5.5 Divide Monomials
Learning Objectives
By the end of this section, you will be able to:
- 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
- Use the definition of a negative exponent
- Simplify expressions with integer exponents
- Simplify expressions by applying several properties with negative exponents
- Divide monomials
Try It
Before you get started, take this readiness quiz:
1) Simplify: [latex]\frac{8}{24}[/latex]
2) Simplify: [latex](2{m}^{3})^{5}[/latex]
3) Simplify: [latex]\frac{12x}{12y}[/latex]
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.
Summary of Exponent Properties for Multiplication
If [latex]a[/latex] and [latex]b[/latex] are real numbers, and [latex]m[/latex] and [latex]n[/latex] are whole numbers, then
| Product Property | [latex]{a}^{m}\times{{a}^{n}}={a}^{m+n}[/latex] |
|---|---|
| Power Property | [latex]({a}^{m})^{n}={a}^{m\times{n}}[/latex] |
| Product to a Power | [latex]{(ab)}^{m}={a}^{m}{b}^{m}[/latex] |
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.
Equivalent Fractions Property
If [latex]a[/latex], [latex]b[/latex], and [latex]c[/latex] are whole numbers where [latex]b\neq{0}, c\neq{0}[/latex] then
[latex]\displaystyle\frac{a}{b}=\frac{a\times{c}}{b\times{c}}[/latex] and [latex]\displaystyle\frac{a\times{c}}{b\times{c}}=\frac{a}{b}[/latex]
As before, we’ll try to discover a property by looking at some examples.
| Example 1 | Example 2 | |
| Consider | [latex]\displaystyle\frac{x^5}{x^2}[/latex] | [latex]\displaystyle\frac{x^2}{x^3}[/latex] |
| What do they mean? | [latex]\displaystyle\frac{{x}\cdot {x}\cdot {x}\cdot {x}\cdot {x}}{{x}\cdot {x}}[/latex] | [latex]\displaystyle\frac{{x}\cdot {x}}{{x}\cdot{x}\cdot{x}}[/latex] |
| Use the Equivalent Fractions Property. | [latex]\displaystyle\frac{\cancel x\cdot\cancel x\cdot x\cdot x\cdot x}{\cancel x\cdot\cancel x}[/latex] | [latex]\displaystyle\frac{\cancel x\cdot\cancel x\cdot1}{\cancel x\cdot\cancel x\cdot x}[/latex] |
| Simplify | [latex]x^3[/latex] | [latex]\displaystyle\frac{1}{x}[/latex] |
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:
| [latex]\displaystyle\frac{x^5}{x^2}[/latex] | [latex]\displaystyle\frac{x^2}{x^3}[/latex] |
| [latex]\displaystyle x^{5-2}[/latex] | [latex]\displaystyle\frac{1}{x^{3-2}}[/latex] |
| [latex]\displaystyle x^3[/latex] | [latex]\displaystyle\frac{1}{x}[/latex] |
This leads to the Quotient Property for Exponents.
Quotient Property for Exponents
If [latex]a[/latex] is a real number, [latex]a\neq{0}[/latex], and [latex]n[/latex] are whole numbers, then
A couple of examples with numbers may help to verify this property.
| [latex]\begin{align*}\frac{3^4}{3^2}&=3^{4-2}\\[2ex]\frac{81}{9}&=3^{2}\\[2ex]9&=9\checkmark\end{align*}[/latex] | [latex]\begin{align*}\frac{5^2}{5^3}&=\frac1{5^{3-2}}\\[2ex]\frac{25}{125}&=\frac{1}{5^{1}}\\[2ex]\frac{1}{5}&=\frac{1}{5}\checkmark\end{align*}[/latex] |
Example 5.5.1
Simplify:
a. [latex]\frac{{x}^{9}}{{x}^{7}}[/latex]
b. [latex]\frac{{3}^{10}}{{3}^{2}}[/latex]
Solution
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
a.
Step 1: Since [latex]9 > 7[/latex], there are more factors of [latex]x[/latex] in the numerator.
[latex]\frac{x^9}{x^7}[/latex]
Step 2: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]
[latex]\begin{eqnarray*}&=&x^{{\color{red}{9-7}}}\\\text{Simplify.}\;\;&=&x^2\end{eqnarray*}[/latex]
b.
Step 1: Since [latex]10 > 2[/latex], there are more factors of [latex]x[/latex] in the numerator.
[latex]\frac{3^{10}}{3^2}[/latex]
Step 2: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex]
[latex]\begin{eqnarray*}&=&3^{\color{red}{{10\;-\;2}}}\\\text{Simplify.}\;\;&=&3^8 \end{eqnarray*}[/latex]
Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.
Try It
4) Simplify:
a. [latex]\frac{{x}^{15}}{{x}^{10}}[/latex]
b. [latex]\frac{{6}^{14}}{{6}^{5}}[/latex]
Solution
a. [latex]{x}^{5}[/latex]
b. [latex]{6}^{9}[/latex]
5) Simplify:
a. [latex]\frac{{y}^{43}}{{y}^{37}}[/latex]
b. [latex]\frac{{10}^{15}}{{10}^{7}}[/latex]
Solution
a. [latex]{y}^{6}[/latex]
b. [latex]{10}^{8}[/latex]
Example 5.5.2
Simplify:
a. [latex]\frac{{b}^{8}}{{b}^{12}}[/latex]
b. [latex]\frac{{7}^{3}}{{7}^{5}}[/latex]
Solution
To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
a.
Step 1: Since [latex]12 > 8[/latex], there are more factors of [latex]b[/latex] in the denominator.
[latex]\frac{b^8}{b^{12}}[/latex]
Step 2: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}[/latex]
[latex]\begin{eqnarray*}&=&\frac{\color{red}{1}}{b^{\color{red}{{12\;-\;8}}}}\\\text{Simplify.}\;\;&=&\frac{1}{b^4}\end{eqnarray*}[/latex]
b.
Step 1: Since [latex]5 > 3[/latex], there are more factors of [latex]3[/latex] in the denominator.
[latex]\frac{7^3}{7^5}[/latex]
Step 2: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}[/latex]
[latex]\begin{eqnarray*}&=&\frac{\color{red}{1}}{7^{\color{red}{5-3}}}\\[1ex]\text{Simplify.}\;\;&=&\frac{1}{7^2}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{49}\end{eqnarray*}[/latex]
Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.
Try It
6) Simplify:
a. [latex]\frac{{x}^{18}}{{x}^{22}}[/latex]
b. [latex]\frac{{12}^{15}}{{12}^{30}}[/latex]
Solution
a. [latex]\frac{1}{{x}^{4}}[/latex]
b. [latex]\frac{1}{{12}^{15}}[/latex]
7) Simplify:
a. [latex]\frac{{m}^{7}}{{m}^{15}}[/latex]
b. [latex]\frac{{9}^{8}}{{9}^{19}}[/latex]
Solution
a. [latex]\frac{1}{{m}^{8}}[/latex]
b. [latex]\frac{1}{{9}^{11}}[/latex]
Notice the difference in the two previous examples:
- 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.
