Mathematics » Polynomials II » Divide Monomials

# 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 Expressions

Dividing Monomials

### Optional Video:

You can also watch this optional video on dividing monomials.