ {z}^{9}}{0.3{z}^{2}})}^{0}\)

Try it.

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

Solution

\(\frac{1}{{r}^{3}}\)

Try it.

\(\frac{{({p}^{4})}^{2}\ {({p}^{3})}^{5}}{{({p}^{2})}^{9}}\)

Try it.

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

Solution

3x⁸

Try it.

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

Divide Monomials

In the following exercises, divide the monomials.

Try it.

\(48{b}^{8}\div 6{b}^{2}\)

Solution

8b⁶

Try it.

\(42{a}^{14}\div 6{a}^{2}\)

Try it.

\(36{x}^{3}\div (-2{x}^{9})\)

Solution

\(\frac{-18}{{x}^{6}}\)

Try it.

\(20{u}^{8}\div (-4{u}^{6})\)

Try it.

\(\frac{18{x}^{3}}{9{x}^{2}}\)

Solution

Try it.

\(\frac{36{y}^{9}}{4{y}^{7}}\)

Try it.

\(\frac{-35{x}^{7}}{-42{x}^{13}}\)

Solution

\(\frac{5}{6{x}^{6}}\)

Try it.

\(\frac{18{x}^{5}}{-27{x}^{9}}\)

Try it.

\(\frac{18{r}^{5}s}{3{r}^{3}{s}^{9}}\)

Solution

\(\frac{6{r}^{2}}{{s}^{8}}\)

Try it.

\(\frac{24{p}^{7}q}{6{p}^{2}{q}^{5}}\)

Try it.

\(\frac{8m{n}^{10}}{64m{n}^{4}}\)

Solution

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

Try it.

\(\frac{10{a}^{4}b}{50{a}^{2}{b}^{6}}\)

Try it.

\(\frac{-12{x}^{4}{y}^{9}}{15{x}^{6}{y}^{3}}\)

Solution

\(-\frac{4{y}^{6}}{5{x}^{2}}\)

Try it.

\(\frac{48{x}^{11}{y}^{9}{z}^{3}}{36{x}^{6}{y}^{8}{z}^{5}}\)

Try it.

\(\frac{64{x}^{5}{y}^{9}{z}^{7}}{48{x}^{7}{y}^{12}{z}^{6}}\)

Solution

\(\frac{4z}{3{x}^{2}{y}^{3}}\)

Try it.

\(\frac{(10{u}^{2}v)(4{u}^{3}{v}^{6})}{5{u}^{9}{v}^{2}}\)

Try it.

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

Solution

\(\frac{10{n}^{2}}{{m}^{4}}\)

Try it.

\(\frac{(6{a}^{4}{b}^{3})(4a{b}^{5})}{(12{a}^{8}b)({a}^{3}b)}\)

Try it.

\(\frac{(4{u}^{5}{v}^{4})(15{u}^{8}v)}{(12{u}^{3}v)({u}^{6}v)}\)

Solution

5u⁴v³

Condensed — the full section is in OpenStax Prealgebra 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 the problem, review .

Cavabı göstər

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

Cavabı göstər

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

Cavabı göstər

\(\frac{x}{y}\)
Simplify:
1. ⓐ \(\ \frac{{x}^{10}}{{x}^{8}}\)
2. ⓑ \(\ \frac{{2}^{9}}{{2}^{2}}\)
Cavabı göstər

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
ⓐ
Since 10 > 8, there are more factors of \(x\) in the numerator. \(\frac{{x}^{10}}{{x}^{8}}\)
Use the quotient property with \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).
Simplify. \({x}^{2}\)
ⓑ
Since 9 > 2, there are more factors of 2 in the numerator. \(\frac{{2}^{9}}{{2}^{2}}\)
Use the quotient property with \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)
Simplify. \({2}^{7}\)
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Simplify:
1. ⓐ \(\ \frac{{x}^{12}}{{x}^{9}}\)
2. ⓑ \(\ \frac{{7}^{14}}{{7}^{5}}\)
Cavabı göstər
1. ⓐ x³
2. ⓑ 7⁹
Simplify:
1. ⓐ \(\ \frac{{y}^{23}}{{y}^{17}}\)
2. ⓑ \(\ \frac{{8}^{15}}{{8}^{7}}\)
Cavabı göstər
1. ⓐ y⁶
2. ⓑ 8⁸

