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Divide Monomials
Simplify expressions using the Quotient Property of 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. 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. Try it. Simplify: ⓐ \({9}^{0}\) ⓑ \({n}^{0}.\) The definition says any non-zero number raised to the zero power is 1. Try it. Simplify: ⓐ \({(5b)}^{0}\) ⓑ \({(-4{a}^{2}b)}^{0}.\) Condensed: the full section is in OpenStax Elementary Algebra 2e.\(\begin{array}{l}\frac{8}{8}=1\end{array}\) 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. \(\begin{array}{l}{2}^{0}=1\end{array}\) Example
Solution
ⓐ
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
Solution
ⓐ \({(5b)}^{0}\) Use the definition of the zero exponent. \(1\) ⓑ \({(-4{a}^{2}b)}^{0}\) Use the definition of the zero exponent. \(1\)
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:
\(\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\ \text{and}\ \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}},n>m\)
- If \(a\) is a real number, \(a\ne 0\), and \(m,n\) are whole numbers, then:
- 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.
- If \(a\) and \(b\) are real numbers, \(b\ne 0,\) and \(m\) is a counting number, then:
- Summary of Exponent Properties
- If \(a,b\) are real numbers and \(m,n\) are whole numbers, then
\(\begin{array}{lllll}\text{Product Property} & & {a}^{m}\cdot {a}^{n} & = & {a}^{m+n} \\ \text{Power Property} & & {({a}^{m})}^{n} & = & {a}^{m\cdot n} \\ \text{Product to a Power} & & {(ab)}^{m} & = & {a}^{m}{b}^{m} \\ \text{Quotient Property} & & \frac{{a}^{m}}{{a}^{n}} & = & {a}^{m-n},a\ne 0,m>n \\ & & \frac{{a}^{m}}{{a}^{n}} & = & \frac{1}{{a}^{n-m}},a\ne 0,n>m \\ \text{Zero Exponent Definition} & & {a}^{o} & = & 1,a\ne 0 \\ \text{Quotient to a Power Property} & & {(\frac{a}{b})}^{m} & = & \frac{{a}^{m}}{{b}^{m}},b\ne 0\end{array}\)
- If \(a,b\) are real numbers and \(m,n\) are whole numbers, then
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
1
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.
ⓐ \(24{a}^{5}+2{a}^{5}\) ⓑ \(24{a}^{5}-2{a}^{5}\) ⓒ \(24{a}^{5}\cdot 2{a}^{5}\) ⓓ \(24{a}^{5}\div 2{a}^{5}\)
Try it.
ⓐ \(15{n}^{10}+3{n}^{10}\) ⓑ \(15{n}^{10}-3{n}^{10}\) ⓒ \(15{n}^{10}\cdot 3{n}^{10}\) ⓓ \(15{n}^{10}\div 3{n}^{10}\)
Solution
ⓐ \(18{n}^{10}\) ⓑ \(12{n}^{10}\) ⓒ \(45{n}^{20}\) ⓓ 5
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.
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.
\({10}^{0}\)
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
7
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
1
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
1
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
2x
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.
ಈಗ ನೀವು ಯಾವುದೇ ಕ್ಯಾಲುಕೇಂಟ್ ಇದನ್ನು ಸಚೇತಗೊಳಿಸುವುದಿಲ್ಲ, ಆದರೆ ಅದರ ತುಂಡುಗಳು ಸಹಕಾರಿಯಾಗುತ್ತವೆ. ಕೆಳಗೆ ಯಾವುದನ್ನು ಪ್ರಯತ್ನಿಸಿ ಅಥವಾ ನಿಮ್ಮದನ್ನು ಟೈಪಿಸಿ.
ರೂಢಿ (40)
Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.
-
Simplify: \(\frac{8}{24}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{1}{3}\)
-
Simplify: \({(2{m}^{3})}^{5}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(32{m}^{15}\)
-
Simplify: \(\frac{12x}{12y}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{x}{y}\)
-
Simplify: ⓐ \(\frac{{x}^{9}}{{x}^{7}}\) ⓑ \(\frac{{3}^{10}}{{3}^{2}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
- ⓐ
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.
