5.2 Use Quotient Property of Exponents

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

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 {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}
Product to a Power {\left(ab\right)}^{m}={a}^{m}{b}^{m}

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\ne 0,c\ne 0,

then \dfrac{a}{b}=\dfrac{a\cdot c}{b\cdot c}  and \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{\overline{)x}\cdot \overline{)x}\cdot x \cdot x \cdot x}{\overline{)x}\cdot \overline{)x}} \dfrac{\overline{)x}\cdot \overline{)x}\cdot 1}{\overline{)x}\cdot \overline{)x}\cdot x}
Simplify. {x}^{3} \dfrac{1}{x}

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{array}{cc} \dfrac{{x}^{5}}{{x}^{2}} & \dfrac{{x}^{2}}{{x}^{3}}\\ {x}^{5-2} & \dfrac{1}{{x}^{3-2}} \\ {x}^{3} & \dfrac{1}{x}\hfill \end{array}

This leads to the Quotient Property for Exponents.

Quotient Property for Exponents

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

\dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n},m > n and \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{array}{rlrl}\dfrac{{3}^{4}}{{3}^{2}}& = {3}^{4-2}\hfill & \hfill \dfrac{{5}^{2}}{{5}^{3}} &= \dfrac{1}{{5}^{3-2}} \\ \dfrac{81}{9} &= {3}^{2}\hfill & \dfrac{25}{125}& = \dfrac{1}{{5}^{1}} \\ 9& = 9\checkmark\hfill & \dfrac{1}{5}& = \dfrac{1}{5}\checkmark \end{array}

EXAMPLE 1

Simplify: a) \frac{{x}^{9}}{{x}^{7}} b) \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.

  1. Since 9 > 7, there are more factors of x in the numerator. \dfrac{x^9}{x^7}
    Use the Quotient Property, \dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}. x^{\color{red}{9-7}}
    Simplify. x^2
  2. Since 10 > 2, there are more factors of x in the numerator. \dfrac{3^{10}}{3^2}
    Use the Quotient Property, \dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n}. 3^{\color{red}{10-2}}
    Simplify. 3^8

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

TRY IT 1.1

Simplify: a) \dfrac{{x}^{15}}{{x}^{10}} b) \dfrac{{6}^{14}}{{6}^{5}}.

Show answer

a) {x}^{5} b) {6}^{9}

TRY IT 1.2

Simplify: a) \dfrac{{y}^{43}}{{y}^{37}} b) \dfrac{{10}^{15}}{{10}^{7}}.

Show answer

a) {y}^{6} b) {10}^{8}

EXAMPLE 2

Simplify: a) \dfrac{{b}^{8}}{{b}^{12}} b) \dfrac{{7}^{3}}{{7}^{5}}.

Solution

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

  1. Since 12 > 8, there are more factors of b in the denominator. \dfrac{b^8}{b^{12}}
    Use the Quotient Property, \dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{n-m}}. \dfrac{{\color{red}{1}}}{b^{\color{red}{12-8}}}
    Simplify. \dfrac{1}{b^4}
  2. Since 5 > 3, there are more factors of 3 in the denominator. \dfrac{7^3}{7^5}
    Use the Quotient Property, \dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{n-m}}. \dfrac{{\color{red}{1}}}{7^{\color{red}{5-3}}}
    Simplify. \dfrac{1}{7^2}
    Simplify. \dfrac{1}{49}

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

TRY IT 2.1

Simplify: a) \dfrac{{x}^{18}}{{x}^{22}} b) \dfrac{{12}^{15}}{{12}^{30}}.

Show answer

a) \dfrac{1}{{x}^{4}} b) \dfrac{1}{{12}^{15}}

TRY IT 2.2

Simplify: a) \dfrac{{m}^{7}}{{m}^{15}} b) \dfrac{{9}^{8}}{{9}^{19}}.