The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator. Later in this chapter, we will explore this further while using negative exponents.
Example 5.5.3
Simplify:
a. [latex]\frac{{a}^{5}}{{a}^{9}}[/latex]
b. [latex]\frac{{x}^{11}}{{x}^{7}}[/latex]
Solution
a. Is the exponent of [latex]a[/latex] larger in the numerator or denominator? Since [latex]9 > 5[/latex], there are more [latex]a[/latex]‘s in the denominator and so we will end up with factors in the denominator.
Step 1: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}[/latex]
[latex]\begin{eqnarray*}&=&\frac{\color{red}{1}}{a^{\color{red}{{9\;-\;5}}}}\\\text{Simplify.}\;\;&=&\frac{1}{a^4}\end{eqnarray*}[/latex]
b. Notice there are more factors of [latex]x[/latex] in the numerator, since [latex]11 > 7[/latex]. So we will end up with factors in the numerator.
Step 1: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}[/latex]
[latex]\begin{eqnarray*}&=&x^{{\color{red}{11\;-\;7}}}\\\text{Simplify.}\;\;&=&x^4\end{eqnarray*}[/latex]
Try It
8) Simplify:
a. [latex]\frac{{b}^{19}}{{b}^{11}}[/latex]
b. [latex]\frac{{z}^{5}}{{z}^{11}}[/latex]
Solution
a. [latex]{b}^{8}[/latex]
b. [latex]\frac{1}{{z}^{6}}[/latex]
9) Simplify:
a. [latex]\frac{{p}^{9}}{{p}^{17}}[/latex]
b. [latex]\frac{{w}^{13}}{{w}^{9}}[/latex]
Solution
a. [latex]\frac{1}{{p}^{8}}[/latex]
b. [latex]{w}^{4}[/latex]
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 [latex]\frac{{a}^{m}}{{a}^{m}}[/latex]. From your earlier work with fractions, you know that:
In words, a number divided by itself is [latex]1[/latex]. So, [latex]\frac{x}{x}=1[/latex], for any [latex]x[/latex] [latex]({x} \neq {0})[/latex], since any number divided by itself is [latex]1[/latex].
The Quotient Property for Exponents shows us how to simplify [latex]\frac{{a}^{m}}{{a}^{n}}[/latex] when [latex]m>n[/latex] and when [latex]n < m[/latex] by subtracting exponents. What if [latex]m=n[/latex]?
Consider [latex]\frac{8}{8}[/latex], which we know is [latex]1[/latex].
[latex]\begin{align*} &\;&\frac{8}{8}&=1\\[1ex] &\text{Write}\;8\;\text{as}\;2^3\;&\frac{{2}^{3}}{{2}^{3}}&=1\\[1ex] &\text{Subtract exponents}\;&{2}^{3-3}&=1\\ &\text{Simplify}\;&2^0&=1 \end{align*}[/latex]
Now we will simplify [latex]\frac{{a}^{m}}{{a}^{m}}[/latex] in two ways to lead us to the definition of the zero exponent. In general, for [latex]{a}\neq{0}[/latex] :
[latex]\begin{array}{cc}\displaystyle\frac{a^m}{a^m}\;\;&\displaystyle\frac{a^m}{a^m}\\[3ex]a^{m-m}\;\;&\displaystyle\frac{\overbrace{\cancel a\cdot \cancel a\cdot...\cdot\cancel a}^{{\color{blue}{m\text{ factors}}}}}{\underbrace{\cancel a\cdot \cancel a\cdot...\cdot\cancel a}_{{\color{blue}{m\text{ factors}}}}}\\[3ex]a^0\;\;&1\end{array}[/latex]
We see [latex]\frac{{a}^{m}}{{a}^{m}}[/latex] simplifies to [latex]{a}^{0}[/latex] and to [latex]1[/latex]. So [latex]{a}^{0}=1[/latex]
Zero Exponent
If [latex]a[/latex] is a non-zero number, then [latex]{a}^{0}=1[/latex].
Any nonzero number raised to the zero power is [latex]1[/latex].
In this text, we assume any variable that we raise to the zero power is not zero.
Example 5.5.4
Simplify:
a. [latex]{9}^{0}[/latex]
b. [latex]{n}^{0}[/latex]
Solution
The definition says any non-zero number raised to the zero power is [latex]1[/latex].
a.
Step 1: Use the definition of the zero exponent.
[latex]\begin{eqnarray*}&=&9^0\\&=&1\end{eqnarray*}[/latex]
b.
Step 1: Use the definition of the zero exponent.
[latex]\begin{eqnarray*}&=&n^0\\&=&1\end{eqnarray*}[/latex]
Try It
10) Simplify:
a. [latex]{15}^{0}[/latex]
b. [latex]{m}^{0}[/latex]
Solution
a. 1
b. 1
11) Simplify:
a. [latex]{k}^{0}[/latex]
b. [latex]{29}^{0}[/latex]
Solution
a. 1
b. 1
Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.
What about raising an expression to the zero power? Let’s look at [latex](2x)^{0}[/latex]. We can use the product to a power rule to rewrite this expression.
[latex]\begin{align*} &\;&(2x)^{0}\\[1ex] &\text{Use the product to a power rule.}\;&{2}^{0}{x}^{0}\\[1ex] &\text{Use the zero exponent property.}\;&1\times{1}\\ &\text{Simplify}\;&1 \end{align*}[/latex]
This tells us that any nonzero expression raised to the zero power is one.
Example 5.5.5
Simplify:
a. [latex](5b)^{0}[/latex]
b. [latex](-4{a}^{2}b)^{0}[/latex]
Solution
a.
Step 1: Use the definition of the zero exponent.
[latex]\begin{eqnarray*}&=&(5b)^0\\&=&1\end{eqnarray*}[/latex]
b.
Step 1: Use the definition of the zero exponent.
[latex]\begin{eqnarray*}&=&(-4a^{2b})^0\\&=&1\end{eqnarray*}[/latex]
Try It
12) Simplify:
a. [latex](11z)^{0}[/latex]
b. [latex](-11p{q}^{3})^{0}[/latex]
Solution
a. [latex]1[/latex]
b. [latex]1[/latex]
13) Simplify:
a. [latex](-6d)^{0}[/latex]
b. [latex](-8{m}^{2}{n}^{3})^{0}[/latex]
Solution
a. [latex]1[/latex]
b. [latex]1[/latex]
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.
[latex]\begin{align*} &\;&\;&\left(\frac{x}{y}\right)^{3}\\[1ex] &\text{This means:}&\;&\frac{x}{y}\cdot\frac{x}{y}\cdot\frac{x}{y}\\[1ex] &\text{Multiply the fractions.}&\;&\frac{x\cdot{x}\cdot{x}}{y\cdot{y}\cdot{y}}\\[1ex] &\text{Write with exponents.}&\;&\frac{{x}^{3}}{{y}^{3}} \end{align*}[/latex]
Notice that the exponent applies to both the numerator and the denominator.
