Divide Monomials

6.5 Divide Monomials

Learning Objectives

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

Simplify expressions using the Quotient Property for Exponents
Simplify expressions with zero exponents
Simplify expressions using the quotient to a Power Property
Simplify expressions by applying several properties
Divide monomials

Be Prepared 6.9

Before you get started, take this readiness quiz.

Simplify: $\frac{8}{24} .$
If you missed this problem, review Example 1.65.

Be Prepared 6.10

Simplify: ${(2 m^{3})}^{5} .$
If you missed this problem, review Example 6.23.

Be Prepared 6.11

Simplify: $\frac{12 x}{12 y} .$
If you missed this problem, review Example 1.67.

Simplify Expressions Using the Quotient Property for Exponents

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

Summary of Exponent Properties for Multiplication

If $a and b$ are real numbers, and $m and n$ are whole numbers, then

Product Property	$a^{m} \cdot a^{n} = a^{m + n}$
Power Property	${(a^{m})}^{n} = a^{m \cdot n}$
Product to a Power	${(a b)}^{m} = a^{m} b^{m}$

This table presents fundamental properties of exponents, including product, power, and product-to-a-power rules, along with their corresponding mathematical formulas.

Now we will look at the exponent properties for division. A quick memory refresher may help before we get started. You have learned to simplify fractions by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property. This property will also help you work with algebraic fractions—which are also quotients.

Equivalent Fractions Property

If $a, b, and c$ are whole numbers where $b \neq 0, c \neq 0$ ,

then \frac{a}{b} = \frac{a \cdot c}{b \cdot c} and \frac{a \cdot c}{b \cdot c} = \frac{a}{b}

As before, we’ll try to discover a property by looking at some examples.

Consider	$\frac{x^{5}}{x^{2}}$	and	$\frac{x^{2}}{x^{3}}$
What do they mean?	$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$		$\frac{x \cdot x}{x \cdot x \cdot x}$
Use the Equivalent Fractions Property.	$\frac{x \cdot x \cdot x \cdot x \cdot x}{x \cdot x}$		$\frac{x \cdot x \cdot 1}{x \cdot x \cdot x}$
Simplify.	$x^{3}$		$\frac{1}{x}$

Step-by-step simplification of rational expressions with exponents using the Equivalent Fractions Property.

Notice, in each case the bases were the same and we subtracted exponents.

When the larger exponent was in the numerator, we were left with factors in the numerator.

When the larger exponent was in the denominator, we were left with factors in the denominator—notice the numerator of 1.

We write:

\begin{matrix} \frac{x^{5}}{x^{2}} & \frac{x^{2}}{x^{3}} \\ x^{5 - 2} & \frac{1}{x^{3 - 2}} \\ x^{3} & \frac{1}{x} \end{matrix}

This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

If $a$ is a real number, $a \neq 0$ , and $m and n$ are whole numbers, then

\frac{a^{m}}{a^{n}} = a^{m - n}, m > n and \frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}, n > m

A couple of examples with numbers may help to verify this property.

\begin{matrix} \frac{3^{4}}{3^{2}} & = & 3^{4 - 2} & \frac{5^{2}}{5^{3}} & = & \frac{1}{5^{3 - 2}} \\ \frac{81}{9} & = & 3^{2} & \frac{25}{125} & = & \frac{1}{5^{1}} \\ 9 & = & 9 ✓ & \frac{1}{5} & = & \frac{1}{5} ✓ \end{matrix}

Example 6.59

Simplify: ⓐ $\frac{x^{9}}{x^{7}}$ ⓑ $\frac{3^{10}}{3^{2}} .$

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

ⓐ

Since 9 > 7, there are more factors of x in the numerator.

Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n}$ .

Simplify.
ⓑ

Since 10 > 2, there are more factors of x in the numerator.

Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n}$ .

Simplify.

Notice that when the larger exponent is in the numerator, we are left with factors in the numerator.