Show answer

a) \dfrac{1}{{m}^{8}} b) \dfrac{1}{{9}^{11}}

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 3

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

Solution
  1. Is the exponent of a larger in the numerator or denominator? Since 9 > 5, there are more a\text{'}\text{s} in the denominator and so we will end up with factors in the denominator.
    \dfrac{a^5}{a^9}
    Use the Quotient Property, \dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{n-m}}. \dfrac{{\color{red}{1}}}{a^{\color{red}{9-5}}}
    Simplify. \dfrac{1}{a^4}
  2. 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^7}
    Use the Quotient Property, \dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{n-m}}. x^{{\color{red}{11-7}}}
    Simplify. x^4

TRY IT 3.1

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

Show answer

a) {b}^{8} b) \dfrac{1}{{z}^{6}}

TRY IT 3.2

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

Show answer

a) \dfrac{1}{{p}^{8}} b) {w}^{4}

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 \dfrac{{a}^{m}}{{a}^{m}}. From your earlier work with fractions, you know that:

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

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

The Quotient Property for 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?

Consider \frac{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

Now we will simplify \dfrac{{a}^{m}}{{a}^{m}} in two ways to lead us to the definition of the zero exponent. In general, for a\ne 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 \dfrac{{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 4

Simplify: a) {9}^{0} b) {n}^{0}.

Solution

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

a)
Use the definition of the zero exponent.		 \begin{array}{c}{9}^{0}\\ =1\end{array}
b)
Use the definition of the zero exponent.		 \begin{array}{c}{n}^{0}\\ =1\end{array}

TRY IT 4.1

Simplify: a) {15}^{0} b) {m}^{0}.

Show answer

a) 1 b) 1

TRY IT 4.2

Simplify: a) {k}^{0} b) {29}^{0}.

Show answer

a) 1 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 {\left(2x\right)}^{0}. We can use the product to a power rule to rewrite this expression.

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

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

EXAMPLE 5

Simplify: a) {\left(5b\right)}^{0} b) {\left(-4{a}^{2}b\right)}^{0}.

Solution
a) {\left(5b\right)}^{0}
Use the definition of the zero exponent. 1
b) {\left(-4{a}^{2}b\right)}^{0}
Use the definition of the zero exponent. 1

TRY IT 5.1

Simplify: a) {\left(11z\right)}^{0} b) {\left(-11p{q}^{3}\right)}^{0}.

Show answer

a) 1 b) 1

TRY IT 5.2

Simplify: a) {\left(-6d\right)}^{0} b) {\left(-8{m}^{2}{n}^{3}\right)}^{0}.

Show answer

a) 1 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 write: {\left(\dfrac{x}{y}\right)}^{3}
\dfrac{{x}^{3}}{{y}^{3}}

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\ne 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{array}{rl}{\left(\dfrac{2}{3}\right)}^{3} =& \dfrac{{2}^{3}}{{3}^{3}}\\\dfrac{2}{3}\cdot \dfrac{2}{3}\cdot \dfrac{2}{3} =& \dfrac{8}{27}\\ \dfrac{8}{27} =& \dfrac{8}{27}\checkmark \end{array}

EXAMPLE 6

Simplify: a) {\left(\dfrac{3}{7}\right)}^{2} b) {\left(\dfrac{b}{3}\right)}^{4} c) {\left(\dfrac{k}{j}\right)}^{3}.

Solution

a)

\left(\dfrac{3}{7}\right)^2
Use the Quotient Property, {\left(\dfrac{a}{b}\right)}^{m}=\dfrac{{a}^{m}}{{b}^{m}}. \dfrac{3^{\color{red}{2}}}{7^{\color{red}{2}}}
Simplify. \dfrac{9}{49}

b)

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

c)

\left(\dfrac{k}{j}\right)^3
Raise the numerator and denominator to the third power. \dfrac{k^{\color{red}{3}}}{j^{\color{red}{3}}}

TRY IT 6.1

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

Show answer

a) \dfrac{25}{64} b) \dfrac{{p}^{4}}{10,000} c) \dfrac{{m}^{7}}{{n}^{7}}

TRY IT 6.2

Simplify: a) {\left(\dfrac{1}{3}\right)}^{3} b) {\left(\dfrac{-2}{q}\right)}^{3} c) {\left(\dfrac{w}{x}\right)}^{4}.

