10.6: Divide Monomials (Part 1)

Learning Objectives

• 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

be prepared!

Before you get started, take this readiness quiz.

1. Simplify: $\dfrac{8}{24}$. If you missed the problem, review Example 4.3.1.
2. Simplify:(2m³)⁵. If you missed the problem, review Example 10.3.13.
3. Simplify: $\dfrac{12x}{12y}$. If you missed the problem, review Example 4.3.5.

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

Product Property	a^m • a^n = a^m + n
Power Property	(a^m)^n = a^m • n
Product to a Power	(ab)^m = a^mb^m

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

Definition: Equivalent Fractions Property

If a, b, c are whole numbers where b ≠ 0, c ≠ 0, then

$$\dfrac{a}{b} = \dfrac{a \cdot c}{b \cdot c}\quad and\quad \dfrac{a \cdot c}{b \cdot c} = \dfrac{a}{b}$$

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

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

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

• When the larger exponent was in the numerator, we were left with factors in the numerator and 1 in the denominator, which we simplified.
• When the larger exponent was in the denominator, we were left with factors in the denominator, and 1 in the numerator, which could not be simplified.

We write:

$$\begin{split} \dfrac{x^{5}}{x^{2}} \qquad &\quad \dfrac{x^{2}}{x^{3}} \\ x^{5-2} \qquad &\; \dfrac{1}{x^{3-2}} \\ x^{3} \qquad \quad &\quad \dfrac{1}{x} \end{split}$$

Definition: Quotient Property of Exponents

If a is a real number, a ≠ 0, and m, n are whole numbers, then

$$\dfrac{a^{m}}{a^{n}} = a^{m-n}, \; m>n \quad and \quad \dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n-m}},\; n>m$$

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

$$\begin{split} \dfrac{3^{4}}{3^{2}} &\stackrel{?}{=} 3^{4-2} \qquad \; \dfrac{5^{2}}{5^{3}} \stackrel{?}{=} \dfrac{1}{5^{3-2}} \\ \dfrac{81}{9} &\stackrel{?}{=} 3^{2} \qquad \; \; \dfrac{25}{125} \stackrel{?}{=} \dfrac{1}{5^{1}} \\ 9 &= 9\; \checkmark \qquad \; \; \; \dfrac{1}{5} = \dfrac{1}{5}\; \checkmark \end{split}$$

When we work with numbers and the exponent is less than or equal to 3, we will apply the exponent. When the exponent is greater than 3, we leave the answer in exponential form.

Example $\PageIndex{1}$:

Simplify: (a) $\dfrac{x^{10}}{x^{8}}$ (b) $\dfrac{2^{9}}{2^{2}}$

Solution

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

(a)

Since 10 > 8, there are more factors of x in the numerator.	$$\dfrac{x^{10}}{x^{8}}$$
Use the quotient property with m > n, $\dfrac{a^{m}}{a^{n}} = a^{m − n}$.	$$x^{\textcolor{red}{10-8}}$$
Simplify.	$$x^{2}$$

(b)

Since 9 > 2, there are more factors of 2 in the numerator.	$$\dfrac{2^{9}}{2^{2}}$$
Use the quotient property with m > n, $\dfrac{a^{m}}{a^{n}} = a^{m − n}$.	$$2^{\textcolor{red}{9-2}}$$
Simplify.	$$2^{7}$$

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

Exercise $\PageIndex{1}$:

Simplify: (a) $\dfrac{x^{12}}{x^{9}}$ (b) $\dfrac{7^{14}}{7^{5}}$

Answer a

$x^3$

Answer b

$7^9$

Exercise $\PageIndex{2}$:

Simplify: (a) $\dfrac{y^{23}}{y^{17}}$ (b) $\dfrac{8^{15}}{8^{7}}$

Answer a

$y^6$

Answer b

$8^8$

Example $\PageIndex{2}$:

Simplify: (a) $\dfrac{b^{10}}{b^{15}}$ (b) $\dfrac{3^{3}}{3^{5}}$

Solution

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

(a)

Since 15 > 10, there are more factors of b in the denominator.	$$\dfrac{b^{10}}{b^{15}}$$
Use the quotient property with n > m, $\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}$.	$$\dfrac{\textcolor{red}{1}}{b^{\textcolor{red}{15-10}}}$$
Simplify.	$$\dfrac{1}{b^{5}}$$