Try It 6.117

Simplify: ⓐ $\frac{x^{15}}{x^{10}}$ ⓑ $\frac{6^{14}}{6^{5}} .$

Try It 6.118

Simplify: ⓐ $\frac{y^{43}}{y^{37}}$ ⓑ $\frac{10^{15}}{10^{7}} .$

Example 6.60

Simplify: ⓐ $\frac{b^{8}}{b^{12}}$ ⓑ $\frac{7^{3}}{7^{5}} .$

Solution

To simplify an expression with a quotient, we need to first compare the exponents in the numerator and denominator.

ⓐ

Since 12 > 8, there are more factors of b in the denominator.

Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .

Simplify.
ⓑ

Since 5 > 3, there are more factors of 3 in the denominator.

Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .

Simplify.

Simplify. $The fraction one forty-ninth is displayed, represented as 1 over 49 with a horizontal line separating the numerator and the denominator.$

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator.

Try It 6.119

Simplify: ⓐ $\frac{x^{18}}{x^{22}}$ ⓑ $\frac{12^{15}}{12^{30}} .$

Try It 6.120

Simplify: ⓐ $\frac{m^{7}}{m^{15}}$ ⓑ $\frac{9^{8}}{9^{19}} .$

Notice the difference in the two previous examples:

If we start with more factors in the numerator, we will end up with factors in the numerator.
If we start with more factors in the denominator, we will end up with factors in the denominator.

The first step in simplifying an expression using the Quotient Property for Exponents is to determine whether the exponent is larger in the numerator or the denominator.

Example 6.61

Simplify: ⓐ $\frac{a^{5}}{a^{9}}$ ⓑ $\frac{x^{11}}{x^{7}} .$

Solution

ⓐ Is the exponent of $a$ larger in the numerator or denominator? Since 9 > 5, there are more $a' s$ in the denominator and so we will end up with factors in the denominator.

Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .

Simplify.
ⓑ Notice there are more factors of $x$ in the numerator, since 11 > 7. So we will end up with factors in the numerator.

Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .

Simplify.

Try It 6.121

Simplify: ⓐ $\frac{b^{19}}{b^{11}}$ ⓑ $\frac{z^{5}}{z^{11}} .$

Try It 6.122

Simplify: ⓐ $\frac{p^{9}}{p^{17}}$ ⓑ $\frac{w^{13}}{w^{9}} .$

Simplify Expressions with an Exponent of Zero

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as an expression like $\frac{a^{m}}{a^{m}}$ . From your earlier work with fractions, you know that:

\frac{2}{2} = 1 \frac{17}{17} = 1 \frac{−43}{−43} = 1

In words, a number divided by itself is 1. So, $\frac{x}{x} = 1$ , for any $x (x \neq 0)$ , since any number divided by itself is 1.

The Quotient Property for Exponents shows us how to simplify $\frac{a^{m}}{a^{n}}$ when $m > n$ and when $n < m$ by subtracting exponents. What if $m = n$ ?

Consider $\frac{8}{8}$ , which we know is 1.

	$\begin{matrix} \frac{8}{8} = 1 \end{matrix}$
Write $8$ as $2^{3}$ .	$\begin{matrix} \frac{2^{3}}{2^{3}} = 1 \end{matrix}$
Subtract exponents.	$\begin{matrix} 2^{3 - 3} = 1 \end{matrix}$
Simplify.	$\begin{matrix} 2^{0} = 1 \end{matrix}$

This table illustrates the derivation of the zero exponent rule, showing step-by-step how any non-zero number raised to the power of zero equals one.

Now we will simplify $\frac{a^{m}}{a^{m}}$ in two ways to lead us to the definition of the zero exponent. In general, for $a \neq 0$ :

$This figure is divided into two columns. At the top of the figure, the left and right columns both contain a to the m power divided by a to the m power. In the next row, the left column contains a to the m minus m power. The right column contains the fraction m factors of a divided by m factors of a, represented in the numerator and denominator by a times a followed by an ellipsis. All the as in the numerator and denominator are canceled out. In the bottom row, the left column contains a to the zero power. The right column contains 1.$

We see $\frac{a^{m}}{a^{m}}$ simplifies to $a^{0}$ and to 1. So $a^{0} = 1$ .

Zero Exponent

If $a$ is a non-zero number, then $a^{0} = 1$ .

Any nonzero number raised to the zero power is 1.

In this text, we assume any variable that we raise to the zero power is not zero.