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
- ⓐ
-
Simplify: ⓐ \(\frac{{x}^{15}}{{x}^{10}}\) ⓑ \(\frac{{6}^{14}}{{6}^{5}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \({x}^{5}\) ⓑ \({6}^{9}\)
-
Simplify: ⓐ \(\frac{{y}^{43}}{{y}^{37}}\) ⓑ \(\frac{{10}^{15}}{{10}^{7}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \({y}^{6}\) ⓑ \({10}^{8}\)
-
Simplify: ⓐ \(\frac{{b}^{8}}{{b}^{12}}\) ⓑ \(\frac{{7}^{3}}{{7}^{5}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
- ⓐ
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.
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
- ⓐ
-
Simplify: ⓐ \(\frac{{x}^{18}}{{x}^{22}}\) ⓑ \(\frac{{12}^{15}}{{12}^{30}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(\frac{1}{{x}^{4}}\) ⓑ \(\frac{1}{{12}^{15}}\)
-
Simplify: ⓐ \(\frac{{m}^{7}}{{m}^{15}}\) ⓑ \(\frac{{9}^{8}}{{9}^{19}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(\frac{1}{{m}^{8}}\) ⓑ \(\frac{1}{{9}^{11}}\)
-
Simplify: ⓐ \(\frac{{a}^{5}}{{a}^{9}}\) ⓑ \(\frac{{x}^{11}}{{x}^{7}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
- ⓐ 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. - ⓑ 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.
- ⓐ 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.
-
Simplify: ⓐ \(\frac{{b}^{19}}{{b}^{11}}\) ⓑ \(\frac{{z}^{5}}{{z}^{11}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \({b}^{8}\) ⓑ \(\frac{1}{{z}^{6}}\)
-
Simplify: ⓐ \(\frac{{p}^{9}}{{p}^{17}}\) ⓑ \(\frac{{w}^{13}}{{w}^{9}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(\frac{1}{{p}^{8}}\) ⓑ \({w}^{4}\)
-
Simplify: ⓐ \({9}^{0}\) ⓑ \({n}^{0}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
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: ⓐ \({15}^{0}\) ⓑ \({m}^{0}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ 1 ⓑ 1
-
Simplify: ⓐ \({k}^{0}\) ⓑ \({29}^{0}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ 1 ⓑ 1
-
Simplify: ⓐ \({(5b)}^{0}\) ⓑ \({(-4{a}^{2}b)}^{0}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \({(5b)}^{0}\) Use the definition of the zero exponent. \(1\) ⓑ \({(-4{a}^{2}b)}^{0}\) Use the definition of the zero exponent. \(1\) -
Simplify: ⓐ \({(11z)}^{0}\) ⓑ \({(-11p{q}^{3})}^{0}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(1\) ⓑ \(1\)
-
Simplify: ⓐ \({(-6d)}^{0}\) ⓑ \({(-8{m}^{2}{n}^{3})}^{0}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(1\) ⓑ \(1\)
-
Simplify: ⓐ \({(\frac{3}{7})}^{2}\) ⓑ \({(\frac{b}{3})}^{4}\) ⓒ \({(\frac{k}{j})}^{3}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ
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: ⓐ \({(\frac{5}{8})}^{2}\) ⓑ \({(\frac{p}{10})}^{4}\) ⓒ \({(\frac{m}{n})}^{7}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(\frac{25}{64}\) ⓑ \(\frac{{p}^{4}}{10,000}\) ⓒ \(\frac{{m}^{7}}{{n}^{7}}\)
-
Simplify: ⓐ \({(\frac{1}{3})}^{3}\) ⓑ \({(\frac{-2}{q})}^{3}\) ⓒ \({(\frac{w}{x})}^{4}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
ⓐ \(\frac{1}{27}\) ⓑ \(\frac{-8}{{q}^{3}}\) ⓒ \(\frac{{w}^{4}}{{x}^{4}}\)
-
Simplify: \(\frac{{({y}^{4})}^{2}}{{y}^{6}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{{({y}^{4})}^{2}}{{y}^{6}}\) Multiply the exponents in the numerator. \(\frac{{y}^{8}}{{y}^{6}}\) Subtract the exponents. \({y}^{2}\) -
Simplify: \(\frac{{({m}^{5})}^{4}}{{m}^{7}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\({m}^{13}\)
-