[latex]\begin{align*} &\text{We write:}\;&\left(\frac{x}{y}\right)^{3}\\[1ex] &\;&\frac{{x}^{3}}{{y}^{3}}\;\;\;\end{align*}[/latex]
This leads to the Quotient to a Power Property for Exponents.
Quotient to a Power Property for Exponents
If [latex]a[/latex] and [latex]b[/latex] are real numbers, [latex]b\neq{0}[/latex], and [latex]m[/latex] is a counting number, then
To raise a fraction to a power, raise the numerator and denominator to that power.
An example with numbers may help you understand this property:
[latex]\begin{eqnarray*}(\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}\checkmark\end{eqnarray*}[/latex]
Example 5.5.6
Simplify:
a. [latex](\frac{3}{7})^{2}[/latex]
b. [latex](\frac{b}{3})^{4}[/latex]
c. [latex]\left(\frac{k}{j}\right)^3[/latex]
Solution
a.
Step 1: Use the Quotient Property, [latex](\frac{a}{b})^{m}=\frac{{a}^{m}}{{b}^{m}}[/latex]
[latex]\begin{eqnarray*}&=&{b}^{0}\\ \text{Simplify.}\;\;&=&1 \end{eqnarray*}[/latex]
b.
Step 1: Use the Quotient Property, [latex](\frac{a}{b})^{m}=\frac{{a}^{m}}{{b}^{m}}[/latex]
[latex]\begin{eqnarray*}&=&\frac{b^{\color{red}{4}}}{3^{\color{red}{4}}}\\\text{Simplify.}\;\;&=&\frac{b^{4}}{81}\end{eqnarray*}[/latex]
c.
Step 1: Raise the numerator and denominator to the third power.
[latex]\displaystyle\frac{k^{\color{red}{3}}}{j^{\color{red}{3}}}[/latex]
Try It
14) Simplify:
a. [latex](\frac{5}{8})^{2}[/latex]
b. [latex](\frac{p}{10})^{4}[/latex]
c. [latex](\frac{m}{n})^{7}[/latex]
Solution
a. [latex]\frac{25}{64}[/latex]
b. [latex]\frac{{p}^{4}}{10,000}[/latex]
c. [latex]\frac{{m}^{7}}{{n}^{7}}[/latex]
15) Simplify:
a. [latex](\frac{1}{3})^{3}[/latex]
b. [latex](\frac{-2}{q})^{3}[/latex]
c. [latex](\frac{w}{x})^{4}[/latex]
Solution
a. [latex]\frac{1}{27}[/latex]
b. [latex]\frac{-8}{{q}^{3}}[/latex]
c. [latex]\frac{{w}^{4}}{{x}^{4}}[/latex]
Simplify Expressions by Applying Several Properties
We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.
Summary of Exponent Properties
If [latex]a[/latex] and [latex]b[/latex] are real numbers, and [latex]m[/latex] and [latex]n[/latex] are whole numbers, then
| Product Property | [latex]{a}^{m}\times{a}^{n}={a}^{m+n}[/latex] |
|---|---|
| Power Property | [latex]({a}^{m})^{n}={a}^{m\times{n}}[/latex] |
| Product to a Power | [latex](ab)^{m}={a}^{m}{b}^{m}[/latex] |
| Quotient Property | [latex]\frac{{a}^{m}}{{b}^{m}}={a}^{m-n}, a\neq{0} , m>n[/latex]
[latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},a\neq{0},n>m[/latex] |
| Zero Exponent Definition | [latex]{a}^{0}=1,a\neq{0}[/latex] |
| Quotient to a Power Property | [latex](\frac{a}{b})^{m}=\frac{{a}^{m}}{{b}^{m}},b\neq{0}[/latex] |
Example 5.5.7
Simplify: [latex]\frac{(y^4)^2}{y^6}[/latex]
Solution
Step 1: Multiply the exponents in the numerator.
[latex]\frac{{y}^{8}}{{y}^{6}}[/latex]
Step 2: Subtract the exponents.
[latex]{y}^{2}[/latex]
Try It
16) Simplify: [latex]\frac{({m}^{5})^4}{m^7}[/latex]
Solution
[latex]{m}^{13}[/latex]
17) Simplify: [latex]\frac{{({k}^{2})}^{6}}{{k}^{7}}[/latex]
Solution
[latex]{k}^{5}[/latex]
Example 5.5.8
Simplify: [latex]\frac{{b}^{12}}{{({b}^{2})}^{6}}[/latex]
Solution
Step 1: Multiply the exponents in the numerator.
[latex]\frac{{b}^{12}}{{b}^{12}}[/latex]
Step 2: Subtract the exponents.
[latex]\begin{eqnarray*}&=&{b}^{0}\\ \text{Simplify.}\;\;&=&1 \end{eqnarray*}[/latex]
Try It
18) Simplify: [latex]\frac{{n}^{12}}{{({n}^{3}})^{4}}[/latex]
Solution
[latex]1[/latex]
19) Simplify: [latex]\frac{{x}^{15}}{(x^3)^{5}}[/latex]
Solution
[latex]1[/latex]
Example 5.5.9
Simplify: [latex]\left(\frac{{y}^{9}}{{y}^{4}}\right)^{2}[/latex]
Solution
Step 1: Remember parentheses come before exponents.
Notice the bases are the same, so we can simplify inside the parentheses.
Subtract the exponents.
[latex](y^5)^2[/latex]
Step 2: Multiply the exponents.
[latex]{y}^{10}[/latex]
Try It
20) Simplify: [latex]\left(\frac{{r}^{5}}{{r}^{3}}\right)^{4}[/latex]
Solution
[latex]{r}^{8}[/latex]
21) Simplify: [latex]\left(\frac{{v}^{6}}{{v}^{4}}\right)^{3}[/latex]
Solution
[latex]{v}^{6}[/latex]
Example 5.5.10
Simplify: [latex]\left(\frac{{j}^{2}}{{k}^{3}}\right)^{4}[/latex]
Solution
Here we cannot simplify inside the parentheses first, since the bases are not the same.
Step 1: Raise the numerator and denominator to the third power using the Quotient to a Power Property, [latex](\frac{a}{b})^{m}=\frac{{a}^{m}}{{b}^{m}}[/latex]
[latex]\left(\frac{{j}^{2}}{{k}^{3}}\right)^{4}[/latex]
Step 2: Use the Power Property and simplify.
[latex]\frac{{j}^{8}}{{k}^{12}}[/latex]
Try It
22) Simplify: [latex]\left(\frac{a^3}{b^2}\right)^{4}[/latex]
Solution
[latex]\frac{{a}^{12}}{{b}^{8}}[/latex]
23) Simplify: [latex]\left(\frac{q^7}{r^5}\right)^3[/latex]
Solution
[latex]\frac{{q}^{21}}{{r}^{15}}[/latex]
Example 5.5.11
Simplify: [latex]\left(\frac{2{m}^{2}}{5n}\right)^{4}[/latex]
Solution
Step 1: Raise the numerator and denominator to the fourth power, using the Quotient to a Power Property, [latex]\left(\frac{a}{b}\right)^m=\;\frac{a^m}{b^m}[/latex]
[latex]\frac{(2m^{2})^{4}}{{(5n)}^{4}}[/latex]
Step 2: Raise each factor to the fourth power.
