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 [latex]a\text{ and }b[/latex] are real numbers and [latex]m\text{ and }n[/latex] are whole numbers, then [latex-display]\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}[/latex-display]Equivalent Fractions Property
If [latex]a,b,c[/latex] are whole numbers where [latex]b\ne 0,c\ne 0[/latex], then [latex-display]{\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}}[/latex-display][latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}[/latex]
You can rewrite the expression as: [latex] \displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}[/latex]. Then you can cancel the common factors of [latex]4[/latex] in the numerator and denominator: [latex] \displaystyle [/latex] Finally, this expression can be rewritten as [latex]4^{3}[/latex] using exponential notation. Notice that the exponent, [latex]3[/latex], is the difference between the two exponents in the original expression, [latex]5[/latex] and [latex]2[/latex]. So, [latex] \displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}[/latex]. Now, let's consider an example in which the base is the variable [latex]x[/latex]. [latex-display]\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}[/latex-display] 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 [latex]1[/latex] in the denominator, which we simplified.
- When the larger exponent was in the denominator, we were left with factors in the denominator, and [latex]1[/latex] in the numerator, which could not be simplified.
[latex]\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}[/latex]
Quotient Property of Exponents
If [latex]a[/latex] is a real number, [latex]a\ne 0[/latex], and [latex]m,n[/latex] are whole numbers, then [latex-display]{\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[/latex-display]example
Simplify: 1. [latex]\Large\frac{{x}^{10}}{{x}^{8}}[/latex] 2. [latex]\Large\frac{{2}^{9}}{{2}^{2}}[/latex] 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 [latex]x[/latex] in the numerator. | [latex]\Large\frac{{x}^{10}}{{x}^{8}}[/latex] |
| Use the quotient property with [latex]m>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={a}^{m-n}[/latex] . | [latex]{x}^{\color{red}{10-8}}[/latex] |
| Simplify. | [latex]{x}^{2}[/latex] |
| 2. | |
| Since 9 > 2, there are more factors of 2 in the numerator. | [latex]\Large\frac{{2}^{9}}{{2}^{2}}[/latex] |
| Use the quotient property with [latex]m>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={a}^{m-n}[/latex]. | [latex]{2}^{\color{red}{9-2}}[/latex] |
| Simplify. | [latex]{2}^{7}[/latex] |
try it
[ohm_question]146219[/ohm_question]example
Simplify: 1. [latex]\Large\frac{{b}^{10}}{{b}^{15}}[/latex] 2. [latex]\Large\frac{{3}^{3}}{{3}^{5}}[/latex]Answer: Solution To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.
| 1. | |
| Since [latex]15>10[/latex], there are more factors of [latex]b[/latex] in the denominator. | [latex]\Large\frac{{b}^{10}}{{b}^{15}}[/latex] |
| Use the quotient property with [latex]n>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}[/latex]. | [latex]\Large\frac{\color{red}{1}}{{b}^{\color{red}{15-10}}}[/latex] |
| Simplify. | [latex]\Large\frac{1}{{b}^{5}}[/latex] |
| 2. | |
| Since [latex]5>3[/latex], there are more factors of [latex]3[/latex] in the denominator. | [latex]\Large\frac{{3}^{3}}{{3}^{5}}[/latex] |
| Use the quotient property with [latex]n>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}[/latex]. | [latex]\Large\frac{\color{red}{1}}{{3}^{\color{red}{5-3}}}[/latex] |
| Simplify. | [latex]\Large\frac{1}{{3}^{2}}[/latex] |
| Apply the exponent. | [latex]\Large\frac{1}{9}[/latex] |
try it
[ohm_question]146220[/ohm_question]example
Simplify: 1. [latex]\Large\frac{{a}^{5}}{{a}^{9}}[/latex] 2. [latex]\Large\frac{{x}^{11}}{{x}^{7}}[/latex]Answer: Solution
| 1. | |
| 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. | [latex]\Large\frac{{a}^{5}}{{a}^{9}}[/latex] |
| Use the Quotient Property for [latex]n>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}[/latex]. | [latex]\Large\frac{\color{red}{1}}{{a}^{\color{red}{9-5}}}[/latex] |
| Simplify. | [latex]\Large\frac{1}{{a}^{4}}[/latex] |
| 2. | |
| Notice there are more factors of [latex]x[/latex] in the numerator, since 11 > 7. So we will end up with factors in the numerator. | [latex]\Large\frac{{x}^{11}}{{x}^{7}}[/latex] |
| Use the Quotient Property for [latex]m>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={a}^{n-m}[/latex]. | [latex]{x}^{\color{red}{11-7}}[/latex] |