Show answer

a) \dfrac{1}{27} b) \dfrac{-8}{{q}^{3}} c) \dfrac{{w}^{4}}{{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 {\left({a}^{m}\right)}^{n}={a}^{m \cdot n}
Product to a Power {\left(ab\right)}^{m}={a}^{m}{b}^{m}
Quotient Property \begin{aligned} \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{aligned}
Zero Exponent Definition {a}^{o}=1,a\ne 0
Quotient to a Power Property {\left(\dfrac{a}{b}\right)}^{m}=\dfrac{{a}^{m}}{{b}^{m}},b\ne 0

EXAMPLE 7

Simplify: \dfrac{{\left({y}^{4}\right)}^{2}}{{y}^{6}}.

Solution
\dfrac{{\left({y}^{4}\right)}^{2}}{{y}^{6}}
Multiply the exponents in the numerator. \dfrac{{y}^{8}}{{y}^{6}}
Subtract the exponents. {y}^{2}

TRY IT 7.1

Simplify: \dfrac{{\left({m}^{5}\right)}^{4}}{{m}^{7}}.

Show answer

{m}^{13}

TRY IT 7.2

Simplify: \dfrac{{\left({k}^{2}\right)}^{6}}{{k}^{7}}.

Show answer

{k}^{5}

EXAMPLE 8

Simplify: \dfrac{{b}^{12}}{{\left({b}^{2}\right)}^{6}}.

Solution
\dfrac{{b}^{12}}{{\left({b}^{2}\right)}^{6}}
Multiply the exponents in the numerator. \dfrac{{b}^{12}}{{b}^{12}}
Subtract the exponents. {b}^{0}
Simplify. 1

TRY IT 8.1

Simplify: \dfrac{{n}^{12}}{{\left({n}^{3}\right)}^{4}}.

Show answer

1

TRY IT 8.2

Simplify: \dfrac{{x}^{15}}{{\left({x}^{3}\right)}^{5}}.

Show answer

1

EXAMPLE 9

Simplify: {\left(\dfrac{{y}^{9}}{{y}^{4}}\right)}^{2}.

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

TRY IT 9.1

Simplify: {\left(\dfrac{{r}^{5}}{{r}^{3}}\right)}^{4}.

Show answer

{r}^{8}

TRY IT 9.2

Simplify: {\left(\dfrac{{v}^{6}}{{v}^{4}}\right)}^{3}.

Show answer

{v}^{6}

EXAMPLE 10

Simplify: {\left(\dfrac{{j}^{2}}{{k}^{3}}\right)}^{4}.

Solution

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

{\left(\dfrac{{j}^{2}}{{k}^{3}}\right)}^{4}
Raise the numerator and denominator to the third power
using the Quotient to a Power Property, {\left(\dfrac{a}{b}\right)}^{m}=\dfrac{{a}^{m}}{{b}^{m}}.		 \dfrac{{j^{\color{red}{2 \cdot 4}}}}{k^{\color{red}{3 \cdot 4}}}}
Use the Power Property and simplify. \dfrac{{j}^8}{k^{12}}

TRY IT 10.1

Simplify: {\left(\dfrac{{a}^{3}}{{b}^{2}}\right)}^{4}.

Show answer

\dfrac{{a}^{12}}{{b}^{8}}

TRY IT 10.2

Simplify: {\left(\dfrac{{q}^{7}}{{r}^{5}}\right)}^{3}.

Show answer

\dfrac{{q}^{21}}{{r}^{15}}

EXAMPLE 11

Simplify: {\left(\dfrac{2{m}^{2}}{5n}\right)}^{4}.

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

TRY IT 11.1

Simplify: {\left(\dfrac{7{x}^{3}}{9y}\right)}^{2}.

Show answer

\dfrac{49{x}^{6}}{81{y}^{2}}

TRY IT 11.2

Simplify: {\left(\dfrac{3{x}^{4}}{7y}\right)}^{2}.

Show answer

\dfrac{9{x}^{8}}{49{y}^{2}}

EXAMPLE 12

Simplify: \dfrac{{\left({x}^{3}\right)}^{4}{\left({x}^{2}\right)}^{5}}{{\left({x}^{6}\right)}^{5}}.

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

TRY IT 12.1

Simplify: \dfrac{{\left({a}^{2}\right)}^{3}{\left({a}^{2}\right)}^{4}}{{\left({a}^{4}\right)}^{5}}.

