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Guias de estudo > ALGEBRA / TRIG I

Simplifying Variable Expressions Using Exponent Properties II

Learning Outcomes

  • Simplify expressions using the Quotient Property of Exponents

Simplify Expressions Using the Quotient Property of Exponents

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

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{\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}\text{ and }{\Large\frac{a\cdot c}{b\cdot c}}={\Large\frac{a}{b}}
As before, we'll try to discover a property by looking at some examples. Let’s look at dividing terms containing exponential expressions. What happens if you divide two numbers in exponential form with the same base? Consider the following expression.

4542 \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}

You can rewrite the expression as: 4⋅4⋅4⋅4⋅44⋅4 \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}. Then you can cancel the common factors of 44 in the numerator and denominator: \displaystyle Finally, this expression can be rewritten as 434^{3} using exponential notation. Notice that the exponent, 33, is the difference between the two exponents in the original expression, 55 and 22. So, 4542=45−2=43 \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}. Now, let's consider an example in which the base is the variable xx. 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 {\Large\frac{{x}^{5}}{{x}^{2}}}\hfill & & & \text{and}\hfill & & & \hfill {\Large\frac{{x}^{2}}{{x}^{3}}}\hfill \\ \text{What do they mean?}\hfill & & & \hfill {\Large\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}}\hfill & & & & & & \hfill {\Large\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 {\Large\frac{1}{x}}\hfill \end{array} Notice that in each case the bases were the same and we subtracted the exponents.  So, to divide two exponential terms with the same base, subtract 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{\Large\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n\text{ and }{\Large\frac{{a}^{m}}{{a}^{n}}}={\Large\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\Large\frac{{x}^{10}}{{x}^{8}} 2. 2922\Large\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\Large\frac{{x}^{10}}{{x}^{8}}
Use the quotient property with m>n,aman=am−nm>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={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\Large\frac{{2}^{9}}{{2}^{2}}
Use the quotient property with m>n,aman=am−nm>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={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\Large\frac{{b}^{10}}{{b}^{15}} 2. 3335\Large\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\Large\frac{{b}^{10}}{{b}^{15}}
Use the quotient property with n>m,aman=1an−mn>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}. 1b15−10\Large\frac{\color{red}{1}}{{b}^{\color{red}{15-10}}}
Simplify. 1b5\Large\frac{1}{{b}^{5}}
2.
Since 5>35>3, there are more factors of 33 in the denominator. 3335\Large\frac{{3}^{3}}{{3}^{5}}
Use the quotient property with n>m,aman=1an−mn>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}. 135−3\Large\frac{\color{red}{1}}{{3}^{\color{red}{5-3}}}
Simplify. 132\Large\frac{1}{{3}^{2}}
Apply the exponent. 19\Large\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\Large\frac{{a}^{5}}{{a}^{9}} 2. x11x7\Large\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\Large\frac{{a}^{5}}{{a}^{9}}
Use the Quotient Property for n>m,aman=1an−mn>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}. 1a9−5\Large\frac{\color{red}{1}}{{a}^{\color{red}{9-5}}}
Simplify. 1a4\Large\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\Large\frac{{x}^{11}}{{x}^{7}}
Use the Quotient Property for m>n,aman=an−mm>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={a}^{n-m}. x11−7{x}^{\color{red}{11-7}}
Simplify. x4{x}^{4}

 

try it

[ohm_question]146889[/ohm_question]
When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.

Example

Simplify. 12x42x \displaystyle \frac{12{{x}^{4}}}{2x}

Answer: Separate into numerical and variable factors.(122)(x4x) \displaystyle \left( \frac{12}{2} \right)\left( \frac{{{x}^{4}}}{x} \right) Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables. 6(x4−1) \displaystyle 6\left( {{x}^{4-1}} \right) Answer 12x42x=6x3 \frac{12{{x}^{4}}}{2x}=6{{x}^{3}}

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 Quotients Raised to a Power

Now we will look at an example that will lead us to the Quotient to a Power Property. Now let’s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have 34 \displaystyle \frac{3}{4} and raise it to the 3<sup>rd</sup>3<sup>rd</sup>power.

(34)3=(34)(34)(34)=3⋅3⋅34⋅4⋅4=3343 \displaystyle {{\left( \frac{3}{4} \right)}^{3}}=\left( \frac{3}{4} \right)\left( \frac{3}{4} \right)\left( \frac{3}{4} \right)=\frac{3\cdot 3\cdot 3}{4\cdot 4\cdot 4}=\frac{{{3}^{3}}}{{{4}^{3}}}

You can see that raising the quotient to the power of 33 can also be written as the numerator (3)(3) to the power of 33, and the denominator (4)(4) to the power of 33. Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.
(xy)3{\left(\Large\frac{x}{y}\normalsize\right)}^{3}
This means xy⋅xy⋅xy\Large\frac{x}{y}\normalsize\cdot\Large\frac{x}{y}\normalsize\cdot \Large\frac{x}{y}
Multiply the fractions. x⋅x⋅xy⋅y⋅y\Large\frac{x\cdot x\cdot x}{y\cdot y\cdot y}
Write with exponents. x3y3\Large\frac{{x}^{3}}{{y}^{3}}
Notice that the exponent applies to both the numerator and the denominator.  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(\Large\frac{a}{b}\normalsize\right)}^{m}=\Large\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:

example

Simplify: 1. (58)2{\left(\Large\frac{5}{8}\normalsize\right)}^{2} 2. (x3)4{\left(\Large\frac{x}{3}\normalsize\right)}^{4} 3. (ym)3{\left(\Large\frac{y}{m}\normalsize\right)}^{3}

Answer: Solution

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

try it

[ohm_question]146227[/ohm_question] [ohm_question]146891[/ohm_question] [ohm_question]146892[/ohm_question]

Example

Simplify. (2x2yx)3 \displaystyle {{\left( \frac{2{x}^{2}y}{x} \right)}^{3}}

Answer: Apply the power to each factor individually.

23(x2)3y3x3 \displaystyle \frac{{{2}^{3}{\left({x}^{2}\right)}^{3}{y}^{3}}}{{{x}^{3}}}

Separate into numerical and variable factors.

23⋅x3⋅2x3⋅y31 \displaystyle {{2}^{3}}\cdot \frac{{{x}^{3\cdot2}}}{{{x}^{3}}}\cdot \frac{{{y}^{3}}}{1}

Simplify by taking 22 to the third power and applying the Power and Quotient Rules for exponents—multiply and subtract the exponents of matching variables.

8⋅x(6−3)⋅y3 \displaystyle 8\cdot {{x}^{(6-3)}}\cdot {{y}^{3}}

Simplify.

8x3y3 \displaystyle 8{{x}^{3}}{{y}^{3}}

Answer

(2x2yx)3=8x3y3 \displaystyle {{\left( \frac{2{x}^{2}y}{x} \right)}^{3}}=8{{x}^{3}}{{y}^{3}}

For more examples of how to simplify a quotient raised to a power, watch the following video. https://youtu.be/BoBe31pRxFM In the following video you will be shown examples of simplifying quotients that are raised to a power. https://youtu.be/ZbxgDRV35dE

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