Simplify:

ⓐ \(\ \frac{{b}^{10}}{{b}^{15}}\)
ⓑ \(\ \frac{{3}^{3}}{{3}^{5}}\)

Cavabı göstər

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

ⓐ
Since 15 > 10, there are more factors of \(b\) in the denominator.	\(\frac{{b}^{10}}{{b}^{15}}\)
Use the quotient property with \(n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
Simplify.	\(\frac{1}{{b}^{5}}\)

ⓑ
Since 5 > 3, there are more factors of 3 in the denominator.	\(\frac{{3}^{3}}{{3}^{5}}\)
Use the quotient property with \(n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
Simplify.	\(\frac{1}{{3}^{2}}\)
Apply the exponent.	\(\frac{1}{9}\)

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and \(1\) in the numerator.

Simplify:
1. ⓐ \(\ \frac{{x}^{8}}{{x}^{15}}\)
2. ⓑ \(\ \frac{{12}^{11}}{{12}^{21}}\)
Cavabı göstər
1. ⓐ \(\ \frac{1}{{x}^{7}}\)
2. ⓑ \(\ \frac{1}{{12}^{10}}\)
Simplify:
1. ⓐ \(\ \frac{{m}^{17}}{{m}^{26}}\)
2. ⓑ \(\ \frac{{7}^{8}}{{7}^{14}}\)
Cavabı göstər
1. ⓐ \(\ \frac{1}{{m}^{9}}\)
2. ⓑ \(\ \frac{1}{{7}^{6}}\)

Simplify:

ⓐ \(\ \frac{{a}^{5}}{{a}^{9}}\)
ⓑ \(\ \frac{{x}^{11}}{{x}^{7}}\)

Cavabı göstər

ⓐ
Since 9 > 5, there are more \(a\)'s in the denominator and so we will end up with factors in the denominator.	\(\frac{{a}^{5}}{{a}^{9}}\)
Use the Quotient Property for \(n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.\)
Simplify.	\(\frac{1}{{a}^{4}}\)

ⓑ
Notice there are more factors of \(x\) in the numerator, since 11 > 7. So we will end up with factors in the numerator.	\(\frac{{x}^{11}}{{x}^{7}}\)
Use the Quotient Property for \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{n-m}.\)
Simplify.	\({x}^{4}\)

Simplify:
1. ⓐ \(\ \frac{{b}^{19}}{{b}^{11}}\)
2. ⓑ \(\ \frac{{z}^{5}}{{z}^{11}}\)
Cavabı göstər
1. ⓐ \(\ {b}^{8}\)
2. ⓑ \(\ \frac{1}{{z}^{6}}\)
Simplify:
1. ⓐ \(\ \frac{{p}^{9}}{{p}^{17}}\)
2. ⓑ \(\ \frac{{w}^{13}}{{w}^{9}}\)
Cavabı göstər
1. ⓐ \(\ \frac{1}{{p}^{8}}\)
2. ⓑ \(\ {w}^{4}\)
Simplify:
1. ⓐ \(\ {12}^{0}\)
2. ⓑ \(\ {y}^{0}\)
Cavabı göstər

The definition says any non-zero number raised to the zero power is \(1.\)
ⓐ
\({12}^{0}\)
Use the definition of the zero exponent. 1
ⓑ
\({y}^{0}\)
Use the definition of the zero exponent. 1
Simplify:
1. ⓐ \(\ {17}^{0}\)
2. ⓑ \(\ {m}^{0}\)
Cavabı göstər
1. ⓐ 1
2. ⓑ 1
Simplify:
1. ⓐ \(\ {k}^{0}\)
2. ⓑ \(\ {29}^{0}\)
Cavabı göstər
1. ⓐ 1
2. ⓑ 1
Simplify: \({(7z)}^{0}.\)

Cavabı göstər

\({(7z)}^{0}\)
Use the definition of the zero exponent. 1
Simplify: \({(-4y)}^{0}.\)