Example 6.62

Simplify: ⓐ $9^{0}$ ⓑ $n^{0} .$

Solution

The definition says any non-zero number raised to the zero power is 1.

ⓐ Use the definition of the zero exponent.	$\begin{matrix} 9^{0} \\ 1 \end{matrix}$
ⓑ Use the definition of the zero exponent.	$\begin{matrix} n^{0} \\ 1 \end{matrix}$

This table demonstrates the zero exponent rule with examples, showing that any non-zero base raised to the power of zero equals one.

Try It 6.123

Simplify: ⓐ $15^{0}$ ⓑ $m^{0} .$

Try It 6.124

Simplify: ⓐ $k^{0}$ ⓑ $29^{0} .$

Now that we have defined the zero exponent, we can expand all the Properties of Exponents to include whole number exponents.

What about raising an expression to the zero power? Let’s look at ${(2 x)}^{0}$ . We can use the product to a power rule to rewrite this expression.

	${(2 x)}^{0}$
Use the product to a power rule.	$2^{0} x^{0}$
Use the zero exponent property.	$1 \cdot 1$
Simplify.	$1$

Table 6.1 Step-by-step simplification of (2x)^0, illustrating the product to a power rule and zero exponent property.

This tells us that any nonzero expression raised to the zero power is one.

Example 6.63

Simplify: ⓐ ${(5 b)}^{0}$ ⓑ ${(−4 a^{2} b)}^{0} .$

Solution

ⓐ	${(5 b)}^{0}$
Use the definition of the zero exponent.	$1$
ⓑ	${(−4 a^{2} b)}^{0}$
Use the definition of the zero exponent.	$1$

Examples demonstrating the zero exponent rule, where any non-zero base raised to the power of zero equals one.

Try It 6.125

Simplify: ⓐ ${(11 z)}^{0}$ ⓑ ${(−11 p q^{3})}^{0} .$

Try It 6.126

Simplify: ⓐ ${(−6 d)}^{0}$ ⓑ ${(−8 m^{2} n^{3})}^{0} .$

Simplify Expressions Using the Quotient to a Power Property

Now we will look at an example that will lead us to the Quotient to a Power Property.

	${(\frac{x}{y})}^{3}$
This means:	$\frac{x}{y} \cdot \frac{x}{y} \cdot \frac{x}{y}$
Multiply the fractions.	$\frac{x \cdot x \cdot x}{y \cdot y \cdot y}$
Write with exponents.	$\frac{x^{3}}{y^{3}}$

Steps demonstrating how to expand and simplify a fraction raised to a power, showing (x/y)^3 equals x^3/y^3.

Notice that the exponent applies to both the numerator and the denominator.

We write:	${(\frac{x}{y})}^{3}$
	$\frac{x^{3}}{y^{3}}$

Illustration of the power of a quotient rule: (x/y)3 = x3/y3.

This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property for Exponents

If $a$ and $b$ are real numbers, $b \neq 0$ , and $m$ is a counting number, then

{(\frac{a}{b})}^{m} = \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:

\begin{matrix} {(\frac{2}{3})}^{3} & = & \frac{2^{3}}{3^{3}} \\ \frac{2}{3} \cdot \frac{2}{3} \cdot \frac{2}{3} & = & \frac{8}{27} \\ \frac{8}{27} & = & \frac{8}{27} ✓ \end{matrix}

Example 6.64

Simplify: ⓐ ${(\frac{3}{7})}^{2}$ ⓑ ${(\frac{b}{3})}^{4}$ ⓒ ${(\frac{k}{j})}^{3} .$

Solution

ⓐ


Use the Quotient Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$ .
Simplify.

ⓑ


Use the Quotient Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$ .
Simplify.

ⓒ


Raise the numerator and denominator to the third power.

Try It 6.127

Simplify: ⓐ ${(\frac{5}{8})}^{2}$ ⓑ ${(\frac{p}{10})}^{4}$ ⓒ ${(\frac{m}{n})}^{7} .$

Try It 6.128

Simplify: ⓐ ${(\frac{1}{3})}^{3}$ ⓑ ${(\frac{−2}{q})}^{3}$ ⓒ ${(\frac{w}{x})}^{4} .$

Simplify Expressions by Applying Several Properties

We’ll now summarize all the properties of exponents so they are all together to refer to as we simplify expressions using several properties. Notice that they are now defined for whole number exponents.

