3.2 Exponential Properties for Division

Learning Outcomes

Simplify Expressions Using the Quotient Property of Exponents
Simplify exponential expressions containing negative exponents
Simplify exponential expressions containing exponents of 0 and 1

Key words

Quotient: the result of dividing
Exponent: the power that a base number is raised to

Simplifying Expressions Using the Quotient Property of Exponents

So far we have developed the properties of exponents for multiplication. We summarize these properties here.

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: $\left (ab\right )^{n}={a}^{n}{b}^{n}$

Now we will look at the exponent properties for division. We previously learned that fractions may be simplified by dividing out common factors from the numerator and denominator using the Equivalent Fractions Property.

Equivalent Fractions Property

If $a,b,c$ are whole numbers where $b\ne 0,c\ne 0$ , then ${\frac{a}{b}}={\frac{a\cdot c}{b\cdot c}}\text{ and }{\frac{a\cdot c}{b\cdot c}}={\frac{a}{b}}$ .

If the numerator and denominator of a fraction are multiplied or divided by the same non-zero number, the resulting fraction is equivalent to the original fraction.

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

Let’s look at dividing terms containing exponential expressions. What happens if we divide two numbers in exponential form with the same base? Consider the following expression.

$\displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}$

We can rewrite the expression as: $\displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}$ . Then we can cancel the common factors of $4$ in the numerator and denominator: $\displaystyle \frac{\color{red}{4}\cdot \color{red}{4}\cdot 4\cdot 4\cdot 4}{\color{red}{4}\cdot \color{red}{4}}$ . This leaves $4^{3}$ on the numerator and $1$ on the denominator, which simplifies to $4^{3}$ using exponential notation. Notice that the exponent, $3$ , is the difference between the two exponents in the original expression, $5$ and $2$ .

$\displaystyle \frac{{{4}^{5}}}{{{4}^{2}}}=4^{5-2}=4^{3}$ .

As another example, consider $\frac{{2}^{5}}{{2}^{2}}$ :

$\frac{{2}^{5}}{{2}^{2}}=\frac{\color{red}{2\cdot 2}\cdot 2\cdot 2\cdot 2}{\color{red}{2\cdot 2}}=\frac{2\cdot 2\cdot 2\cdot}{1}=2^3$

If we subtract the exponents and keep the common base we get $\frac{{2}^{5}}{{2}^{2}}=2^{5-2}=2^3$ . The same answer we got when we expanded the exponents into multiplication.

Notice that in each case the bases were the same and we subtracted the exponents. So, to divide two exponential terms with the same base, subtract the exponents.

Quotient Property of Exponents

If $a$ is a real number, $a\ne 0$ , and $m,n$ are whole numbers, then ${\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n},m>n$ .

To divide two exponential terms with the same base, keep the base and subtract the exponents.

Examples

Simplify:

1. $\frac{7^8}{7^3}$

$=7^{8-3}=7^5$

2. $\frac{(-5)^9}{(-5)^4}$

$=(-5)^{9-4}=(-5)^5$

Try It

Simplify:

1. $\frac{4^5}{4^3}$

2. $\frac{(-8)^7}{(-8)^5}$

Show Answer

1. $\frac{4^5}{4^3}=4^2$

2. $\frac{(-8)^7}{(-8)^5}=(-8)^{2}$

One as an Exponent

What does $3^1$ or ${(-5)}^1$ equal?

Consider $\frac{2^4}{2^3}$ . If we use the quotient property, $\frac{2^4}{2^3}=2^1$ .

Alternatively, we could expand the exponential terms: $\frac{2^4}{2^3}=\frac{2\cdot 2\cdot 2\cdot 2}{2\cdot 2\cdot 2}$ . Then by cancelling out common factors, we get $\frac{\color{red}{2\cdot 2\cdot 2\cdot} 2}{\color{red}{2\cdot 2\cdot 2}}=2$ .

This means that $2^1=2$ , and leads us to the property that, for any integer $a\text{, }a^1=a$ .

Examples

Simplify:

1. $\frac{7^4}{7^3}$

$=7^{4-3}=7^1=7$

2. $\frac{(-5)^5}{(-5)^4}$

$=(-5)^{5-4}=(-5)^1=-5$

Zero as an Exponent

A special case of the Quotient Property is when the exponents of the numerator and denominator are equal, such as $\frac{{5}^{4}}{{5}^{4}}$ . From earlier work with fractions, we know that,

$\frac{2}{2} =\frac{17}{17} =\frac{-43}{-43} =1$

In words, a non-zero integer divided by itself is $1$ . Remember that $\frac{0}{0}$ is undefined.

We also know that $a\text{, }a^1=a$ , for any non-zero integer $a$ .