(b)

Since 5 > 3, there are more factors of 3 in the denominator.	$$\dfrac{3^{3}}{3^{5}}$$
Use the quotient property with n > m, $\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}$.	$$\dfrac{\textcolor{red}{1}}{3^{\textcolor{red}{5-3}}}$$
Simplify.	$$\dfrac{1}{3^{2}}$$
Apply the exponent.	$$\dfrac{1}{9}$$

Notice that when the larger exponent is in the denominator, we are left with factors in the denominator and 1 in the numerator.

Exercise $\PageIndex{3}$:

Simplify: (a) $\dfrac{x^{8}}{x^{15}}$ (b) $\dfrac{12^{11}}{12^{21}}$

Answer a

$\frac{1}{x^7}$

Answer b

$\frac{1}{12^10}$

Exercise $\PageIndex{4}$:

Simplify: (a) $\dfrac{m^{17}}{m^{26}}$ (b) $\dfrac{7^{8}}{7^{14}}$

Answer a

$\frac{1}{m^9}$

Answer b

$\frac{1}{7^6}$

Example $\PageIndex{3}$:

Simplify: (a) $\dfrac{a^{5}}{a^{9}}$ (b) $\dfrac{x^{11}}{x^{7}}$

Solution

(a)

Since 9 > 5, there are more a's in the denominator and so we will end up with factors in the denominator.	$$\dfrac{a^{5}}{a^{9}}$$
Use the quotient property with n > m, $\dfrac{a^{m}}{a^{n}} = \dfrac{1}{a^{n − m}}$.	$$\dfrac{\textcolor{red}{1}}{a^{\textcolor{red}{9-5}}}$$
Simplify.	$$\dfrac{1}{a^{4}}$$

(b)

Notice there are more factors of x in the numerator, since 11 > 7. So we will end up with factors in the numerator.	$$\dfrac{x^{11}}{x^{97}}$$
Use the quotient property with m > n, $\dfrac{a^{m}}{a^{n}} = a^{m − n}$.	$$a^{\textcolor{red}{11-7}}$$
Simplify.	$$x^{4}$$

Exercise $\PageIndex{5}$:

Simplify: (a) $\dfrac{b^{19}}{b^{11}}$ (b) $\dfrac{z^{5}}{z^{11}}$

Answer a

$b^8$

Answer b

$\frac{1}{z^6}$

Exercise $\PageIndex{6}$:

Simplify: (a) $\dfrac{p^{9}}{p^{17}}$ (b) $\dfrac{w^{13}}{w^{9}}$

Answer a

$\frac{1}{p^8}$

Answer b

$w^4$

Simplify Expressions with Zero Exponents

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

$$\dfrac{2}{2} = 1 \qquad \dfrac{17}{17} = 1 \qquad \dfrac{-43}{-43} = 1$$

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

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

Now we will simplify $\dfrac{a^{m}}{a^{m}}$ in two ways to lead us to the definition of the zero exponent. Consider first $\dfrac{8}{8}$, which we know is 1.

	$$\dfrac{8}{8} = 1$$
Write 8 as 2^3.	$$\dfrac{2^{3}}{2^{3}} = 1$$
Subtract exponents.	$$2^{3-3} = 1$$
Simplify.	$$2^{0} = 1$$

We see $\dfrac{a^{m}}{a^{n}}$ simplifies to a⁰ and to 1. So a⁰ = 1.

Definition: Zero Exponent

If a is a non-zero number, then a⁰ = 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 $\PageIndex{4}$:

Simplify: (a) 12⁰ (b) y⁰

Solution

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

(a) 12⁰

Use the definition of the zero exponent.

1

(b) y⁰

Use the definition of the zero exponent.

1

Exercise $\PageIndex{7}$:

Simplify: (a) 17⁰ (b) m⁰

Answer a

1

Answer b

1

Exercise $\PageIndex{8}$:

Simplify: (a) k⁰ (b) 29⁰

Answer a

1

Answer b

1

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 (2x)⁰. We can use the product to a power rule to rewrite this expression.

