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Guias de estudo > Prealgebra

Using the Quotient Property

Learning Outcomes

  • Simplify a polynomial expression using the quotient property of exponents
  • Simplify expressions with exponents equal to zero
  • Simplify quotients raised to a power
 

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.

Summary of Exponent Properties for Multiplication

If a and ba\text{ and }b are real numbers and m and nm\text{ and }n are whole numbers, then Product Propertyam⋅an=am+nPower Property(am)n=am⋅nProduct to a Power(ab)m=ambm\begin{array}{cccc}\text{Product Property}\hfill & & & \hfill {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \text{Power Property}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \text{Product to a Power}\hfill & & & \hfill {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \end{array}
  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.

Equivalent Fractions Property

If a,b,ca,b,c are whole numbers where b≠0,c≠0b\ne 0,c\ne 0, then ab=a⋅cb⋅c and a⋅cb⋅c=ab\frac{a}{b}=\frac{a\cdot c}{b\cdot c}\text{ and }\frac{a\cdot c}{b\cdot c}=\frac{a}{b}
  As before, we'll try to discover a property by looking at some examples. Considerx5x2andx2x3What do they mean?x⋅x⋅x⋅x⋅xx⋅xx⋅xx⋅x⋅xUse the Equivalent Fractions Property.)x‾⋅)x‾⋅x⋅x⋅x)x‾⋅)x‾⋅1)x‾⋅)x‾⋅1)x‾⋅)x‾⋅xSimplify.x31x\begin{array}{cccccccccc}\text{Consider}\hfill & & & \hfill \frac{{x}^{5}}{{x}^{2}}\hfill & & & \text{and}\hfill & & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ \text{What do they mean?}\hfill & & & \hfill \frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}\hfill & & & & & & \hfill \frac{x\cdot x}{x\cdot x\cdot x}\hfill \\ \text{Use the Equivalent Fractions Property.}\hfill & & & \hfill \frac{\overline{)x}\cdot \overline{)x}\cdot x\cdot x\cdot x}{\overline{)x}\cdot \overline{)x}\cdot 1}\hfill & & & & & & \hfill \frac{\overline{)x}\cdot \overline{)x}\cdot 1}{\overline{)x}\cdot \overline{)x}\cdot x}\hfill \\ \text{Simplify.}\hfill & & & \hfill {x}^{3}\hfill & & & & & & \hfill \frac{1}{x}\hfill \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 11 in the denominator, which we simplified.
  • When the larger exponent was in the denominator, we were left with factors in the denominator, and 11 in the numerator, which could not be simplified.
We write: x5x2x2x3x5−21x3−2x31x\begin{array}{ccccc}\frac{{x}^{5}}{{x}^{2}}\hfill & & & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ {x}^{5 - 2}\hfill & & & & \hfill \frac{1}{{x}^{3 - 2}}\hfill \\ {x}^{3}\hfill & & & & \hfill \frac{1}{x}\hfill \end{array}

Quotient Property of Exponents

If aa is a real number, a≠0a\ne 0, and m,nm,n are whole numbers, then aman=am−n,m>n and aman=1an−m,n>m\frac{{a}^{m}}{{a}^{n}}={a}^{m-n},m>n\text{ and }\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},n>m
  A couple of examples with numbers may help to verify this property. 3432=?34−25253=?153−2819=?3225125=?1519=915=15\begin{array}{cccc}\frac{{3}^{4}}{{3}^{2}}\stackrel{?}{=}{3}^{4 - 2}\hfill & & & \hfill \frac{{5}^{2}}{{5}^{3}}\stackrel{?}{=}\frac{1}{{5}^{3 - 2}}\hfill \\ \frac{81}{9}\stackrel{?}{=}{3}^{2}\hfill & & & \hfill \frac{25}{125}\stackrel{?}{=}\frac{1}{{5}^{1}}\hfill \\ 9=9 \hfill & & & \hfill \frac{1}{5}=\frac{1}{5}\hfill \end{array} When we work with numbers and the exponent is less than or equal to 33, we will apply the exponent. When the exponent is greater than 33 , we leave the answer in exponential form.  