| Simplify. | [latex]{x}^{4}[/latex] |
try it
[ohm_question]146889[/ohm_question]Example
Simplify. [latex] \displaystyle \frac{12{{x}^{4}}}{2x}[/latex]Answer: Separate into numerical and variable factors.[latex] \displaystyle \left( \frac{12}{2} \right)\left( \frac{{{x}^{4}}}{x} \right)[/latex] 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. [latex-display] \displaystyle 6\left( {{x}^{4-1}} \right)[/latex-display] Answer [latex-display] \frac{12{{x}^{4}}}{2x}=6{{x}^{3}}[/latex-display]
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 [latex] \displaystyle \frac{3}{4}[/latex] and raise it to the [latex]3<sup>rd</sup>[/latex]power.[latex] \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}}}[/latex]
You can see that raising the quotient to the power of [latex]3[/latex] can also be written as the numerator [latex](3)[/latex] to the power of [latex]3[/latex], and the denominator [latex](4)[/latex] to the power of [latex]3[/latex]. 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.| [latex]{\left(\Large\frac{x}{y}\normalsize\right)}^{3}[/latex] | |
| This means | [latex]\Large\frac{x}{y}\normalsize\cdot\Large\frac{x}{y}\normalsize\cdot \Large\frac{x}{y}[/latex] |
| Multiply the fractions. | [latex]\Large\frac{x\cdot x\cdot x}{y\cdot y\cdot y}[/latex] |
| Write with exponents. | [latex]\Large\frac{{x}^{3}}{{y}^{3}}[/latex] |
Quotient to a Power Property of Exponents
If [latex]a[/latex] and [latex]b[/latex] are real numbers, [latex]b\ne 0[/latex], and [latex]m[/latex] is a counting number, then [latex-display]{\left(\Large\frac{a}{b}\normalsize\right)}^{m}=\Large\frac{{a}^{m}}{{b}^{m}}[/latex-display] To raise a fraction to a power, raise the numerator and denominator to that power.example
Simplify: 1. [latex]{\left(\Large\frac{5}{8}\normalsize\right)}^{2}[/latex] 2. [latex]{\left(\Large\frac{x}{3}\normalsize\right)}^{4}[/latex] 3. [latex]{\left(\Large\frac{y}{m}\normalsize\right)}^{3}[/latex]Answer: Solution
| 1. | |
| [latex]\Large(\frac{5}{8})^2[/latex] | |
| Use the Quotient to a Power Property, [latex]{\Large\left(\frac{a}{b}\right)}^{m}\normalsize =\Large\frac{{a}^{m}}{{b}^{m}}[/latex] . | [latex]\Large\frac{5^{\color{red}{2}}}{8^{\color{red}{2}}}[/latex] |
| Simplify. | [latex]\Large\frac{25}{64}[/latex] |
| 2. | |
| [latex]\Large(\frac{x}{3})^4[/latex] | |
| Use the Quotient to a Power Property, [latex]{\Large\left(\frac{a}{b}\right)}^{m}\normalsize =\Large\frac{{a}^{m}}{{b}^{m}}[/latex] . | [latex]\Large\frac{x^{\color{red}{4}}}{3^{\color{red}{4}}}[/latex] |
| Simplify. | [latex]\Large\frac{x^4}{81}[/latex] |
| 3. | |
| [latex]\Large(\frac{y}{m})^3[/latex] | |
| Raise the numerator and denominator to the third power. | [latex]\Large\frac{y^{3}}{m^{3}}[/latex] |
try it
[ohm_question]146227[/ohm_question] [ohm_question]146891[/ohm_question] [ohm_question]146892[/ohm_question]Example
Simplify. [latex] \displaystyle {{\left( \frac{2{x}^{2}y}{x} \right)}^{3}}[/latex]Answer: Apply the power to each factor individually.
[latex] \displaystyle \frac{{{2}^{3}{\left({x}^{2}\right)}^{3}{y}^{3}}}{{{x}^{3}}}[/latex]
Separate into numerical and variable factors.[latex] \displaystyle {{2}^{3}}\cdot \frac{{{x}^{3\cdot2}}}{{{x}^{3}}}\cdot \frac{{{y}^{3}}}{1}[/latex]
Simplify by taking [latex]2[/latex] to the third power and applying the Power and Quotient Rules for exponents—multiply and subtract the exponents of matching variables.[latex] \displaystyle 8\cdot {{x}^{(6-3)}}\cdot {{y}^{3}}[/latex]
Simplify.[latex] \displaystyle 8{{x}^{3}}{{y}^{3}}[/latex]
Answer
[latex-display] \displaystyle {{\left( \frac{2{x}^{2}y}{x} \right)}^{3}}=8{{x}^{3}}{{y}^{3}}[/latex-display]Contribute!
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- Question ID: 146892, 146891, 146227, 146222, 146223, 146890, 146221, 146889, 146220, 146219 . Authored by: Alyson Day. License: CC BY: Attribution. License terms: IMathAS Community License CC-BY + GPL.
CC licensed content, Shared previously
- Ex: Simplify Exponential Expressions Using the Quotient Property of Exponents. Authored by: Lumen Learning. License: CC BY: Attribution.
- Ex 3: Exponent Properties (Zero Exponent). Provided by: q Authored by: Lumen Learning. License: CC BY: Attribution.
- Simplify Expressions Using Exponent Rules (Power of a Quotient). Authored by: James Sousa (mathispower4u.com). License: CC BY: Attribution.
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