Summary of Exponent Properties

If $a and b$ are real numbers, and $m and n$ are whole numbers, then

Product Property	$a^{m} \cdot a^{n} = a^{m + n}$
Power Property	${(a^{m})}^{n} = a^{m \cdot n}$
Product to a Power	${(a b)}^{m} = a^{m} b^{m}$
Quotient Property	$\begin{matrix} \frac{a^{m}}{a^{n}} & = & a^{m - n}, a \neq 0, m > n \\ \frac{a^{m}}{a^{n}} & = & \frac{1}{a^{n - m}}, a \neq 0, n > m \end{matrix}$
Zero Exponent Definition	$a^{o} = 1, a \neq 0$
Quotient to a Power Property	${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0$

This table summarizes fundamental exponent properties, including product, power, quotient, and zero exponent rules, presented with their corresponding mathematical formulas.

Example 6.65

Simplify: $\frac{{(y^{4})}^{2}}{y^{6}} .$

Solution

	$\frac{{(y^{4})}^{2}}{y^{6}}$
Multiply the exponents in the numerator.	$\frac{y^{8}}{y^{6}}$
Subtract the exponents.	$y^{2}$

Steps to simplify the exponential expression (y^4)^2 / y^6 using exponent rules.

Try It 6.129

Simplify: $\frac{{(m^{5})}^{4}}{m^{7}} .$

Try It 6.130

Simplify: $\frac{{(k^{2})}^{6}}{k^{7}} .$

Example 6.66

Simplify: $\frac{b^{12}}{{(b^{2})}^{6}} .$

Solution

	$\frac{b^{12}}{{(b^{2})}^{6}}$
Multiply the exponents in the numerator.	$\frac{b^{12}}{b^{12}}$
Subtract the exponents.	$b^{0}$
Simplify.	$1$

This table illustrates the step-by-step simplification of the exponential expression b^12 / (b^2)^6 using exponent rules, resulting in a final value of 1.

Try It 6.131

Simplify: $\frac{n^{12}}{{(n^{3})}^{4}} .$

Try It 6.132

Simplify: $\frac{x^{15}}{{(x^{3})}^{5}} .$

Example 6.67

Simplify: ${(\frac{y^{9}}{y^{4}})}^{2} .$

Solution

	${(\frac{y^{9}}{y^{4}})}^{2}$
Remember parentheses come before exponents. Notice the bases are the same, so we can simplify inside the parentheses. Subtract the exponents.	${(y^{5})}^{2}$
Multiply the exponents.	$y^{10}$

This table demonstrates the step-by-step simplification of the algebraic expression (y^9 / y^4)^2, applying exponent rules for division and power of a power.

Try It 6.133

Simplify: ${(\frac{r^{5}}{r^{3}})}^{4} .$

Try It 6.134

Simplify: ${(\frac{v^{6}}{v^{4}})}^{3} .$

Example 6.68

Simplify: ${(\frac{j^{2}}{k^{3}})}^{4} .$

Solution

Here we cannot simplify inside the parentheses first, since the bases are not the same.

	${(\frac{j^{2}}{k^{3}})}^{4}$
Raise the numerator and denominator to the fourth power using the Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$ .
Use the Power Property and simplify.	$\frac{j^{8}}{k^{12}}$

Step-by-step simplification of a rational expression using the Quotient to a Power Property and other exponent rules.

Try It 6.135

Simplify: ${(\frac{a^{3}}{b^{2}})}^{4} .$

Try It 6.136

Simplify: ${(\frac{q^{7}}{r^{5}})}^{3} .$

Example 6.69

Simplify: ${(\frac{2 m^{2}}{5 n})}^{4} .$

Solution

	${(\frac{2 m^{2}}{5 n})}^{4}$
Raise the numberator and denominator to the fourth power, using the Quotient to a Power Property, ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}$ .	$\frac{{(2 m^{2})}^{4}}{{(5 n)}^{4}}$
Raise each factor to the fourth power.	$\frac{{(2 m^{2})}^{4}}{{(5 n)}^{4}}$
Use the Power Property and simplify.	$\frac{16 m^{8}}{625 n^{4}}$

Step-by-step simplification of a rational expression raised to a power using exponent properties.