example

Simplify: 1. x10x8\frac{{x}^{10}}{{x}^{8}} 2. 2922\frac{{2}^{9}}{{2}^{2}} Solution To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
1.
Since 10 > 8, there are more factors of xx in the numerator. x10x8\frac{{x}^{10}}{{x}^{8}}
Use the quotient property with m>n,aman=am−nm>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n} . x10−8{x}^{\color{red}{10-8}}
Simplify. x2{x}^{2}
2.
Since 9 > 2, there are more factors of 2 in the numerator. 2922\frac{{2}^{9}}{{2}^{2}}
Use the quotient property with m>n,aman=am−nm>n,\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}. 29−2{2}^{\color{red}{9-2}}
Simplify. 27{2}^{7}
  Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

try it

[ohm_question]146219[/ohm_question]
   

example

Simplify: 1. b10b15\frac{{b}^{10}}{{b}^{15}} 2. 3335\frac{{3}^{3}}{{3}^{5}}

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

1.
Since 15>1015>10, there are more factors of bb in the denominator. b10b15\frac{{b}^{10}}{{b}^{15}}
Use the quotient property with n>m,aman=1an−mn>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}. 1b15−10\frac{\color{red}{1}}{{b}^{\color{red}{15-10}}}
Simplify. 1b5\frac{1}{{b}^{5}}
2.
Since 5>35>3, there are more factors of [/latex]3[/latex] in the denominator. 3335\frac{{3}^{3}}{{3}^{5}}
Use the quotient property with n>m,aman=1an−mn>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}. 135−3\frac{\color{red}{1}}{{3}^{\color{red}{5-3}}}
Simplify. 132\frac{1}{{3}^{2}}
Apply the exponent. 19\frac{1}{9}

  Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and 11 in the numerator.

try it

[ohm_question]146220[/ohm_question]
Now let's see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.

example

Simplify: 1. a5a9\frac{{a}^{5}}{{a}^{9}} 2. x11x7\frac{{x}^{11}}{{x}^{7}}

Answer: Solution

1.
Since 9>59>5, there are more aa 's in the denominator and so we will end up with factors in the denominator. a5a9\frac{{a}^{5}}{{a}^{9}}
Use the Quotient Property for n>m,aman=1an−mn>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}. 1a9−5\frac{\color{red}{1}}{{a}^{\color{red}{9-5}}}
Simplify. 1a4\frac{1}{{a}^{4}}
2.
Notice there are more factors of xx in the numerator, since 11 > 7. So we will end up with factors in the numerator. x11x7\frac{{x}^{11}}{{x}^{7}}
Use the Quotient Property for m>n,aman=an−mm>n,\frac{{a}^{m}}{{a}^{n}}={a}^{n-m}. x11−7{x}^{\color{red}{11-7}}
Simplify. x4{x}^{4}

 

try it

[ohm_question]146889[/ohm_question]
Watch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient. https://youtu.be/Jmf-CPhm3XM

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 amam\frac{{a}^{m}}{{a}^{m}}. From earlier work with fractions, we know that 22=11717=1−43−43=1\frac{2}{2}=1\frac{17}{17}=1\frac{-43}{-43}=1 In words, a number divided by itself is 11. So xx=1\frac{x}{x}=1, for any xx ( x≠0x\ne 0 ), since any number divided by itself is 11. The Quotient Property of Exponents shows us how to simplify aman\frac{{a}^{m}}{{a}^{n}} when m>nm>n and when n<mn<m by subtracting exponents. What if m=nm=n ? Now we will simplify amam\frac{{a}^{m}}{{a}^{m}} in two ways to lead us to the definition of the zero exponent. Consider first 88\frac{8}{8}, which we know is 11.
88=1\frac{8}{8}=1
Write 88 as 23{2}^{3} . 2323=1\frac{{2}^{3}}{{2}^{3}}=1
Subtract exponents. 23−3=1{2}^{3 - 3}=1
Simplify. 20=1{2}^{0}=1
. We see aman\frac{{a}^{m}}{{a}^{n}} simplifies to a a0{a}^{0} and to 11 . So a0=1{a}^{0}=1 .

Zero Exponent

If aa is a non-zero number, then a0=1{a}^{0}=1. Any nonzero number raised to the zero power is 11.
  In this text, we assume any variable that we raise to the zero power is not zero.  

example

Simplify: 1. 120{12}^{0} 2. y0{y}^{0}

Answer: Solution The definition says any non-zero number raised to the zero power is 11.