Show answer

\dfrac{1}{{a}^{6}}

TRY IT 12.2

Simplify: \dfrac{{\left({p}^{3}\right)}^{4}{\left({p}^{5}\right)}^{3}}{{\left({p}^{7}\right)}^{6}}.

Show answer

\dfrac{1}{{p}^{15}}

EXAMPLE 13

Simplify: \dfrac{{\left(10{p}^{3}\right)}^{2}}{{\left(5p\right)}^{3}{\left(2{p}^{5}\right)}^{4}}.

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

TRY IT 13.1

Simplify: \dfrac{{\left(3{r}^{3}\right)}^{2}{\left({r}^{3}\right)}^{7}}{{\left({r}^{3}\right)}^{3}}.

Show answer

9{r}^{18}

TRY IT 13.2

Simplify: \dfrac{{\left(2{x}^{4}\right)}^{5}}{{\left(4{x}^{3}\right)}^{2}{\left({x}^{3}\right)}^{5}}.

Show answer

\dfrac{2}{x}

Divide 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 14

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

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

TRY IT 14.1

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

Show answer

7{y}^{6}

TRY IT 14.2

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

Show answer

6{z}^{6}

EXAMPLE 15

Find the quotient: \dfrac{45{a}^{2}{b}^{3}}{-5a{b}^{5}}.

Solution

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

TRY IT 15.1

Find the quotient: \dfrac{-72{a}^{7}{b}^{3}}{8{a}^{12}{b}^{4}}.

Show answer

-\dfrac{9}{{a}^{5}b}

TRY IT 15.2

Find the quotient: \dfrac{-63{c}^{8}{d}^{3}}{7{c}^{12}{d}^{2}}.

Show answer

\dfrac{-9d}{{c}^{4}}

EXAMPLE 16

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

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

TRY IT 16.1

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

Show answer

\dfrac{2{a}^{6}}{3{b}^{2}}

TRY IT 16.2

Find the quotient: \dfrac{27{p}^{4}{q}^{7}}{-45{p}^{12}q}.

Show answer

-\dfrac{3{q}^{6}}{5{p}^{8}}

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 17

Find the quotient: \dfrac{14{x}^{7}{y}^{12}}{21{x}^{11}{y}^{6}}.

Solution

Be very careful to simplify \dfrac{14}{21} by dividing out a common factor, and to simplify the variables by subtracting their exponents.

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

TRY IT 17.1

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

Show answer

\dfrac{4{y}^{2}}{7{x}^{4}}

TRY IT 17.2

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

Show answer

\dfrac{5}{8{m}^{5}{n}^{3}}

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 18

Find the quotient: \dfrac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}.

Solution
\dfrac{\left(6{x}^{2}{y}^{3}\right)\left(5{x}^{3}{y}^{2}\right)}{\left(3{x}^{4}{y}^{5}\right)}
Simplify the numerator. \dfrac{30{x}^{5}{y}^{5}}{3{x}^{4}{y}^{5}}
Simplify. 10x

TRY IT 18.1

Find the quotient: \dfrac{\left(6{a}^{4}{b}^{5}\right)\left(4{a}^{2}{b}^{5}\right)}{12{a}^{5}{b}^{8}}.

Show answer

2a{b}^{2}

TRY IT 18.2

Find the quotient: \dfrac{\left(-12{x}^{6}{y}^{9}\right)\left(-4{x}^{5}{y}^{8}\right)}{-12{x}^{10}{y}^{12}}.

Show answer

-4x{y}^{5}

Additional Online Resources

Key Concepts

  • Quotient Property for Exponents:
    • If a is a real number, a\ne 0, and m,n are whole numbers, then:
      \dfrac{{a}^{m}}{{a}^{n}}={a}^{m-n},m > n\text{ and }\dfrac{{a}^{m}}{{a}^{n}}=\dfrac{1}{{a}^{n-m}},n > m
  • Zero Exponent
    • If a is a non-zero number, then {a}^{0}=1.
  • Quotient to a Power Property for Exponents:
    • If a and b are real numbers, b\ne 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.
  • Summary of Exponent Properties
    • If a,b are real numbers and m,n are whole numbers, then
      Product Property {a}^{m}\cdot{a}^{n}={a}^{m+n}
      Power Property {\left({a}^{m}\right)}^{n}={a}^{m \cdot n}
      Product to a Power {\left(ab\right)}^{m}={a}^{m}{b}^{m}
      Quotient Property \begin{aligned} \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{aligned}
      Zero Exponent Definition {a}^{o}=1,a\ne 0
      Quotient to a Power Property {\left(\dfrac{a}{b}\right)}^{m}=\dfrac{{a}^{m}}{{b}^{m}},b\ne 0

