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Study Guides > Prealgebra

Introduction to Dividing Monomials

Divide Monomials

Learning Outcomes

By the end of this section, you will be able to:
  • 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
 

readiness quiz

Before you get started, take this readiness quiz.
  1. Simplify: [latex]\frac{8}{24}[/latex].If you missed the problem, review [link].
  2. Simplify: [latex]{\left(2{m}^{3}\right)}^{5}[/latex].If you missed the problem, review [link].
  3. Simplify: [latex]\frac{12{x}^{}}{12y}[/latex].If you missed the problem, review [link].
 
   

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. Find the quotient: [latex]56{x}^{5}\div 7{x}^{2}[/latex]. Solution
[latex]56{x}^{5}\div 7{x}^{2}[/latex]
Rewrite as a fraction. [latex]\frac{56{x}^{5}}{7{x}^{2}}[/latex]
Use fraction multiplication to separate the number part from the variable part. [latex]\frac{56}{7}\cdot \frac{{x}^{5}}{{x}^{2}}[/latex]
Use the Quotient Property. [latex]8{x}^{3}[/latex]
Find the quotient: [latex]63{x}^{8}\div 9{x}^{4}[/latex]. [latex]7{x}^{4}[/latex]  Find the quotient: [latex]96{y}^{11}\div 6{y}^{8}[/latex]. [latex-display]16{y}^{3}[/latex-display] When we divide monomials with more than one variable, we write one fraction for each variable. Find the quotient: [latex]\frac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}[/latex]. Solution
[latex]\frac{42{x}^{2}{y}^{3}}{-7x{y}^{5}}[/latex]
Use fraction multiplication. [latex]\frac{42}{-7}\cdot \frac{{x}^{2}}{x}\cdot \frac{{y}^{3}}{{y}^{5}}[/latex]
Simplify and use the Quotient Property. [latex]-6\cdot x\cdot \frac{1}{{y}^{2}}[/latex]
Multiply. [latex]-\frac{6x}{{y}^{2}}[/latex]
Find the quotient: [latex]\frac{-84{x}^{8}{y}^{3}}{7{x}^{10}{y}^{2}}[/latex]. [latex-display]-\frac{12y}{{x}^{2}}[/latex-display] Find the quotient: [latex]\frac{-72{a}^{4}{b}^{5}}{-8{a}^{9}{b}^{5}}[/latex]. [latex-display]\frac{9}{{a}^{5}}[/latex-display] Find the quotient: [latex]\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}[/latex]. Solution
[latex]\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}[/latex]
Use fraction multiplication. [latex]\frac{24}{48}\cdot \frac{{a}^{5}}{a}\cdot \frac{{b}^{3}}{{b}^{4}}[/latex]
Simplify and use the Quotient Property. [latex]\frac{1}{2}\cdot {a}^{4}\cdot \frac{1}{b}[/latex]
Multiply. [latex]\frac{{a}^{4}}{2b}[/latex]
Find the quotient: [latex]\frac{16{a}^{7}{b}^{6}}{24a{b}^{8}}[/latex]. [latex-display]\frac{2{a}^{6}}{3{b}^{2}}[/latex-display] Find the quotient: [latex]\frac{27{p}^{4}{q}^{7}}{-45{p}^{12}{q}^{}}[/latex]. [latex-display]-\frac{3{q}^{6}}{5{p}^{8}}[/latex-display] 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. Find the quotient: [latex]\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}[/latex]. Solution
[latex]\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}[/latex]
Simplify and use the Quotient Property. [latex]\frac{2{y}^{6}}{3{x}^{4}}[/latex]
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. Find the quotient: [latex]\frac{28{x}^{5}{y}^{14}}{49{x}^{9}{y}^{12}}[/latex]. [latex-display]\frac{4{y}^{2}}{7{x}^{4}}[/latex-display] Find the quotient: [latex]\frac{30{m}^{5}{n}^{11}}{48{m}^{10}{n}^{14}}[/latex]. [latex-display]\frac{5}{8{m}^{5}{n}^{3}}[/latex-display] 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. Find the quotient: [latex]\frac{\left(3{x}^{3}{y}^{2}\right)\left(10{x}^{2}{y}^{3}\right)}{6{x}^{4}{y}^{5}}[/latex]. Solution Remember, the fraction bar is a grouping symbol. We will simplify the numerator first.
[latex]\frac{\left(3{x}^{3}{y}^{2}\right)\left(10{x}^{2}{y}^{3}\right)}{6{x}^{4}{y}^{5}}[/latex]
Simplify the numerator. [latex]\frac{30{x}^{5}{y}^{5}}{6{x}^{4}{y}^{5}}[/latex]
Simplify, using the Quotient Rule. [latex]5x[/latex]
Find the quotient: [latex]\frac{\left(3{x}^{4}{y}^{5}\right)\left(8{x}^{2}{y}^{5}\right)}{12{x}^{5}{y}^{8}}[/latex]. [latex-display]2{xy}^{2}[/latex-display] Find the quotient: [latex]\frac{\left(-6{a}^{6}{b}^{9}\right)\left(-8{a}^{5}{b}^{8}\right)}{-12{a}^{10}{b}^{12}}[/latex]. [latex-display]-4{ab}^{5}[/latex-display]    
 

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