1.
120{12}^{0}
Use the definition of the zero exponent. 11
2.
y0{y}^{0}
Use the definition of the zero exponent. 11

 

try it

[ohm_question]146221[/ohm_question] [ohm_question]146890[/ohm_question]
  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 (2x)0{\left(2x\right)}^{0}. We can use the product to a power rule to rewrite this expression.
(2x)0{\left(2x\right)}^{0}
Use the Product to a Power Rule. 20x0{2}^{0}{x}^{0}
Use the Zero Exponent Property. 1⋅11\cdot 1
Simplify. 11
This tells us that any non-zero expression raised to the zero power is one.  

example

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

Answer: Solution

(7z)0{\left(7z\right)}^{0}
Use the definition of the zero exponent. 11

 

try it

[ohm_question]146222[/ohm_question]
Now let's compare the difference between the previous example, where the entire expression was raised to a zero exponent, and what happens when only one factor is raised to a zero exponent.

example

Simplify: 1. (−3x2y)0{\left(-3{x}^{2}y\right)}^{0} 2. −3x2y0-3{x}^{2}{y}^{0}

Answer: Solution

1.
The product is raised to the zero power. (−3x2y)0{\left(-3{x}^{2}y\right)}^{0}
Use the definition of the zero exponent. 11
2.
Notice that only the variable yy is being raised to the zero power. −3x2y0{-3{x}^{2}y}^{0}
Use the definition of the zero exponent. −3x2⋅1-3{x}^{2}\cdot 1
Simplify. −3x2-3{x}^{2}

Now you can try a similar problem to make sure you see the difference between raising an entire expression to a zero power and having only one factor raised to a zero power.

try it

[ohm_question]146223[/ohm_question] [ohm_question]146222[/ohm_question]
In the next video we show some different examples of how you can apply the zero exponent rule. https://youtu.be/zQJy1aBm1dQ

Simplify Quotients Raised to a Power

Now we will look at an example that will lead us to the Quotient to a Power Property.
(xy)3{\left(\frac{x}{y}\right)}^{3}
This means xy⋅xy⋅xy\frac{x}{y}\cdot \frac{x}{y}\cdot \frac{x}{y}
Multiply the fractions. x⋅x⋅xy⋅y⋅y\frac{x\cdot x\cdot x}{y\cdot y\cdot y}
Write with exponents. x3y3\frac{{x}^{3}}{{y}^{3}}
Notice that the exponent applies to both the numerator and the denominator. We see that (xy)3{\left(\frac{x}{y}\right)}^{3} is x3y3\frac{{x}^{3}}{{y}^{3}}. We write:(xy)3x3y3\begin{array}{ccccc}\text{We write:}\hfill & & & & {\left(\frac{x}{y}\right)}^{3}\hfill \\ & & & & \frac{{x}^{3}}{{y}^{3}}\hfill \end{array} This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property of Exponents

If aa and bb are real numbers, b≠0b\ne 0, and mm is a counting number, then (ab)m=ambm{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}} To raise a fraction to a power, raise the numerator and denominator to that power.
  An example with numbers may help you understand this property: 233[/latex]= [latex]2333[/latex]=[latex]827\frac{2}{3}^3[/latex] = [latex]\frac{{2}^{3}}{{3}^{3}}[/latex] = [latex]\frac{8}{27}  

example

Simplify: 1. (58)2{\left(\frac{5}{8}\right)}^{2} 2. (x3)4{\left(\frac{x}{3}\right)}^{4} 3. (ym)3{\left(\frac{y}{m}\right)}^{3} Solution
1.
(58)2(\frac{5}{8})^2
Use the Quotient to a Power Property, (ab)m=ambm{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}} . 5282\frac{5^{\color{red}{2}}}{8^{\color{red}{2}}}
Simplify. 2564\frac{25}{64}
2.
(x3)4(\frac{x}{3})^4
Use the Quotient to a Power Property, (ab)m=ambm{\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}} . x434\frac{x^{\color{red}{4}}}{3^{\color{red}{4}}}
Simplify. x481\frac{x^4}{81}
3.
(ym)3(\frac{y}{m})^3
Raise the numerator and denominator to the third power. y3m3\frac{y^{\color{red}{3}}}{m^{\color{red}{3}}}
 

try it

[ohm_question]146227[/ohm_question] [ohm_question]146891[/ohm_question] [ohm_question]146892[/ohm_question]  
For more examples of how to simplify a quotient raised to a power, watch the following video. https://youtu.be/BoBe31pRxFM  

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