[latex]\frac{(2m^{2})^{4}}{{(5n)}^{4}}[/latex]
Step 3: Use the Power Property and simplify.
[latex]\frac{16{m}^{8}}{625{n}^{4}}[/latex]
Try It
24) Simplify: [latex]\left(\frac{7x^3}{9y}\right)^2[/latex]
Solution
[latex]\frac{49{x}^{6}}{81{y}^{2}}[/latex]
25) Simplify: [latex]\left(\frac{3x^4}{7y}\right)^2[/latex]
Solution
[latex]\frac{9{x}^{8}}{49{y}^{2}}[/latex]
Example 5.5.12
Simplify: [latex]\frac{\left(x^3\right)^4\times\left(x^2\right)^5}{\left(x^6\right)^5}[/latex]
Solution
Step 1: Use the Power Property, [latex]{(a^m)}^{n}={a}^{mn}[/latex]
[latex]\frac{\left(x^{12}\right)\times\left(x^{10}\right)}{\left(x^{30}\right)}[/latex]
Step 2: Add the exponents in the numerator.
[latex]\frac{x^{22}}{x^{30}}[/latex]
Step 3: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}[/latex]
[latex]\frac{1}{{x}^{8}}[/latex]
Try It
26) Simplify: [latex]\frac{{({a}^{2})}^{3}{({a}^{2})}^{4}}{({a}^{4})^{5}}[/latex]
Solution
[latex]\frac{1}{{a}^{6}}[/latex]
27) Simplify: [latex]\frac{{({p}^{3})}^{4}{({p}^{5})}^{3}}{{({p}^{7})}^{6}}[/latex]
Solution
[latex]\frac{1}{{p}^{15}}[/latex]
Example 5.5.13
Simplify: [latex]\frac{(10p^3)^2}{(5p)^3\cdot\left(2p^5\right)^4}[/latex]
Solution
Step 1: Use the Product to a Power Property, [latex]{(ab)}^{m}={a}^{m}{b}^{m}[/latex]
[latex]\frac{{(10)}^{2}{({p}^{3})}^{2}}{(5)^3(p)^{3}\cdot (2)^4(p^5)^4}[/latex]
Step 2: Use the Power Property, [latex]{{a}^{m}}^{n}={a}^{m\times{n}}[/latex]
[latex]\frac{100{p}^{6}}{125p^3\cdot 16{p}^{20}}[/latex]
Step 3: Add the exponents in the denominator.
[latex]\frac{100{p}^{6}}{125\times{16}{p}^{23}}[/latex]
Step 4: Use the Quotient Property, [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}[/latex]
[latex]\begin{eqnarray*}&=&\frac{100}{125\times{16{p}}^{17}}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{20{p}^{17}} \end{eqnarray*}[/latex]
Try It
28) Simplify: [latex]\frac{{(3{r}^{3}}^{2}{({r}^{3})}^{7}}{{({r}^{3})}^{3}}[/latex]
Solution
[latex]9{r}^{18}[/latex]
29) Simplify: [latex]\frac{{(2{x}^{4})}^{5}}{{(4{x}^{3})}^{2}{({x}^{3})}^{5}}[/latex]
Solution
[latex]\frac{2}{x}[/latex]
Use the Definition of a Negative Exponent
Up until this point, we have been careful to minimize our use of negative exponents. Before taking on division of polynomials, we will revisit some of the properties we’ve seen in the previous sections but, this time, with negative exponents.
Quotient Property for Exponents
If [latex]a[/latex] is a real number, [latex]{a}\neq 0[/latex] , and [latex]m[/latex] and [latex]n[/latex] are whole numbers, then
[latex]\frac{a^m}{a^n}=a^{m-n}[/latex], for [latex]m>n[/latex] and [latex]\frac{a^m}{a^n}=\frac{1}{a^{n-m}}[/latex], for [latex]n>m[/latex].
What if we just subtract exponents regardless of which is larger?
Let’s consider [latex]\frac{{x}^{2}}{{x}^{5}}[/latex].
We subtract the exponent in the denominator from the exponent in the numerator.
We can also simplify [latex]\frac{{x}^{2}}{{x}^{5}}[/latex] by dividing out common factors:
[latex]\begin{eqnarray*}&=&\frac{{\color{red}{\cancel x}}\cdot{\color{red}{\cancel x}}}{{\color{red}{\cancel x}}\cdot{\color{red}{\cancel x}}\cdot x\cdot x\cdot x}\\&=&\frac1{x^3}\end{eqnarray*}[/latex]
This implies that [latex]{x}^{-3}=\frac{1}{{x}^{3}}[/latex] and it leads us to the definition of a negative exponent.
Negative Exponents
If [latex]n[/latex] is an integer and [latex]a\neq{0}[/latex], then [latex]{a}^{\text{−}n}=\frac{1}{{a}^{n}}[/latex].
The negative exponent tells us we can re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent.
Any expression that has negative exponents is not considered to be in simplest form. We will use the definition of a negative exponent and other properties of exponents to write the expression with only positive exponents.
For example, if after simplifying an expression we end up with the expression [latex]{x}^{-3}[/latex], we will take one more step and write [latex]\frac{1}{{x}^{3}}[/latex]. The answer is considered to be in simplest form when it has only positive exponents.
Example 5.5.14
Simplify:
a. [latex]{4}^{-2}[/latex]
b. [latex]{10}^{-3}[/latex]
Solution
a.
Step 1: Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex].
[latex]\begin{eqnarray*}&=&\frac{1}{{4}^{2}}\\ \text{Simplify.}\;\;&=&\frac{1}{16} \end{eqnarray*}[/latex]
b.
Step 1: Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex].
[latex]\begin{eqnarray*}&=&\frac{1}{{10}^{3}}\\ \text{Simplify.}\;\;&=&\frac{1}{1000}\end{eqnarray*}[/latex]
Try It
30) Simplify:
a. [latex]{2}^{-3}[/latex]
b. [latex]{10}^{-7}[/latex]
Solution
a. [latex]\frac{1}{8}[/latex]
b. [latex]\frac{1}{{10}^{7}}[/latex]
31) Simplify:
a. [latex]{3}^{-2}[/latex]
b. [latex]{10}^{-4}[/latex]
Solution
a. [latex]\frac{1}{9}[/latex]
b. [latex]\frac{1}{10,000}[/latex]
In example 5.2.13 we raised an integer to a negative exponent. What happens when we raise a fraction to a negative exponent? We’ll start by looking at what happens to a fraction whose numerator is one and whose denominator is an integer raised to a negative exponent.