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1
Simplify: \({(\frac{2}{3}\ x)}^{0}.\)

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1
Simplify:
1. ⓐ \(\ {(-3{x}^{2}y)}^{0}\)
2. ⓑ \(\ -3{x}^{2}{y}^{0}\)
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ⓐ
The product is raised to the zero power. \({(-3{x}^{2}y)}^{0}\)
Use the definition of the zero exponent. \(1\)
ⓑ
Notice that only the variable \(y\) is being raised to the zero power. \({-3{x}^{2}y}^{0}\)
Use the definition of the zero exponent. \(-3{x}^{2}⋅1\)
Simplify. \(-3{x}^{2}\)
Simplify:
1. ⓐ \(\ {(7{x}^{2}y)}^{0}\)
2. ⓑ \(\ 7{x}^{2}{y}^{0}\)
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1. ⓐ 1
2. ⓑ 7x²
Simplify:
1. ⓐ \(\ -23{x}^{2}{y}^{0}\)
2. ⓑ \(\ {(-23{x}^{2}y)}^{0}\)
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1. ⓐ −23x²
2. ⓑ 1
Simplify:
1. ⓐ \(\ {(\frac{5}{8})}^{2}\)
2. ⓑ \(\ {(\frac{x}{3})}^{4}\)
3. ⓒ \(\ {(\frac{y}{m})}^{3}\)
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ⓐ
Use the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\ \frac{{a}^{m}}{{b}^{m}}\).
Simplify.
ⓑ
Use the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\ \frac{{a}^{m}}{{b}^{m}}\).
Simplify.
ⓒ
Raise the numerator and denominator to the third power.
Simplify:
1. ⓐ \(\ {(\frac{7}{9})}^{2}\)
2. ⓑ \(\ {(\frac{y}{8})}^{3}\)
3. ⓒ \(\ {(\frac{p}{q})}^{6}\)
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1. ⓐ \(\ \frac{49}{81}\)
2. ⓑ \(\ \frac{{y}^{3}}{512}\)
3. ⓒ \(\ \frac{{p}^{6}}{{q}^{6}}\)
Simplify:
1. ⓐ \(\ {(\frac{1}{8})}^{2}\)
2. ⓑ \(\ {(\frac{-5}{m})}^{3}\)
3. ⓒ \(\ {(\frac{r}{s})}^{4}\)
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1. ⓐ \(\ \frac{1}{64}\)
2. ⓑ \(\ -\frac{125}{{m}^{3}}\)
3. ⓒ \(\ \frac{{r}^{4}}{{s}^{4}}\)
Simplify: \(\frac{{({x}^{2})}^{3}}{{x}^{5}}.\)

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\(\frac{{({x}^{2})}^{3}}{{x}^{5}}\)
Multiply the exponents in the numerator, using the
Power Property. \(\frac{{x}^{6}}{{x}^{5}}\)
Subtract the exponents. \(x\)
Simplify: \(\frac{{({a}^{4})}^{5}}{{a}^{9}}.\)

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a¹¹
Simplify: \(\frac{{({b}^{5})}^{6}}{{b}^{11}}.\)

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b¹⁹
Simplify: \(\frac{{m}^{8}}{{({m}^{2})}^{4}}.\)

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\(\frac{{m}^{8}}{{({m}^{2})}^{4}}\)
Multiply the exponents in the numerator, using the
Power Property. \(\frac{{m}^{8}}{{m}^{8}}\)
Subtract the exponents. \({m}^{0}\)
Zero power property \(1\)
Simplify: \(\frac{{k}^{11}}{{({k}^{3})}^{3}}.\)

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k²
Simplify: \(\frac{{d}^{23}}{{({d}^{4})}^{6}}.\)

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\(\frac{1}{d}\)

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

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	\({(\frac{{x}^{7}}{{x}^{3}})}^{2}\)
Remember parentheses come before exponents, and the bases are the same so we can simplify inside the parentheses. Subtract the exponents.	\({({x}^{7-3})}^{2}\)
Simplify.	\({({x}^{4})}^{2}\)
Multiply the exponents.	\({x}^{8}\)