Now consider simplifying the term $\frac{8}{8}$ in two different ways.

We know that $\frac{8}{8}=1$ by division.

We also know that $\frac{8}{8}=\frac{8^1}{8^1}=8^{1-1}=8^0$ using the Quotient Property..

This means that $8^0=1$ .

Exponents of 0 or 1

Any integer raised to a power of $1$ is the number itself.

$n^{1}=n$

Any non-zero number integer raised to a power of $0$ is equal to $1$ .

$n^{0}=1$

The quantity $0^{0}$ is undefined.

example

Simplify:

1. ${12}^{0}$

2. ${(-7)}^{0}$

Show Solution

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

1.
	${12}^{0}$
Use the definition of the zero exponent.	$1$

2.
	${(-7)}^{0}$
Use the definition of the zero exponent.	$1$

try it

Negative Exponents

Now let’s see what happens when the denominator has a larger exponent than the numerator, so that when we subtract the exponents, we get a negative integer:

$\frac{{2}^{2}}{{2}^{3}}=\frac{\color{red}{2\cdot 2}}{\color{red}{2\cdot 2}\cdot 2}=\frac{1}{2}$

When we subtract the exponents and keep the common base we get:

$\frac{{2}^{2}}{{2}^{3}}=2^{2-3}=2^{-1}$

This means that $2^{-1}=\frac{1}{2}$ .

Let’s consider one more example:

$\frac{{7}^{2}}{{7}^{5}}=\frac{\color{red}{7\cdot 7}}{\color{red}{7\cdot 7\cdot} 7\cdot 7\cdot 7}=\frac{1}{7^3}$ .

On the other hand, subtracting the exponents and keeping the common base gives us $\frac{{7}^{2}}{{7}^{5}}=7^{2-5}=7^{-3}$ .

So, $\frac{1}{7^3}=7^{-3}$ .

This leads us to the meaning of negative exponents.

NEGATIVE Exponents

If $a$ is a real number, $a\ne 0$ , and $n$ is a whole numbers, then $a^{-n}=\frac{1}{a^n}$ .

A negative exponent is equivalent to a positive exponent on the reciprocal of the number.

When the exponent in the denominator is larger than the exponent in the numerator, the exponent of the quotient will be negative. If the result gives us a negative exponent, the negative exponent tells us to re-write the expression by taking the reciprocal of the base and then changing the sign of the exponent. We rewrite it by using the definition of negative exponents, ${a}^{-n}={\frac{1}{{a}^{n}}}$ . Any expression that has negative exponents is not considered to be in simplest form, so we will use the definition of a negative exponent and other properties of exponents to write an expression with only positive exponents.

Example

Evaluate the expression ${4}^{-3}$ .

Show Solution

First, write the expression with positive exponents by putting the term with the negative exponent in the denominator.

${4}^{-3} = \frac{1}{{4}^{3}} = \frac{1}{4\cdot4\cdot4}$

Now that we have an expression that looks somewhat familiar.

$\frac{1}{4\cdot4\cdot4} = \frac{1}{64}$

Answer

${4}^{-3}=\frac{1}{64}$

example

Simplify:

1. ${4}^{-2}$

2. $\left ( \frac{1}{2}\right ) ^{-3}$

Solution

1.
	${4}^{-2}$
Use the definition of a negative exponent, ${a}^{-n}={\frac{1}{{a}^{n}}}$ .	${\frac{1}{{4}^{2}}}$
Simplify.	${\frac{1}{16}}$

2.
	$\left ( \frac{1}{2}\right ) ^{-3}$
Take the reciprocal and turn the exponent positive.	$\left ( \frac{2}{1}\right )^{3}$
Simplify.	$2^3=8$

try it

In the following video you will see examples of simplifying expressions with negative exponents.

We can now update the quotient property of exponents so that it includes negative and zero exponents by removing the condition that $m\gt n$ .

Quotient Property of Exponents

If $a$ is a real number, $a\ne 0$ , and $m,n$ are integers, then ${\frac{{a}^{m}}{{a}^{n}}}={a}^{m-n}$ .

example

Simplify:

$\frac{{2}^{9}}{{2}^{2}}$

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


	$\frac{{2}^{9}}{{2}^{2}}$
Use the quotient property with $,\frac{{a}^{m}}{{a}^{n}} ={a}^{m-n}$ .	${2}^{\color{red}{9-2}}$
Simplify.	${2}^{7}$

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

Try It

Simplify:

1. $\frac{(6)^9}{(6)^7}$

2. $\frac{(-3)^7}{(-3)^4}$

Show Answer

1. $\frac{(6)^9}{(6)^7}$

$=(6)^{9-7}=6^2$

2. $\frac{(-3)^7}{(-3)^4}$

$={(-3)}^{7-4}={(-3)}^{3}$

example

Simplify:

$\frac{{3}^{3}}{{3}^{5}}$

Solution

Both bases are $3$ so we can subtract the exponents:

$\frac{{3}^{3}}{{3}^{5}}=3^{3-5}=3^{-2}=\frac{1}{3^2}=\frac{1}{9}$

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

Try It

Simplify the terms:

1. $\frac{(-5)^4}{(-5)^7}$

2. $\frac{2^5}{2^{-3}}$

3. $\frac{7^{-4}}{7^{-6}}$

4. $\frac{(-3)^{-6}}{(-3)^{-4}}$

Show Answer

1. $\frac{(-5)^4}{(-5)^7}$

$={(-5)}^{4-7}={(-5)}^{-3}=\left ( \frac{-1}{5}\right )^3=\frac{(-1)^{3}}{5^3}=-\frac{1}{125}$

2. $\frac{2^5}{2^{-3}}$

$=2^{5-(-3)}=2^8=256$

3. $\frac{7^{-4}}{7^{-6}}$

$=7^{-4-(-6)}=7^2=49$

4. $\frac{(-3)^{-6}}{(-3)^{-4}}$

$=(-3)^{-6-(-4)}=(-3)^{-2}=\left ( \frac{-1}{3}\right ) ^2=\frac{1}{9}$

Simplifying Fractions Raised to a Power

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

Let’s look at what happens if you raise a fraction to a power. Remember that a fraction bar means divide. Suppose we have $\displaystyle \frac{5}{4}$ and raise it to the power $3$ .

$\displaystyle {{\left( \frac{5}{4} \right)}^{3}}=\left( \frac{5}{4} \right)\left( \frac{5}{4} \right)\left( \frac{5}{4} \right)=\frac{5\cdot 5\cdot 5}{4\cdot 4\cdot 4}=\frac{{{5}^{3}}}{{{4}^{3}}}$

Raising the fraction to the power of $3$ can also be written as the numerator $5$ to the power of $3$ , and the denominator $4$ to the power of $3$ .

Notice that the exponent applies to both the numerator and the denominator. This leads to the Quotient to a Power Property for Exponents.

Quotient to a Power Property of Exponents

If $a$ and $b$ are real numbers, $b\ne 0$ , and $n$ is an integer number, then ${\left(\frac{a}{b}\right)}^{m}=\frac{{a}^{m}}{{b}^{m}}$ .

To raise a fraction to a power, raise the numerator and denominator to that power.

example

Simplify:

1. ${\left(\frac{5}{8}\right)}^{2}$

2. ${\left(\frac{2}{3}\right)}^{4}$

Show Solution

Solution

1.
	$(\frac{5}{8})^2$
Use the Quotient to a Power Property, ${\left(\frac{a}{b}\right)}^{m} =\frac{{a}^{m}}{{b}^{m}}$ .	$\frac{5^{\color{red}{2}}}{8^{\color{red}{2}}}$
Simplify.	$\frac{25}{64}$

2.
	$(\frac{2}{3})^4$
Use the Quotient to a Power Property, ${\left(\frac{a}{b}\right)}^{m} =\frac{{a}^{m}}{{b}^{m}}$ .	$\frac{2^{\color{red}{4}}}{3^{\color{red}{4}}}$
Simplify.	$\frac{16}{81}$

try it

Let’s looks at some examples of how this rule applies under different circumstances.

Example

Simplify ${\left(\frac{1}{3}\right)}^{-2}$ .

Show Solution

Apply the power property of exponents.

$\frac{{1}^{-2}}{{3}^{-2}}$

Write each term with a positive exponent, the numerator will go to the denominator and the denominator will go to the numerator.

$\frac{{3}^{2}}{{1}^{2}}$

Simplify.

$= \frac{9}{1}{ = }{9}$

Answer

$\frac{{1}^{-2}}{{3}^{-2}}=9$

Example

Simplify. $\frac{1}{4^{-2}}$ Write your answer using positive exponents.

Show Solution

Write each term with a positive exponent, the denominator will go to the numerator.

$\frac{1}{4^{-2}}=1\cdot {4}^{2}=16$

Answer

$\frac{1}{4^{-2}}=16$

Try It

Simplify:

1. ${\left(\frac{5}{2}\right)}^{-2}$

2. $-\frac{3}{5^{-2}}$

Show Answer

1. ${\left(\frac{5}{2}\right)}^{-2}={\left(\frac{2}{5}\right)}^{2}=\frac{4}{25}$

2. $\frac{-3}{5^{-2}}=-3\cdot\frac{1}{5^{-2}}=-3\cdot\frac{5^{2}}{1}=-75$

NEW CHAPTER 3 Properties of Exponents and Scientific Notation