[latex]\begin{align*} &\;&\;&\frac{1}{a^{-n}}\\[1ex] &\text{Use the definition of a negative exponent,}\;{a}^{-n}=\frac{1}{a^n}&\;&\frac{1}{\frac{1}{{a}^{n}}}\\[1ex] &\text{Simplify the complex fraction.}&\;&1\cdot \frac{{a}^{n}}{1}\\[1ex] &\text{Multiply.}&\;&{a}^{n} \end{align*}[/latex]
This leads to the Property of Negative Exponents.
Property of Negative Exponents
If [latex]n[/latex] is an integer and [latex]{a}\neq 0[/latex], then [latex]\frac{1}{{a}^{-n}}={a}^{n}[/latex].
Example 5.5.15
Simplify:
a. [latex]\frac{1}{{y}^{-4}}[/latex]
b. [latex]\frac{1}{{3}^{-2}}[/latex]
Solution
a.
Step 1: Use the property of a negative exponent, [latex]\frac{1}{{a}^{-n}}={a}^{n}[/latex].
[latex]{y}^{4}[/latex]
b.
Step 1: Use the property of a negative exponent, [latex]\frac{1}{{a}^{-n}}={a}^{n}[/latex].
[latex]\begin{eqnarray*}&=&{3}^{2}\\\text{Simplify.}\;\;&=&9\end{eqnarray*}[/latex]
Try It
32) Simplify:
a. [latex]\frac{1}{{p}^{-8}}[/latex]
b. [latex]\frac{1}{{4}^{-3}}[/latex]
Solution
a. [latex]{p}^{8}[/latex]
b. [latex]64[/latex]
33) Simplify:
a. [latex]\frac{1}{{q}^{-7}}[/latex]
b. [latex]\frac{1}{{2}^{-4}}[/latex]
Solution
a. [latex]{q}^{7}[/latex]
b. [latex]16[/latex]
Suppose now we have a fraction raised to a negative exponent. Let’s use our definition of negative exponents to lead us to a new property.
| [latex](\frac{3}{4})^{-2}[/latex] | |
| Use the definition of a negative exponent, [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]. | [latex]\frac {1}{(\frac{3}{4})^{2}}[/latex] |
| Simplify the denominator. | [latex]\frac{1}{\frac{9}{16}}[/latex] |
| Simplify the complex fraction. | [latex]\frac{16}{9}[/latex] |
| But we know that [latex]\frac{16}{9}[/latex] is [latex](\frac{4}{3})^{2}[/latex]. | [latex]\frac{4^2}{3^2}[/latex] |
| This tells us that: | [latex](\frac{3}{4})^{-2}=(\frac{4}{3})^{2}[/latex] |
To get from the original fraction raised to a negative exponent to the final result, we took the reciprocal of the base—the fraction—and changed the sign of the exponent.
This leads us to the Quotient to a Negative Power Property.
Quotient to a Negative Exponent Property
If [latex]a[/latex] and [latex]b[/latex] are real numbers, [latex]a \neq 0[/latex], [latex]b\neq 0[/latex], and [latex]n[/latex] is an integer, then [latex](\frac{a}{b})^{-n}= (\frac{b}{a})^{n}[/latex].
Example 5.5.16
Simplify:
a. [latex]\left(\frac{5}{7}\right)^{-2}[/latex]
b. [latex]\left(-\frac{2x}{y}\right)^{-3}[/latex]
Solution
a.
Step 1: Use the Quotient to a Negative Exponent Property, [latex](\frac{a}{b})^{-n}={(\frac{b}{a})}^{n}[/latex].
Step 2: Take the reciprocal of the fraction and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&\left(\frac{7}{5}\right)^{2}\;\;\\\text{Simplify.}\;\;&=&\frac{49}{25} \end{eqnarray*}[/latex]
b.
Step 1: Use the Quotient to a Negative Exponent Property, [latex](\frac{a}{b})^{-n}={(\frac{b}{a})}^{n}[/latex]
Step 2: Take the reciprocal of the fraction and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&\left(-\frac{y}{2x}\right)^{3}\;\;\\[1ex]\text{Simplify.}\;\;&=&-\frac{{y}^{3}}{8{x}^{3}}\end{eqnarray*}[/latex]
Try It
34) Simplify:
a. [latex]\left(\frac{2}{3}\right)^{-4}[/latex]
b. [latex](-\frac{6m}{n})^{-2}[/latex]
Solution
a. [latex]\frac{81}{16}[/latex]
b. [latex]\frac{{n}^{2}}{36{m}^{2}}[/latex]
35) Simplify:
a. [latex]\left(\frac{3}{5}\right)^{-3}[/latex]
b. [latex]\left(-\frac{a}{2b}\right)^{-4}[/latex]
Solution
a. [latex]\frac{125}{27}[/latex]
b. [latex]\frac{16{b}^{4}}{{a}^{4}}[/latex]
When simplifying an expression with exponents, we must be careful to correctly identify the base.
Example 5.5.17
Simplify:
a. [latex](-3)^{-2}[/latex]
b. [latex]-3^{-2}[/latex]
c. [latex]\left(-\frac{1}{3}\right)^{-2}[/latex]
d. [latex]-\left(\frac{1}{3}\right)^{-2}[/latex]
Solution
a. Here the exponent applies to the base [latex]-3[/latex].
Step 1: Take the reciprocal of the base and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&\frac{1}{(-3)^2}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{9}\end{eqnarray*}[/latex]
b. The expression [latex]-{3}^{-2}[/latex] means “find the opposite of [latex]{3}^{-2}[/latex].”
Step 1: Rewrite as a product with [latex]-1[/latex].
[latex]-1\times 3^{-2}[/latex]
Step 2: Take the reciprocal of the base and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&-1\cdot \frac{1}{{3}^{2}}\\[1ex]\text{Simplify.}\;\;&=&-\frac{1}{9}\end{eqnarray*}[/latex]
c. Here the exponent applies to the base [latex]-\frac{1}{3}[/latex].
Step 1: Take the reciprocal of the base and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&\left(-\frac{3}{1}\right)^2\\[1ex]\text{Simplify.}\;\;&=&9\end{eqnarray*}[/latex]
d. The expression [latex]-(\frac{1}{3})^{-2}[/latex] means “find the opposite of [latex](\frac{1}{3})^{-2}[/latex].” Here the exponent applies to the base [latex]\frac{1}{3}[/latex].
Step 1: Rewrite as a product with [latex]-1[/latex].
[latex]-1\cdot \left(\frac{3}{1}\right)^{2}[/latex]
Step 2: Take the reciprocal of the base and change the sign of the exponent.
[latex]-9[/latex]
Try It
36) Simplify:
a. [latex](-5)^{-2}[/latex]
b. [latex]{−}{5}^{-2}[/latex]
c. [latex]\left(-\frac{1}{5}\right)^{-2}[/latex]
d. [latex]{−}\left(\frac{1}{5}\right)^{-2}[/latex]
Solution
a. [latex]\frac{1}{25}[/latex]
b. [latex]-\frac{1}{25}[/latex]
c. [latex]25[/latex]
d. [latex]-25[/latex]
37) Simplify:
a. [latex](-7)^{-2}[/latex]
b. [latex]{−}{7}^{-2}[/latex]
c. [latex]\left(-\frac{1}{7}\right)^{-2}[/latex]
d. [latex]\left(-\frac{1}{7}\right)^{-2}[/latex]
Solution
a. [latex]\frac{1}{49}[/latex]
b. [latex]-\frac{1}{49}[/latex]
c. [latex]49[/latex]
d. [latex]-49[/latex]
We must be careful to follow the Order of Operations. In the next example, parts (a) and (b) look similar, but the results are different.