Simplify: \({(\frac{{f}^{14}}{{f}^{8}})}^{2}.\)

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f¹²
Simplify: \({(\frac{{b}^{6}}{{b}^{11}})}^{2}.\)

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\(\frac{1}{{b}^{10}}\)

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

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

	\({(\frac{{p}^{2}}{{q}^{5}})}^{3}\)
Raise the numerator and denominator to the third power using the Quotient to a Power Property, \({(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}}\)	\(\frac{{({p}^{2})}^{3}}{{({q}^{5})}^{3}}\)
Use the Power Property, \({({a}^{m})}^{n}={a}^{m⋅n}.\)	\(\frac{{p}^{6}}{{q}^{15}}\)

Simplify: \({(\frac{{m}^{3}}{{n}^{8}})}^{5}.\)

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\(\frac{{m}^{15}}{{n}^{40}}\)
Simplify: \({(\frac{{t}^{10}}{{u}^{7}})}^{2}.\)

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\(\frac{{t}^{20}}{{u}^{14}}\)

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

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	\({(\frac{2{x}^{3}}{3y})}^{4}\)
Raise the numerator and denominator to the fourth power using the Quotient to a Power Property.	\(\frac{{(2{x}^{3})}^{4}}{{(3y)}^{4}}\)
Raise each factor to the fourth power, using the Power to a Power Property.	\(\frac{{2}^{4}{({x}^{3})}^{4}}{{3}^{4}{y}^{4}}\)
Use the Power Property and simplify.	\(\frac{16{x}^{12}}{81{y}^{4}}\)

Simplify: \({(\frac{5b}{9{c}^{3}})}^{2}.\)

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\(\frac{25{b}^{2}}{81{c}^{6}}\)
Simplify: \({(\frac{4{p}^{4}}{7{q}^{5}})}^{3}.\)

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\(\frac{64{p}^{12}}{343{q}^{15}}\)

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

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	\(\frac{{({y}^{2})}^{3}{({y}^{2})}^{4}}{{({y}^{5})}^{4}}\)
Use the Power Property.	\(\frac{({y}^{6})({y}^{8})}{{y}^{20}}\)
Add the exponents in the numerator, using the Product Property.	\(\frac{{y}^{14}}{{y}^{20}}\)
Use the Quotient Property.	\(\frac{1}{{y}^{6}}\)

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.

\approx

approximately equal
Equal to the precision shown, not exactly.

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

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

a \bmod n

remainder
What is left after dividing a by n.

per cent
Per hundred: 15% = 15/100.

a : b,\ \frac{a}{b}

ratio, fraction
a for every b; a divided by b.

How to: Divide Monomials

Simplify expressions using the Quotient Property of Exponents
Simplify expressions with zero exponents
Simplify expressions using the Quotient to a Power Property
Simplify expressions by applying several properties
Divide monomials
When the larger exponent was in the numerator, we were left with factors in the numerator and
When the larger exponent was in the denominator, we were left with factors in the denominator, and
If

Questions people ask

Why does the order of operations matter?

Because 2 + 3 × 4 would otherwise be two different numbers. The convention (brackets, exponents, multiplication and division, addition and subtraction) exists so every reader gets the same value from the same expression.

How do I check an arithmetic answer?

Estimate first (round every number and compute roughly), then compare. If the estimate and the exact answer disagree by more than a little, one of them is wrong. The solver shows every operation, so you can find which line went astray.

Why are fractions harder than decimals?

They are not harder, they are more exact: 1/3 is a precise number, 0.333 is an approximation. Fractions need a common denominator to add, which is the one extra step people trip on.

Özün sına

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

Daha çox Arithmetic

Order of operations Fractions Percentages Powers and roots

ⓐ
Since 10 > 8, there are more factors of \(x\) in the numerator.	\(\frac{{x}^{10}}{{x}^{8}}\)
Use the quotient property with \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}\).
Simplify.	\({x}^{2}\)

ⓑ
Since 9 > 2, there are more factors of 2 in the numerator.	\(\frac{{2}^{9}}{{2}^{2}}\)
Use the quotient property with \(m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}.\)
Simplify.	\({2}^{7}\)