Practice Makes Perfect

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

1. a) \frac{{x}^{18}}{{x}^{3}} b) \frac{{5}^{12}}{{5}^{3}} 2. a) \frac{{y}^{20}}{{y}^{10}} b) \frac{{7}^{16}}{{7}^{2}}
3. a) \frac{{p}^{21}}{{p}^{7}} b) \frac{{4}^{16}}{{4}^{4}} 4. a) \frac{{u}^{24}}{{u}^{3}} b) \frac{{9}^{15}}{{9}^{5}}
5. a) \frac{{q}^{18}}{{q}^{36}} b) \frac{{10}^{2}}{{10}^{3}} 6. a) \frac{{t}^{10}}{{t}^{40}} b) \frac{{8}^{3}}{{8}^{5}}
7. a) \frac{b}{{b}^{9}} b) \frac{4}{{4}^{6}} 8. a) \frac{x}{{x}^{7}} b) \frac{10}{{10}^{3}}

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

9.

a) {20}^{0}
b) {b}^{0}

10.

a) {13}^{0}
b) {k}^{0}

11.

a) -{27}^{0}
b) -\left({27}^{0}\right)

12.

a) -{15}^{0}
b) -\left({15}^{0}\right)

13.

a) {\left(25x\right)}^{0}
b) 25{x}^{0}

14.

a) {\left(6y\right)}^{0}
b) 6{y}^{0}

15.

a) {\left(12x\right)}^{0}
b) {\left(-56{p}^{4}{q}^{3}\right)}^{0}

16.

a) 7{y}^{0}{\left(17y\right)}^{0}
b) {\left(-93{c}^{7}{d}^{15}\right)}^{0}

17.

a) 12{n}^{0}-18{m}^{0}
b) {\left(12n\right)}^{0}-{\left(18m\right)}^{0}

18.

a) 15{r}^{0}-22{s}^{0}
b) {\left(15r\right)}^{0}-{\left(22s\right)}^{0}

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

19.

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

20.

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

21.

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

22.

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

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

23. \frac{{\left({a}^{2}\right)}^{3}}{{a}^{4}} 24. \frac{{\left({p}^{3}\right)}^{4}}{{p}^{5}}
25. \frac{{\left({y}^{3}\right)}^{4}}{{y}^{10}} 26. \frac{{\left({x}^{4}\right)}^{5}}{{x}^{15}}
27. \frac{{u}^{6}}{{\left({u}^{3}\right)}^{2}} 28. \frac{{v}^{20}}{{\left({v}^{4}\right)}^{5}}
29. \frac{{m}^{12}}{{\left({m}^{8}\right)}^{3}} 30. \frac{{n}^{8}}{{\left({n}^{6}\right)}^{4}}
31. {\left(\frac{{p}^{9}}{{p}^{3}}\right)}^{5} 32. {\left(\frac{{q}^{8}}{{q}^{2}}\right)}^{3}
33. {\left(\frac{{r}^{2}}{{r}^{6}}\right)}^{3} 34. {\left(\frac{{m}^{4}}{{m}^{7}}\right)}^{4}
35. {\left(\frac{p}{{r}^{11}}\right)}^{2} 36. {\left(\frac{a}{{b}^{6}}\right)}^{3}
37. {\left(\frac{{w}^{5}}{{x}^{3}}\right)}^{8} 38. {\left(\frac{{y}^{4}}{{z}^{10}}\right)}^{5}
39. {\left(\frac{2{j}^{3}}{3k}\right)}^{4} 40. {\left(\frac{3{m}^{5}}{5n}\right)}^{3}
41. {\left(\frac{3{c}^{2}}{4{d}^{6}}\right)}^{3} 42. {\left(\frac{5{u}^{7}}{2{v}^{3}}\right)}^{4}
43. {\left(\frac{{k}^{2}{k}^{8}}{{k}^{3}}\right)}^{2} 44. {\left(\frac{{j}^{2}{j}^{5}}{{j}^{4}}\right)}^{3}
45. \frac{{\left({t}^{2}\right)}^{5}{\left({t}^{4}\right)}^{2}}{{\left({t}^{3}\right)}^{7}} 46. \frac{{\left({q}^{3}\right)}^{6}{\left({q}^{2}\right)}^{3}}{{\left({q}^{4}\right)}^{8}}
47. \frac{{\left(-2{p}^{2}\right)}^{4}{\left(3{p}^{4}\right)}^{2}}{{\left(-6{p}^{3}\right)}^{2}} 48. \frac{{\left(-2{k}^{3}\right)}^{2}{\left(6{k}^{2}\right)}^{4}}{{\left(9{k}^{4}\right)}^{2}}
49. \frac{{\left(-4{m}^{3}\right)}^{2}{\left(5{m}^{4}\right)}^{3}}{{\left(-10{m}^{6}\right)}^{3}} 50. \frac{{\left(-10{n}^{2}\right)}^{3}{\left(4{n}^{5}\right)}^{2}}{{\left(2{n}^{8}\right)}^{2}}