Example 5.5.18
Simplify:
a. [latex]4\times{2}^{-1}[/latex]
b. [latex](4\times{2})^{-1}[/latex]
Solution
a.
Step 1: Do exponents before multiplication.
[latex]4\times{2}^{-1}[/latex]
Step 2: Use [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]
[latex]\begin{eqnarray*}&=&4\times \frac{1}{{2}^{1}}\\[1ex] \text{Simplify.}\;\;&=&2\end{eqnarray*}[/latex]
b.
Step 1: Simplify inside the parentheses first.
[latex](8)^{-1}[/latex]
Step 2: Use [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]
[latex]\begin{eqnarray*}&=&\frac{1}{{8}^{1}}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{8} \end{eqnarray*}[/latex]
Try It
38) Simplify:
a. [latex]6\times{3}^{-1}[/latex]
b. [latex](6\times 3)^{-1}[/latex]
Solution
a. [latex]2[/latex]
b. [latex]\frac{1}{18}[/latex]
39) Simplify:
a. [latex]8\times {2}^{-2}[/latex]
b. [latex]8\times {2}^{-2}[/latex]
Solution
a. [latex]2[/latex]
b. [latex]\frac{1}{256}[/latex]
When a variable is raised to a negative exponent, we apply the definition the same way we did with numbers. We will assume all variables are non-zero.
Example 5.5.19
Simplify:
a. [latex]{x}^{-6}[/latex]
b. [latex]({u}^{4})^{-3}[/latex]
Solution
a.
Step 1: Use the definition of a negative exponent [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex]
[latex]\frac{1}{{x}^{6}}[/latex]
b.
Step 1: Use the definition of a negative exponent [latex]{a}^{-n}=\frac{1}{{a}^{n}}[/latex].
[latex]\begin{eqnarray*}&=&\frac{1}{(u^4)^3}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{u^{12}} \end{eqnarray*}[/latex]
Try It
40) Simplify:
a. [latex]{y}^{-7}[/latex]
b. [latex]({z}^{3})^{-5}[/latex]
Solution
a. [latex]\frac{1}{{y}^{7}}[/latex]
b. [latex]\frac{1}{{z}^{15}}[/latex]
41) Simplify:
a. [latex]{p}^{-9}[/latex]
b. [latex]({q}^{4})^{-6}[/latex]
Solution
a. [latex]\frac{1}{p}^{9}[/latex]
b. [latex]\frac{1}{q}^{24}[/latex]
When there is a product and an exponent we have to be careful to apply the exponent to the correct quantity. According to the Order of Operations, we simplify expressions in parentheses before applying exponents. We’ll see how this works in the next example.
Example 5.5.20
Simplify:
a. [latex]5{y}^{-1}[/latex]
b. [latex](5y)^{-1}[/latex]
c. [latex](-5y)^{-1}[/latex]
Solution
a.
Step 1: Notice the exponent applies to just the base [latex]y[/latex].
Step 2: Take the reciprocal of [latex]y[/latex] and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&5\cdot \left(\frac{1}{y^1}\right)\\[1ex] \text{Simplify.}\;\;&=&\frac{5}{y} \end{eqnarray*}[/latex]
b.
Step 1: Here the parentheses make the exponent apply to the base [latex]5y[/latex].
Step 2: Take the reciprocal of [latex]5y[/latex] and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&\frac{1}{{5y}^1}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{5y} \end{eqnarray*}[/latex]
c.
Step 1: The base here is [latex]−5y[/latex].
Step 2: Take the reciprocal of [latex]−5y[/latex] and change the sign of the exponent.
[latex]\begin{eqnarray*}&=&\frac{1}{-5y}\\[1ex]\text{Simplify.}\;\;&=&\left(-\frac{1}{5y}\right) \end{eqnarray*}[/latex]
Try It
42) Simplify:
a. [latex](8{p}^{-1})[/latex]
b. [latex]{(8p)}^{-1}[/latex]
c. [latex]{(-8p)}^{-1}[/latex]
Solution
a. [latex]\frac{8}{p}[/latex]
b. [latex]\frac{1}{8p}[/latex]
c. [latex]-\frac{1}{8p}[/latex]
43) Simplify:
a. [latex]{11q}^{-1}[/latex]
b. [latex]{(11q)}^{-1}[/latex]
c. [latex]{(-11q)}^{-1}[/latex]
Solution
a. [latex]\frac{1}{11q}[/latex]
b. [latex]\frac{1}{11q}[/latex]
c. [latex]-\frac{1}{11q}[/latex]
With negative exponents, the Quotient Rule needs only one form [latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], for [latex]a\neq{0}[/latex]. When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative.
Simplify Expressions with Integer Exponents
All of the exponent properties we developed earlier in the chapter with whole number exponents apply to integer exponents, too. We restate them here for reference.
Summary of Exponent Properties
If [latex]a[/latex] and [latex]b[/latex] are real numbers, and [latex]m[/latex] and [latex]n[/latex] are integers, then
| Product Property | [latex]{a^m}\times{a^n}=a^{m+n}[/latex] |
|---|---|
| Power Property | [latex]({a^m})^n={a}{mn}[/latex] |
| Product to a Power | [latex](ab)^m=a^m b^m[/latex] |
| Quotient Property | [latex]\frac{a^m}{a^n}={a}^{m-n}[/latex] |
| Zero Exponent Property | [latex]a^0=1 , {a}\neq{1}[/latex] |
| Quotient to a Power Property | [latex](\frac {a}{b})^m=\frac{{a}^{m}}{{b}^{m}}, {b}\neq{0}[/latex] |
| Properties of Negative Exponents | [latex]{a}^{-n}=\frac{1}{a^n}[/latex] and [latex]\frac{1}{a^{-n}}=a^{n}[/latex] |
| Quotient to a Negative Exponent | [latex](\frac{a}{b})^{-n}=(\frac{b}{a})^{n}[/latex] |
Example 5.5.21
Simplify:
a. [latex]{x}^{-4}\times{x}^{6}[/latex]
b. [latex]{y}^{-6}\times{y}^{4}[/latex]
c. [latex]{z}^{-5}\times{z}^{-3}[/latex]
Solution
a.
Step 1: Use the Product Property, [latex]a^m\cdot a^n=a^{m+n}[/latex]
[latex]\begin{eqnarray*}&=&{x}^{-4+6}\\[1ex] \text{Simplify.}\;\;&=&x^2 \end{eqnarray*}[/latex]
b.
Step 1: Notice the same bases, so add the exponents.