Try It 6.137

Simplify: ${(\frac{7 x^{3}}{9 y})}^{2} .$

Try It 6.138

Simplify: ${(\frac{3 x^{4}}{7 y})}^{2} .$

Example 6.70

Simplify: $\frac{{(x^{3})}^{4} {(x^{2})}^{5}}{{(x^{6})}^{5}} .$

Solution

	$\frac{{(x^{3})}^{4} {(x^{2})}^{5}}{{(x^{6})}^{5}}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n}$ .	$\frac{(x^{12}) (x^{10})}{(x^{30})}$
Add the exponents in the numerator.	$\frac{x^{22}}{x^{30}}$
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .	$\frac{1}{x^{8}}$

Step-by-step simplification of an algebraic expression using power and quotient properties of exponents.

Try It 6.139

Simplify: $\frac{{(a^{2})}^{3} {(a^{2})}^{4}}{{(a^{4})}^{5}} .$

Try It 6.140

Simplify: $\frac{{(p^{3})}^{4} {(p^{5})}^{3}}{{(p^{7})}^{6}} .$

Example 6.71

Simplify: $\frac{{(10 p^{3})}^{2}}{{(5 p)}^{3} {(2 p^{5})}^{4}} .$

Solution

	$\frac{{(10 p^{3})}^{2}}{{(5 p)}^{3} {(2 p^{5})}^{4}}$
Use the Product to a Power Property, ${(a b)}^{m} = a^{m} b^{m}$ .	$\frac{{(10)}^{2} {(p^{3})}^{2}}{{(5)}^{3} {(p)}^{3} {(2)}^{4} {(p^{5})}^{4}}$
Use the Power Property, ${(a^{m})}^{n} = a^{m \cdot n}$ .	$\frac{100 p^{6}}{125 p^{3} \cdot 16 p^{20}}$
Add the exponents in the denominator.	$\frac{100 p^{6}}{125 \cdot 16 p^{23}}$
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .	$\frac{100}{125 \cdot 16 p^{17}}$
Simplify.	$\frac{1}{20 p^{17}}$

Step-by-step simplification of a complex algebraic expression using properties of exponents.

Try It 6.141

Simplify: $\frac{{(3 r^{3})}^{2} {(r^{3})}^{7}}{{(r^{3})}^{3}} .$

Try It 6.142

Simplify: $\frac{{(2 x^{4})}^{5}}{{(4 x^{3})}^{2} {(x^{3})}^{5}} .$

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 6.72

Find the quotient: $56 x^{7} \div 8 x^{3} .$

Solution

	$56 x^{7} \div 8 x^{3}$
Rewrite as a fraction.	$\frac{56 x^{7}}{8 x^{3}}$
Use fraction multiplication.	$\frac{56}{8} \cdot \frac{x^{7}}{x^{3}}$
Simplify and use the Quotient Property.	$7 x^{4}$

Steps for simplifying 56x^7 8x^3 to 7x^4, illustrating algebraic division and exponent properties.

Try It 6.143

Find the quotient: $42 y^{9} \div 6 y^{3} .$

Try It 6.144

Find the quotient: $48 z^{8} \div 8 z^{2} .$

Example 6.73

Find the quotient: $\frac{45 a^{2} b^{3}}{−5 a b^{5}} .$

Solution

	$\frac{45 a^{2} b^{3}}{−5 a b^{5}}$
Use fraction multiplication.	$\frac{45}{−5} \cdot \frac{a^{2}}{a} \cdot \frac{b^{3}}{b^{5}}$
Simplify and use the Quotient Property.	$−9 \cdot a \cdot \frac{1}{b^{2}}$
Multiply.	$- \frac{9 a}{b^{2}}$

Steps to simplify a rational algebraic expression using fraction multiplication and quotient properties.