Divide Monomials

In the following exercises, divide the monomials.

51. 56{b}^{8}\div 7{b}^{2} 52. 63{v}^{10}\div 9{v}^{2}
53. -88{y}^{15}\div 8{y}^{3} 54. -72{u}^{12}\div 12{u}^{4}
55. \frac{45{a}^{6}{b}^{8}}{-15{a}^{10}{b}^{2}} 56. \frac{54{x}^{9}{y}^{3}}{-18{x}^{6}{y}^{15}}
57. \frac{15{r}^{4}{s}^{9}}{18{r}^{9}{s}^{2}} 58. \frac{20{m}^{8}{n}^{4}}{30{m}^{5}{n}^{9}}
59. \frac{18{a}^{4}{b}^{8}}{-27{a}^{9}{b}^{5}} 60. \frac{45{x}^{5}{y}^{9}}{-60{x}^{8}{y}^{6}}
61. \frac{64{q}^{11}{r}^{9}{s}^{3}}{48{q}^{6}{r}^{8}{s}^{5}} 62. \frac{65{a}^{10}{b}^{8}{c}^{5}}{42{a}^{7}{b}^{6}{c}^{8}}
63. \frac{\left(10{m}^{5}{n}^{4}\right)\left(5{m}^{3}{n}^{6}\right)}{25{m}^{7}{n}^{5}} 64. \frac{\left(-18{p}^{4}{q}^{7}\right)\left(-6{p}^{3}{q}^{8}\right)}{-36{p}^{12}{q}^{10}}
65. \frac{\left(6{a}^{4}{b}^{3}\right)\left(4a{b}^{5}\right)}{\left(12{a}^{2}b\right)\left({a}^{3}b\right)} 66. \frac{\left(4{u}^{2}{v}^{5}\right)\left(15{u}^{3}v\right)}{\left(12{u}^{3}v\right)\left({u}^{4}v\right)}

Mixed Practice

67.

a) 24{a}^{5}+2{a}^{5}
b) 24{a}^{5}-2{a}^{5}
c) 24{a}^{5}\cdot 2{a}^{5}
d) 24{a}^{5}\div 2{a}^{5}

68.

a) 15{n}^{10}+3{n}^{10}
b) 15{n}^{10}-3{n}^{10}
c) 15{n}^{10}\cdot 3{n}^{10}
d) 15{n}^{10}\div 3{n}^{10}

69.

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

70.

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

71.