[latex]\begin{eqnarray*}&=&y^{-6+4}\\[1ex] \text{Simplify.}\;\;&=&y^{-2}\end{eqnarray*}[/latex]
Step 3: Use the definition of a negative exponent, [latex]a^{-n}=\frac{1}{a^n}[/latex]
[latex]\frac{1}{y^2}[/latex]
c.
Step 1: Add the exponents, since the bases are the same.
[latex]\begin{eqnarray*}&=&z^{-5-3}\\[1ex] \text{Simplify.}\;\;&=&z^{-8}\end{eqnarray*}[/latex]
Step 3: Take the reciprocal and change the sign of the exponent, using the definition of a negative exponent.
[latex]\frac{1}{z^{8}}[/latex]
Try It
44) Simplify:
a. [latex]{{x}^{-3}}\times{{x}^{7}}[/latex]
b. [latex]{{y}^{-7}}\times{{y}^{2}}[/latex]
c. [latex]{{z}^{-4}}\times{{z}^{-5}}[/latex]
Solution
a. [latex]{x}^{4}[/latex]
b. [latex]\frac{1}{{y}^{5}}[/latex]
c. [latex]\frac{1}{{z}^{9}}[/latex]
45) Simplify:
a. [latex]{{a}^{-1}}\times{{a}^{6}}[/latex]
b. [latex]b^{-8}\times b^{4}[/latex]
c. [latex]{c}^{-8}\times{{c}^{-7}}[/latex]
Solution
a. [latex]{a}^{5}[/latex]
b. [latex]\frac{1}{{b}^{4}}[/latex]
c. [latex]\frac{1}{{c}^{15}}[/latex]
In the next two examples, we’ll start by using the Commutative Property to group the same variables together. This makes it easier to identify the like bases before using the Product Property.
Example 5.5.22
Simplify: [latex]({m}^{4}{n}^{-3})({m}^{-5}{n}^{-2})[/latex]
Solution
Step 1: Use the Commutative Property to get like bases together.
[latex]m^4 m^{-5} n^{-2} n^{-3}[/latex]
Step 2: Add the exponents for each base.
[latex]m^{-1} n^{-5}[/latex]
Step 3: Take reciprocals and change the signs of the exponents.
[latex]\begin{eqnarray*}&=&\frac{1}{m^1} \frac{1}{n^5}\\[1ex] \text{Simplify.}\;\;&=&\frac{1}{mn^5}\end{eqnarray*}[/latex]
Try It
46) Simplify: [latex]({p}^{6}{q}^{-2})({p}^{-9}{q}^{-1})[/latex]
Solution
[latex]\frac{1}{{p}^{3}{q}^{3}}[/latex]
47) Simplify: [latex]({r}^{5}{s}^{-3})({r}^{-7}{s}^{-5})[/latex]
Solution
[latex]\frac{1}{{r}^{2}{s}^{8}}[/latex]
If the monomials have numerical coefficients, we multiply the coefficients, just like we did earlier.
Example 5.5.23
Simplify: [latex](2{x}^{-6}{y}^{8})(-5{x}^{5}{y}^{-3})[/latex]
Solution
Step 1: Rewrite with the like bases together
[latex]2(-5)(x^{-6} x^{5}) (y^8 y^{-3})[/latex]
Step 2: Multiply the coefficients and add the exponents of each variable.
[latex]-10x^{-1}y^5[/latex]
Step 3: Use the definition of a negative exponent, [latex]a^{-n}=\frac{1}{a^n}[/latex]
[latex]\begin{eqnarray*}&=&-10\cdot \left(\frac{1}{x^1}\right)\cdot y^5\\[1ex] \text{Simplify.}\;\;&=&-\frac{10y^5}{x}\end{eqnarray*}[/latex]
Try It
48) Simplify: [latex](3{u}^{-5}{v}^{7})(-4{u}^{4}{v}^{-2})[/latex]
Solution
[latex]-\frac{12{v}^{5}}{u}[/latex]
49) Simplify: [latex](-6{c}^{-6}{d}^{4})(-5{c}^{-2}{d}^{-1})[/latex]
Solution
[latex]\frac{30{d}^{3}}{{c}^{8}}[/latex]
In the next two examples, we’ll use the Power Property and the Product to a Power Property.
Example 5.5.24
Simplify: [latex](6{k}^{3})^{-2}[/latex]
Solution
Step 1: Use the Product to a Power Property, [latex](ab)^m=a^m b^m[/latex]
[latex](6)^{(-2)}(k^3)^{-2}[/latex]
Step 2: Use the Power Property, [latex](a^m)^n= {a}^{mn}[/latex]
[latex]6^{-2}k^{-6}[/latex]
Step 3: Use the Definition of a Negative Exponent, [latex]a^{-n}=\frac{1}{a^n}[/latex]
[latex]\begin{eqnarray*}&=&\frac{1}{6^2}\cdot \frac{1}{k^6}\\[1ex]\text{Simplify.}\;\;&=&\frac{1}{36k^6}\end{eqnarray*}[/latex]
Try It
50) Simplify: [latex](-4{x}^{4})^{-2}[/latex]
Solution
[latex]\frac{1}{16{x}^{8}}[/latex]
51) Simplify: [latex]\left(2{b}^{3}\right)^{-4}[/latex]
Solution
[latex]\frac{1}{16{b}^{12}}[/latex]
Example 5.5.25
Simplify: [latex](5{x}^{-3})^{2}[/latex]
Solution
Step 1: Use the Product to a Power property, [latex](ab)^m=a^m b^m[/latex]
[latex]5^2(x^{-3})^2[/latex]
Step 2: Simplify [latex]5^2[/latex] and multiply the exponents of [latex]x[/latex] using the Power Property, [latex](a^m)^n=a^{mn}[/latex]
[latex]25x^{-6}[/latex]
Step 3: Rewrite [latex]x^{-6}[/latex] by using the Definition of a Negative Exponent, [latex]a^{-n}=\frac{1}{a^n}[/latex]
[latex]\begin{eqnarray*}&=&25\cdot \left(\frac{1}{x^6}\right)\\[1ex] \text{Simplify.}\;\;&=&\frac{25}{x^6} \end{eqnarray*}[/latex]
Try It
52) Simplify: [latex](8{a}^{-4})^{2}[/latex]
Solution
[latex]\frac{64}{{a}^{8}}[/latex]
53) Simplify: [latex](2{c}^{-4})^{3}[/latex]
Solution
[latex]\frac{8}{{c}^{12}}[/latex]
To simplify a fraction, we use the Quotient Property and subtract the exponents.
Example 5.5.26
Simplify:[latex]\frac{{r}^{5}}{{r}^{-4}}[/latex]
Solution
Step 1: Use the Quotient Property, [latex]\frac{a^m}{a^n}=a^{m-n}[/latex]c
[latex]\begin{eqnarray*}&=&r^{5-(-4)}\\[1ex]\text{Simplify.}\;\;&=&r^9 \end{eqnarray*}[/latex]
Try It
54) Simplify: [latex]\frac{{x}^{8}}{{x}^{-3}}[/latex]
Solution
[latex]{x}^{11}[/latex]
55) Simplify: [latex]\frac{{y}^{8}}{{y}^{-6}}[/latex]
Solution
[latex]{y}^{14}[/latex]
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 5.5.27
Find the quotient: [latex]{56x^7}\div{8x^3}[/latex]
Solution
Step 1: Rewrite as a fraction.