Try It 6.145

Find the quotient: $\frac{−72 a^{7} b^{3}}{8 a^{12} b^{4}} .$

Try It 6.146

Find the quotient: $\frac{−63 c^{8} d^{3}}{7 c^{12} d^{2}} .$

Example 6.74

Find the quotient: $\frac{24 a^{5} b^{3}}{48 a b^{4}} .$

Solution

	$\frac{24 a^{5} b^{3}}{48 a b^{4}}$
Use fraction multiplication.	$\frac{24}{48} \cdot \frac{a^{5}}{a} \cdot \frac{b^{3}}{b^{4}}$
Simplify and use the Quotient Property.	$\frac{1}{2} \cdot a^{4} \cdot \frac{1}{b}$
Multiply.	$\frac{a^{4}}{2 b}$

Step-by-step simplification of an algebraic rational expression using fraction multiplication and quotient properties.

Try It 6.147

Find the quotient: $\frac{16 a^{7} b^{6}}{24 a b^{8}} .$

Try It 6.148

Find the quotient: $\frac{27 p^{4} q^{7}}{−45 p^{12} q} .$

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 6.75

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.

	$\frac{14 x^{7} y^{12}}{21 x^{11} y^{6}}$
Simplify and use the Quotient Property.	$\frac{2 y^{6}}{3 x^{4}}$

This table demonstrates the simplification of a rational expression using the Quotient Property, showing the original expression and its simplified form.

Try It 6.149

Find the quotient: $\frac{28 x^{5} y^{14}}{49 x^{9} y^{12}} .$

Try It 6.150

Find the quotient: $\frac{30 m^{5} n^{11}}{48 m^{10} n^{14}} .$

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 6.76

Find the quotient: $\frac{(6 x^{2} y^{3}) (5 x^{3} y^{2})}{(3 x^{4} y^{5})} .$

Solution

	$\frac{(6 x^{2} y^{3}) (5 x^{3} y^{2})}{(3 x^{4} y^{5})}$
Simplify the numerator.	$\frac{30 x^{5} y^{5}}{3 x^{4} y^{5}}$
Simplify.	$10 x$

Step-by-step simplification of a rational algebraic expression.

Try It 6.151

Find the quotient: $\frac{(6 a^{4} b^{5}) (4 a^{2} b^{5})}{12 a^{5} b^{8}} .$

Try It 6.152

Find the quotient: $\frac{(−12 x^{6} y^{9}) (−4 x^{5} y^{8})}{−12 x^{10} y^{12}} .$

Media

Access these online resources for additional instruction and practice with dividing monomials:

Section 6.5 Exercises

Practice Makes Perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

356.

ⓐ $\frac{x^{18}}{x^{3}}$ ⓑ $\frac{5^{12}}{5^{3}}$

357.

ⓐ $\frac{y^{20}}{y^{10}}$ ⓑ $\frac{7^{16}}{7^{2}}$

358.

ⓐ $\frac{p^{21}}{p^{7}}$ ⓑ $\frac{4^{16}}{4^{4}}$

359.

ⓐ $\frac{u^{24}}{u^{3}}$ ⓑ $\frac{9^{15}}{9^{5}}$

360.

ⓐ $\frac{q^{18}}{q^{36}}$ ⓑ $\frac{10^{2}}{10^{3}}$

361.

ⓐ $\frac{t^{10}}{t^{40}}$ ⓑ $\frac{8^{3}}{8^{5}}$

362.

ⓐ $\frac{b}{b^{9}}$ ⓑ $\frac{4}{4^{6}}$

363.

ⓐ $\frac{x}{x^{7}}$ ⓑ $\frac{10}{10^{3}}$

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

364.

ⓐ $20^{0}$ ⓑ $b^{0}$

365.

ⓐ $13^{0}$ ⓑ $k^{0}$

366.

ⓐ $− 27^{0}$ ⓑ $− (27^{0})$

367.

ⓐ $− 15^{0}$ ⓑ $− (15^{0})$

368.

ⓐ ${(25 x)}^{0}$ ⓑ $25 x^{0}$

369.

ⓐ ${(6 y)}^{0}$ ⓑ $6 y^{0}$

370.

ⓐ ${(12 x)}^{0}$ ⓑ ${(−56 p^{4} q^{3})}^{0}$

371.

ⓐ $7 y^{0}$ ${(17 y)}^{0}$ ⓑ ${(−93 c^{7} d^{15})}^{0}$

372.