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

72.

a) \frac{{z}^{6}}{{z}^{5}}
b) \frac{{z}^{5}}{{z}^{6}}
73. \left(8{x}^{5}\right)\left(9x\right)\div 6{x}^{3} 74. \left(4y\right)\left(12{y}^{7}\right)\div 8{y}^{2}
75. \frac{27{a}^{7}}{3{a}^{3}}+\frac{54{a}^{9}}{9{a}^{5}} 76. \frac{32{c}^{11}}{4{c}^{5}}+\frac{42{c}^{9}}{6{c}^{3}}
77. \frac{32{y}^{5}}{8{y}^{2}}-\frac{60{y}^{10}}{5{y}^{7}} 78. \frac{48{x}^{6}}{6{x}^{4}}-\frac{35{x}^{9}}{7{x}^{7}}
 79. \frac{63{r}^{6}{s}^{3}}{9{r}^{4}{s}^{2}}-\frac{72{r}^{2}{s}^{2}}{6s} 80. \frac{56{y}^{4}{z}^{5}}{7{y}^{3}{z}^{3}}-\frac{45{y}^{2}{z}^{2}}{5y}

Everyday Math

81. Memory One megabyte is approximately {10}^{6} bytes. One gigabyte is approximately {10}^{9} bytes. How many megabytes are in one gigabyte? 82. Memory One gigabyte is approximately {10}^{9} bytes. One terabyte is approximately {10}^{12} bytes. How many gigabytes are in one terabyte?

Writing Exercises

83. Jennifer thinks the quotient \frac{{a}^{24}}{{a}^{6}} simplifies to {a}^{4}. What is wrong with her reasoning? 84. Maurice simplifies the quotient \frac{{d}^{7}}{d} by writing \frac{{\overline{)d}}^{7}}{\overline{)d}}=7. What is wrong with his reasoning?
85. When Drake simplified -{3}^{0} and {\left(-3\right)}^{0} he got the same answer. Explain how using the Order of Operations correctly gives different answers. 86. Robert thinks {x}^{0} simplifies to 0. What would you say to convince Robert he is wrong?

Answers

1. a) {x}^{15}  b) {5}^{9} 3. a) {p}^{14} b) {4}^{12} 5. \frac{1}{{q}^{18}} b) \frac{1}{10}
7. a) \frac{1}{{b}^{8}} b) \frac{1}{{4}^{5}} 9. a) 1 b) 1 11. a) -1 b) -1
13. a) 1 b) 25 15. a) 1 b) 1 17. a) -6 b) 0
19. a) \frac{27}{64} b) \frac{{p}^{5}}{32} c) \frac{{x}^{6}}{{y}^{6}} 21. a) \frac{{a}^{4}}{81{b}^{4}} b) \frac{25}{16{m}^{2}} 23. {a}^{2}
25. \frac{1}{{y}^{4}} 27. 1 29. \frac{1}{{m}^{12}}
31. {p}^{30} 33. \frac{1}{{r}^{12}} 35. \frac{{p}{2}}{{r}^{22}}
37. \frac{{w}^{40}}{{x}^{24}} 39. \frac{16{j}^{12}}{81{k}^{4}} 41. \frac{27{c}^{6}}{64{d}^{18}}
43. {k}^{14} 45. \frac{1}{{t}^{3}} 47. 4{p}^{10}
49. -2 51. 8{b}^{6} 53. -11{u}^{12}
55. -\frac{3{b}^{6}}{{a}^{4}} 57. \frac{3{s}^{7}}{5{r}^{5}} 59. -\frac{2{b}^{3}}{3{a}^{5}}
61. \frac{4{q}^{5}{r}}{3{s}^{2}} 63. 2m{n}^{5} 65. 2{b}^{6}
67.

a) 26{a}^{5}
b) 22{a}^{5}
c) 48{a}^{10}
d) 12

 69.

a) {p}^{10}
b) {p}^{24}

 71.

a) {y}^{2}
b) \frac{1}{{y}^{2}}
73. 12{x}^{3} 75. 15{a}^{4} 77. -8{y}^{3}
81. 1000 83. Answers will vary. 85. Answers will vary.

Attributions

This chapter has been adapted from “Divide Monomials” in Elementary Algebra (OpenStax) by Lynn Marecek and MaryAnne Anthony-Smith, which is under a CC BY 4.0 Licence. Adapted by Izabela Mazur. See the Copyright page for more information.

License

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Intermediate Algebra II Copyright © 2021 by Pooja Gupta is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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