[latex]\frac{56{x}^{7}}{8{x}^{3}}[/latex]
Step 2: Use fraction multiplication.
[latex]\frac{56}{8}\times\frac{x^7}{x^3}[/latex]
Step 3: Simplify and use the Quotient Property.
[latex]7x^4[/latex]
Try It
56) Find the quotient: [latex]42{y}^{9}\div{6{y}^{3}}[/latex]
Solution
[latex]7{y}^{6}[/latex]
57) Find the quotient: [latex]48{z}^{8}\div{8{z}^{2}}[/latex]
Solution
[latex]6{z}^{6}[/latex]
Example 5.5.28
Find the quotient: [latex]\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}[/latex]
Solution
Step 1: Use fraction multiplication.
[latex]{\frac{45}{-5}}{\frac{{a}^{2}}{a}}{\frac{{b}^{3}}{{b}^{5}}}[/latex]
Step 2: Simplify and use the Quotient Property.
[latex]\begin{eqnarray*}&=&-9\cdot {a}\cdot {\frac{1}{{b}^{2}}}\\[1ex] \text{Multiply.}\;\;&=&-\frac{9a}{{b}^{2}} \end{eqnarray*}[/latex]
Try It
58) Find the quotient: [latex]\frac{-72{a}^{7}{b}^{3}}{8{a}^{12}{b}^{4}}[/latex]
Solution
[latex]-\frac{9}{{a}^{5}b}[/latex]
59) Find the quotient: [latex]\frac{-63{c}^{8}{d}^{3}}{7{c}^{12}{d}^{2}}[/latex]
Solution
[latex]\frac{-9d}{{c}^{4}}[/latex]
Example 5.5.29
Find the quotient: [latex]\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}[/latex]
Solution
Step 1: Use fraction multiplication.
[latex]\frac{24}{48}\cdot \frac{a^5}{a}\cdot \frac{b^3}{b^4}[/latex]
Step 2: Simplify and use the Quotient Property.
[latex]\begin{eqnarray*}&=&\frac{1}{2}\cdot {a}^{4}\cdot \frac{1}{b}\\[1ex] \text{Multiply.}\;\;&=&\frac{{a}^{4}}{2b} \end{eqnarray*}[/latex]
Try It
60) Find the quotient:[latex]\frac{16{a}^{7}{b}^{6}}{24a{b}^{8}}[/latex]
Solution
[latex]\frac{2{a}^{6}}{3{b}^{2}}[/latex]
61) Find the quotient: [latex]\frac{27{p}^{4}{q}^{7}}{-45{p}^{12}q}[/latex]
Solution
[latex]-\frac{3{q}^{6}}{5{p}^{8}}[/latex]
Once you become familiar with the process and have practised it step by step several times, you may be able to simplify a fraction in one step.
Example 5.5.30
Find the quotient: [latex]\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}[/latex]
Solution
Be very careful to simplify [latex]\frac{14}{21}[/latex] by dividing out a common factor, and to simplify the variables by subtracting their exponents.
Step 1: Simplify and use the Quotient Property.
[latex]\frac{2{y}^{6}}{3{x}^{4}}[/latex]
Try It
62) Find the quotient: [latex]\frac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}[/latex]
Solution
[latex]\frac{4{y}^{2}}{7{x}^{4}}[/latex]
63) Find the quotient: [latex]\frac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}[/latex]
Solution
[latex]\frac{5}{8{m}^{5}{n}^{3}}[/latex]
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. This follows the order of operations. Remember, a fraction bar is a grouping symbol.
Example 5.5.31
Find the quotient:[latex]\frac{(6{x}^{2}{y}^{3})(5{x}^{3}{y}^{2})}{(3{x}^{4}{y}^{5})}[/latex]
Solution
Step 1: Simplify the numerator.
[latex]\begin{eqnarray*}&=&\frac{30{x}^{5}{y}^{5}}{3{x}^{4}{y}^{5}}\\[1ex]\text{Simplify.}\;\;&=&10x \end{eqnarray*}[/latex]
Try It
64) Find the quotient: [latex]\frac{(6{a}^{4}{b}^{5})(4{a}^{2}{b}^{5})}{12{a}^{5}{b}^{8}}[/latex]
Solution
[latex]2a{b}^{2}[/latex]
65) Find the quotient: [latex]\frac{(-12{x}^{6}{y}^{9})(-4{x}^{5}{y}^{8})}{-12{x}^{10}{y}^{12}}[/latex]
Solution
[latex]-4x{y}^{5}[/latex]
Access these online resources for additional instruction and practice with dividing monomials:
Key Concepts
- Quotient Property for Exponents:
- If [latex]a[/latex] is a real number, [latex]a\neq{0}[/latex], and [latex]m[/latex], [latex]n[/latex] are whole numbers, then:
[latex]\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}[/latex], [latex]m>n[/latex] and [latex]\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{m-n}}[/latex], [latex]n>m[/latex]
- If [latex]a[/latex] is a real number, [latex]a\neq{0}[/latex], and [latex]m[/latex], [latex]n[/latex] are whole numbers, then:
- Zero Exponent
- If [latex]a[/latex] is a non-zero number, then [latex]{a}^{0}=1[/latex].
- If [latex]a[/latex] is a non-zero number, then [latex]{a}^{0}=1[/latex].
- Quotient to a Power Property for Exponents:
- If [latex]a[/latex] and [latex]b[/latex] are real numbers, [latex]b\neq{0}[/latex] and [latex]m[/latex] is a counting number, then:
[latex](\frac{a}{b})^{m}=\frac{{a}^{m}}{{b}^{m}}[/latex] - To raise a fraction to a power, raise the numerator and denominator to that power.
- If [latex]a[/latex] and [latex]b[/latex] are real numbers, [latex]b\neq{0}[/latex] and [latex]m[/latex] is a counting number, then:
- Summary of Exponent Properties
- If [latex]a[/latex], [latex]b[/latex] are real numbers and [latex]m[/latex], [latex]n[/latex] are whole numbers, then
| Product Property | [latex]a^m a^n=a^{m+n}[/latex] |
|---|---|
| Power Property | [latex]{(a^m)^n}=a^{mn}[/latex] |
| Product to a Power | [latex](ab)^m=a^m b^m[/latex] |
| Quotient Property | [latex]\frac{a^m}{b^m}=a^{m-n},a\neq{0} , m>n[/latex] |
| [latex]\frac{a^m}{a^n}=\frac{1}{a^{n-m}},a\neq{0},n>n=m[/latex] | |
| Zero Exponent Definition | [latex]a^{0}=1,a\neq{0}[/latex] |
| Quotient to a Power Property | [latex](\frac{a}{b})^m\frac{a^m}{b^m},b\neq{0}[/latex] |
After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?
