Dividing Monomials
Dividing 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
Find the quotient: \(56{x}^{7}÷8{x}^{3}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}56{x}^{7}÷8{x}^{3}\hfill \\ \text{Rewrite as a fraction.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{56{x}^{7}}{8{x}^{3}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{56}{8}\cdot \frac{{x}^{7}}{{x}^{3}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}7{x}^{4}\hfill \end{array}\)
Example
Find the quotient: \(\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}.\)
Solution
When we divide monomials with more than one variable, we write one fraction for each variable.
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{45}{-5}·\frac{{a}^{2}}{a}·\frac{{b}^{3}}{{b}^{5}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}-9·a·\frac{1}{{b}^{2}}\hfill \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}-\frac{9a}{{b}^{2}}\hfill \end{array}\)
Example
Find the quotient: \(\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{24{a}^{5}{b}^{3}}{48a{b}^{4}}\hfill \\ \text{Use fraction multiplication.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{24}{48}·\frac{{a}^{5}}{a}·\frac{{b}^{3}}{{b}^{4}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{1}{2}·{a}^{4}·\frac{1}{b}\hfill \\ \text{Multiply.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{{a}^{4}}{2b}\hfill \end{array}\)
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.
Example
Find the quotient: \(\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.\)
Solution
Be very careful to simplify \(\frac{14}{21}\) by dividing out a common factor, and to simplify the variables by subtracting their exponents.
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}\hfill \\ \text{Simplify and use the Quotient Property.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{2{y}^{6}}{3{x}^{4}}\hfill \end{array}\)
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
Find the quotient: \(\frac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}.\)
Solution
\(\begin{array}{cccc}& & & \hfill \phantom{\rule{4em}{0ex}}\frac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}\hfill \\ \text{Simplify the numerator.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}\frac{30{x}^{5}{y}^{5}}{3{x}^{4}{y}^{5}}\hfill \\ \text{Simplify.}\hfill & & & \hfill \phantom{\rule{4em}{0ex}}10x\hfill \end{array}\)
Resources:
Access these online resources for additional instruction and practice with dividing monomials:
Rational ExpressionsOptional Video:
You can also watch this optional video on dividing monomials.
This lesson is part of:
Polynomials II