ⓐ $12 n^{0} - 18 m^{0}$ ⓑ ${(12 n)}^{0} - {(18 m)}^{0}$

373.

ⓐ $15 r^{0} - 22 s^{0}$ ⓑ ${(15 r)}^{0} - {(22 s)}^{0}$

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

374.

ⓐ ${(\frac{3}{4})}^{3}$ ⓑ ${(\frac{p}{2})}^{5}$ ⓒ ${(\frac{x}{y})}^{6}$

375.

ⓐ ${(\frac{2}{5})}^{2}$ ⓑ ${(\frac{x}{3})}^{4}$ ⓒ ${(\frac{a}{b})}^{5}$

376.

ⓐ ${(\frac{a}{3 b})}^{4}$ ⓑ ${(\frac{5}{4 m})}^{2}$

377.

ⓐ ${(\frac{x}{2 y})}^{3}$ ⓑ ${(\frac{10}{3 q})}^{4}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

378.

$\frac{{(a^{2})}^{3}}{a^{4}}$

379.

$\frac{{(p^{3})}^{4}}{p^{5}}$

380.

$\frac{{(y^{3})}^{4}}{y^{10}}$

381.

$\frac{{(x^{4})}^{5}}{x^{15}}$

382.

$\frac{u^{6}}{{(u^{3})}^{2}}$

383.

$\frac{v^{20}}{{(v^{4})}^{5}}$

384.

$\frac{m^{12}}{{(m^{8})}^{3}}$

385.

$\frac{n^{8}}{{(n^{6})}^{4}}$

386.

${(\frac{p^{9}}{p^{3}})}^{5}$

387.

${(\frac{q^{8}}{q^{2}})}^{3}$

388.

${(\frac{r^{2}}{r^{6}})}^{3}$

389.

${(\frac{m^{4}}{m^{7}})}^{4}$

390.

${(\frac{p}{r^{11}})}^{2}$

391.

${(\frac{a}{b^{6}})}^{3}$

392.

${(\frac{w^{5}}{x^{3}})}^{8}$

393.

${(\frac{y^{4}}{z^{10}})}^{5}$

394.

${(\frac{2 j^{3}}{3 k})}^{4}$

395.

${(\frac{3 m^{5}}{5 n})}^{3}$

396.

${(\frac{3 c^{2}}{4 d^{6}})}^{3}$

397.

${(\frac{5 u^{7}}{2 v^{3}})}^{4}$

398.

${(\frac{k^{2} k^{8}}{k^{3}})}^{2}$

399.

${(\frac{j^{2} j^{5}}{j^{4}})}^{3}$

400.

$\frac{{(t^{2})}^{5} {(t^{4})}^{2}}{{(t^{3})}^{7}}$

401.

$\frac{{(q^{3})}^{6} {(q^{2})}^{3}}{{(q^{4})}^{8}}$

402.

$\frac{{(−2 p^{2})}^{4} {(3 p^{4})}^{2}}{{(−6 p^{3})}^{2}}$

403.

$\frac{{(−2 k^{3})}^{2} {(6 k^{2})}^{4}}{{(9 k^{4})}^{2}}$

404.

$\frac{{(−4 m^{3})}^{2} {(5 m^{4})}^{3}}{{(−10 m^{6})}^{3}}$

405.

$\frac{{(−10 n^{2})}^{3} {(4 n^{5})}^{2}}{{(2 n^{8})}^{2}}$

Divide Monomials

In the following exercises, divide the monomials.

406.

$56 b^{8} \div 7 b^{2}$

407.

$63 v^{10} \div 9 v^{2}$

408.

$−88 y^{15} \div 8 y^{3}$

409.

$−72 u^{12} \div 12 u^{4}$

410.

$\frac{45 a^{6} b^{8}}{−15 a^{10} b^{2}}$

411.

$\frac{54 x^{9} y^{3}}{−18 x^{6} y^{15}}$

412.

$\frac{15 r^{4} s^{9}}{18 r^{9} s^{2}}$

413.

$\frac{20 m^{8} n^{4}}{30 m^{5} n^{9}}$

414.

$\frac{18 a^{4} b^{8}}{−27 a^{9} b^{5}}$

415.