	(2x)^0
Use the Product to a Power Rule.	2^0x^0
Use the Zero Exponent Property.	1 • 1
Simplify.	1

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

Example $\PageIndex{5}$:

Simplify: (7z)⁰.

Solution

Use the definition of the zero exponent.

1

Exercise $\PageIndex{9}$:

Simplify: (−4y)⁰.

Answer

1

Exercise $\PageIndex{10}$:

Simplify: $\left(\dfrac{2}{3} x\right)^{0}$.

Answer

1

Example $\PageIndex{6}$:

Simplify: (a) (−3x²y)⁰ (b) −3x²y⁰

Solution

(a) (−3x²y)⁰

The product is raised to the zero power.	(−3x^2y)^0
Use the definition of the zero exponent.	1

(b) −3x²y⁰

Notice that only the variable y is being raised to the zero power.	−3x^2y^0
Use the definition of the zero exponent.	−3x^2• 1
Simplify.	−3x^2

Exercise $\PageIndex{11}$:

Simplify: (a) (7x²y)⁰ (b) 7x²y⁰

Answer a

1

Answer b

$7x^2$

Exercise $\PageIndex{12}$:

Simplify: (a) −23x²y⁰ (b) (−23x²y)⁰

Answer a

$-23x^2$

Answer b

1

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.

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

Notice that the exponent applies to both the numerator and the denominator. We see that $\left(\dfrac{x}{y}\right)^{3}$ is $\dfrac{x^{3}}{y^{3}}$. We write:

$$\left(\dfrac{x}{y}\right)^{3} = \dfrac{x^{3}}{y^{3}}$$

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

Definition: Quotient to a Power Property of Exponents

If a and b are real numbers, b ≠ 0, and m is a counting number, then

$$\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$$

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:

$$\begin{split} \left(\dfrac{2}{3}\right)^{3} &\stackrel{?}{=} \dfrac{2^{3}}{3^{3}} \\ \dfrac{2}{3} \cdot \dfrac{2}{3} \cdot \dfrac{2}{3} &\stackrel{?}{=} \dfrac{8}{27} \\ \dfrac{8}{27} &= \dfrac{8}{27}\; \checkmark \end{split}$$

Example $\PageIndex{7}$:

Simplify: (a) $\left(\dfrac{5}{8}\right)^{2}$ (b) $\left(\dfrac{x}{3}\right)^{4}$ (c) $\left(\dfrac{y}{m}\right)^{3}$

Solution

(a) $\left(\dfrac{5}{8}\right)^{2}$

Use the Quotient to a Power Property, $\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$.	$$\dfrac{5^{\textcolor{red}{2}}}{8^{\textcolor{red}{2}}}$$
Simplify.	$$\dfrac{25}{64}$$

(b) $\left(\dfrac{x}{3}\right)^{4}$

Use the Quotient to a Power Property, $\left(\dfrac{a}{b}\right)^{m} = \dfrac{a^{m}}{b^{m}}$.	$$\dfrac{x^{\textcolor{red}{4}}}{3^{\textcolor{red}{4}}}$$
Simplify.	$$\dfrac{x^{4}}{81}$$

(c) $\left(\dfrac{y}{m}\right)^{3}$

Raise the numerator and denominator to the third power.

$$\dfrac{y^{\textcolor{red}{3}}}{m^{\textcolor{red}{3}}}$$

Exercise $\PageIndex{13}$:

Simplify: (a) $\left(\dfrac{7}{9}\right)^{2}$ (b) $\left(\dfrac{y}{8}\right)^{3}$ (c) $\left(\dfrac{p}{q}\right)^{6}$

Answer a

$\dfrac{49}{81}$

Answer b

$\dfrac{y^3}{512}$

Answer c

$\dfrac{p^6}{q^6}$

Exercise $\PageIndex{14}$:

Simplify: (a) $\left(\dfrac{1}{8}\right)^{2}$ (b) $\left(\dfrac{-5}{m}\right)^{3}$ (c) $\left(\dfrac{r}{s}\right)^{4}$

Answer a

$\dfrac{1}{64}$

Answer b

$-\dfrac{125}{m^3}$

Answer c

$\dfrac{r^4}{s^4}$

Contributors and Attributions

• Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed under Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@5.191."