Simplify: \(\frac{{({k}^{2})}^{6}}{{k}^{7}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\({k}^{5}\)
-
Simplify: \(\frac{{b}^{12}}{{({b}^{2})}^{6}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{{b}^{12}}{{({b}^{2})}^{6}}\) Multiply the exponents in the numerator. \(\frac{{b}^{12}}{{b}^{12}}\) Subtract the exponents. \({b}^{0}\) Simplify. \(1\) -
Simplify: \(\frac{{n}^{12}}{{({n}^{3})}^{4}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
1
-
Simplify: \(\frac{{x}^{15}}{{({x}^{3})}^{5}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
1
-
Simplify: \({(\frac{{y}^{9}}{{y}^{4}})}^{2}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\({(\frac{{y}^{9}}{{y}^{4}})}^{2}\) 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: \({(\frac{{r}^{5}}{{r}^{3}})}^{4}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\({r}^{8}\)
-
Simplify: \({(\frac{{v}^{6}}{{v}^{4}})}^{3}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\({v}^{6}\)
-
Simplify: \({(\frac{{j}^{2}}{{k}^{3}})}^{4}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
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: \({(\frac{{a}^{3}}{{b}^{2}})}^{4}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{{a}^{12}}{{b}^{8}}\)
-
Simplify: \({(\frac{{q}^{7}}{{r}^{5}})}^{3}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{{q}^{21}}{{r}^{15}}\)
-
Simplify: \({(\frac{2{m}^{2}}{5n})}^{4}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\({(\frac{2{m}^{2}}{5n})}^{4}\) 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: \({(\frac{7{x}^{3}}{9y})}^{2}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{49{x}^{6}}{81{y}^{2}}\)
-
Simplify: \({(\frac{3{x}^{4}}{7y})}^{2}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{9{x}^{8}}{49{y}^{2}}\)
-
Simplify: \(\frac{{({x}^{3})}^{4}{({x}^{2})}^{5}}{{({x}^{6})}^{5}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{{({x}^{3})}^{4}{({x}^{2})}^{5}}{{({x}^{6})}^{5}}\) 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: \(\frac{{({a}^{2})}^{3}{({a}^{2})}^{4}}{{({a}^{4})}^{5}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{1}{{a}^{6}}\)
-
Simplify: \(\frac{{({p}^{3})}^{4}{({p}^{5})}^{3}}{{({p}^{7})}^{6}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{1}{{p}^{15}}\)
-
Simplify: \(\frac{{(10{p}^{3})}^{2}}{{(5p)}^{3}{(2{p}^{5})}^{4}}.\)
ಉತ್ತರವನ್ನು ತಿಳಿಸಿ
\(\frac{{(10{p}^{3})}^{2}}{{(5p)}^{3}{(2{p}^{5})}^{4}}\) 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. \(\frac{1}{20{p}^{17}}\)
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ಮೇಲೆ ಗುರುತು ಹಾಕು ಪ್ರವೇಶ (ಲಾಗಿನ್)ಇಲ್ಲಿ ಬಳಸಲಾದ ಸಂಕೇತಗಳು
ಈ ವಚನದಲ್ಲಿ ಬಳಸಲಾಗಿರುವ ಗ್ರೀಕ್ ಪದವು, ಚಿತ್ರ ಮತ್ತು ಚಿತ್ರಗಳನ್ನು ನೀವು ಸುಲಭವಾಗಿ ಅರ್ಥಮಾಡಿಕೊಳ್ಳಲು ಸಹಾಯ ಮಾಡುತ್ತದೆ.
ಹೇಗೆ: 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
ಜನರು ಕೇಳುವುದು
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.
ಈ ಪುಟದ ಭಾಗಗಳು ಹೊಂದಿಕೆಯಾಗಿ ಮಾಡಲ್ಪಡುತ್ತವೆ OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0), OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). ಇಲ್ಲಿ ಕೊರೆದು ಪುನಃ ಮರುಕಳಿಸಲಾಗಿದೆ; ದೋಷಗಳು ನಮ್ಮದಾಗಿವೆ. ಅದರ ಮೂಲವು ಲೈಸನ್ಸ್ ಆಗಿರುವುದರಿಂದ, ಈ ಪಾಠ ಉಚಿತವಾಗಿರುತ್ತದೆ, ಯಾವುದನ್ನೂ ಮಾರುವುದಿಲ್ಲ ಮತ್ತು ಅದೇ ಪರವಾನಗಿಗಳ ಕೆಳಗೆ ಹಂಚಲಾಗುತ್ತದೆ.
ಇನ್ನಷ್ಟು Algebra
Linear equationsQuadratic equationsSystems of equationsInequalitiesFactoringExpandingSimplifying expressionsFunctions and graphsExponential and logarithmic equationsPolynomial equationsAbsolute value