$\frac{45 x^{5} y^{9}}{−60 x^{8} y^{6}}$

416.

$\frac{64 q^{11} r^{9} s^{3}}{48 q^{6} r^{8} s^{5}}$

417.

$\frac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$

418.

$\frac{(10 m^{5} n^{4}) (5 m^{3} n^{6})}{25 m^{7} n^{5}}$

419.

$\frac{(−18 p^{4} q^{7}) (−6 p^{3} q^{8})}{−36 p^{12} q^{10}}$

420.

$\frac{(6 a^{4} b^{3}) (4 a b^{5})}{(12 a^{2} b) (a^{3} b)}$

421.

$\frac{(4 u^{2} v^{5}) (15 u^{3} v)}{(12 u^{3} v) (u^{4} v)}$

Mixed Practice

422.

423.

424.

ⓐ $p^{4} \cdot p^{6}$ ⓑ ${(p^{4})}^{6}$

425.

ⓐ $q^{5} \cdot q^{3}$ ⓑ ${(q^{5})}^{3}$

426.

ⓐ $\frac{y^{3}}{y}$ ⓑ $\frac{y}{y^{3}}$

427.

ⓐ $\frac{z^{6}}{z^{5}}$ ⓑ $\frac{z^{5}}{z^{6}}$

428.

$(8 x^{5}) (9 x) \div 6 x^{3}$

429.

$(4 y) (12 y^{7}) \div 8 y^{2}$

430.

$\frac{27 a^{7}}{3 a^{3}} + \frac{54 a^{9}}{9 a^{5}}$

431.

$\frac{32 c^{11}}{4 c^{5}} + \frac{42 c^{9}}{6 c^{3}}$

432.

$\frac{32 y^{5}}{8 y^{2}} - \frac{60 y^{10}}{5 y^{7}}$

433.

$\frac{48 x^{6}}{6 x^{4}} - \frac{35 x^{9}}{7 x^{7}}$

434.

$\frac{63 r^{6} s^{3}}{9 r^{4} s^{2}} - \frac{72 r^{2} s^{2}}{6 s}$

435.

$\frac{56 y^{4} z^{5}}{7 y^{3} z^{3}} - \frac{45 y^{2} z^{2}}{5 y}$

Everyday Math

436.

Memory One megabyte is approximately $10^{6}$ bytes. One gigabyte is approximately $10^{9}$ bytes. How many megabytes are in one gigabyte?

437.

Memory One gigabyte is approximately $10^{9}$ bytes. One terabyte is approximately $10^{12}$ bytes. How many gigabytes are in one terabyte?

Writing Exercises

438.

Jennifer thinks the quotient $\frac{a^{24}}{a^{6}}$ simplifies to $a^{4}$ . What is wrong with her reasoning?

439.

Maurice simplifies the quotient $\frac{d^{7}}{d}$ by writing $\frac{d^{7}}{d} = 7$ . What is wrong with his reasoning?

440.

When Drake simplified $− 3^{0}$ and ${(−3)}^{0}$ he got the same answer. Explain how using the Order of Operations correctly gives different answers.

441.

Robert thinks $x^{0}$ simplifies to 0. What would you say to convince Robert he is wrong?

Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

This is a table that has six rows and four columns. In the first row, which is a header row, the cells read from left to right “I can…,” “Confidently,” “With some help,” and “No-I don’t get it!” The first column below “I can…” reads “simplify expressions using the Quotient Property for Exponents,” “simplify expressions with zero exponents,” “simplify expressions using the Quotient to a Power Property,” “simplify expressions by applying several properties,” and “divide monomials.” The rest of the cells are blank.

ⓑ On a scale of 1-10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?

Since 9 > 7, there are more factors of x in the numerator.
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n}$ .
Simplify.

Since 10 > 2, there are more factors of x in the numerator.
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = a^{m - n}$ .
Simplify.

Since 12 > 8, there are more factors of b in the denominator.
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .
Simplify.

Since 5 > 3, there are more factors of 3 in the denominator.
Use the Quotient Property, $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}$ .
Simplify.
Simplify.	$The fraction one forty-ninth is displayed, represented as 1 over 49 with a horizontal line separating the numerator and the denominator.$