Simplifying Variable Expressions Using Exponent Properties II

Learning Outcomes

Simplify expressions using the Quotient Property of Exponents

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 $a\text{ and }b$ are real numbers and $m\text{ and }n$ are whole numbers, then

$\begin{array}{cccc}\text{Product Property}\hfill & & & \hfill {a}^{m}\cdot {a}^{n}={a}^{m+n}\hfill \\ \text{Power Property}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}={a}^{m\cdot n}\hfill \\ \text{Product to a Power}\hfill & & & \hfill {\left(ab\right)}^{m}={a}^{m}{b}^{m}\hfill \end{array}$

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.

Equivalent Fractions Property

If $a,b,c$ are whole numbers where $b\ne 0,c\ne 0$ , then

${\Large\frac{a}{b}}={\Large\frac{a\cdot c}{b\cdot c}}\text{ and }{\Large\frac{a\cdot c}{b\cdot c}}={\Large\frac{a}{b}}$

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 you divide two numbers in exponential form with the same base? Consider the following expression.

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

You can rewrite the expression as: $\displaystyle \frac{4\cdot 4\cdot 4\cdot 4\cdot 4}{4\cdot 4}$ . Then you can cancel the common factors of $4$ in the numerator and denominator: $\displaystyle$

Finally, this expression can be rewritten as $4^{3}$ using exponential notation. Notice that the exponent, $3$ , is the difference between the two exponents in the original expression, $5$ and $2$ .

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

Now, let’s consider an example in which the base is the variable $x$ .

$\begin{array}{cccccccccc}\text{Consider}\hfill & & & \hfill {\Large\frac{{x}^{5}}{{x}^{2}}}\hfill & & & \text{and}\hfill & & & \hfill {\Large\frac{{x}^{2}}{{x}^{3}}}\hfill \\ \text{What do they mean?}\hfill & & & \hfill {\Large\frac{x\cdot x\cdot x\cdot x\cdot x}{x\cdot x}}\hfill & & & & & & \hfill {\Large\frac{x\cdot x}{x\cdot x\cdot x}}\hfill \\ \text{Use the Equivalent Fractions Property.}\hfill & & & \hfill \frac{\overline{)x}\cdot \overline{)x}\cdot x\cdot x\cdot x}{\overline{)x}\cdot \overline{)x}\cdot 1}\hfill & & & & & & \hfill \frac{\overline{)x}\cdot \overline{)x}\cdot 1}{\overline{)x}\cdot \overline{)x}\cdot x}\hfill \\ \text{Simplify.}\hfill & & & \hfill {x}^{3}\hfill & & & & & & \hfill {\Large\frac{1}{x}}\hfill \end{array}$

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.

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:

$\begin{array}{ccccc}\frac{{x}^{5}}{{x}^{2}}\hfill & & & & \hfill \frac{{x}^{2}}{{x}^{3}}\hfill \\ {x}^{5 - 2}\hfill & & & & \hfill \frac{1}{{x}^{3 - 2}}\hfill \\ {x}^{3}\hfill & & & & \hfill \frac{1}{x}\hfill \end{array}$

Quotient Property of Exponents

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

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

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

$\begin{array}{cccc}\frac{{3}^{4}}{{3}^{2}}\stackrel{?}{=}{3}^{4 - 2}\hfill & & & \hfill \frac{{5}^{2}}{{5}^{3}}\stackrel{?}{=}\frac{1}{{5}^{3 - 2}}\hfill \\ \frac{81}{9}\stackrel{?}{=}{3}^{2}\hfill & & & \hfill \frac{25}{125}\stackrel{?}{=}\frac{1}{{5}^{1}}\hfill \\ 9=9 \hfill & & & \hfill \frac{1}{5}=\frac{1}{5}\hfill \end{array}$

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

Simplify:

1. $\Large\frac{{x}^{10}}{{x}^{8}}$
2. $\Large\frac{{2}^{9}}{{2}^{2}}$

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

1.
Since 10 > 8, there are more factors of $x$ in the numerator.	$\Large\frac{{x}^{10}}{{x}^{8}}$
Use the quotient property with $m>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={a}^{m-n}$ .	${x}^{\color{red}{10-8}}$
Simplify.	${x}^{2}$

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

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

try it

example

Simplify:

1. $\Large\frac{{b}^{10}}{{b}^{15}}$
2. $\Large\frac{{3}^{3}}{{3}^{5}}$

Show Solution

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

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

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

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

Now let’s see if you can determine when you will end up with factors in the denominator, and when you will end up with factors in the numerator.

example

Simplify:

1. $\Large\frac{{a}^{5}}{{a}^{9}}$
2. $\Large\frac{{x}^{11}}{{x}^{7}}$

Show Solution

Solution

1.
Since $9>5$ , there are more $a$ ‘s in the denominator and so we will end up with factors in the denominator.	$\Large\frac{{a}^{5}}{{a}^{9}}$
Use the Quotient Property for $n>m,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize =\Large\frac{1}{{a}^{n-m}}$ .	$\Large\frac{\color{red}{1}}{{a}^{\color{red}{9-5}}}$
Simplify.	$\Large\frac{1}{{a}^{4}}$

2.
Notice there are more factors of $x$ in the numerator, since 11 > 7. So we will end up with factors in the numerator.	$\Large\frac{{x}^{11}}{{x}^{7}}$
Use the Quotient Property for $m>n,\Large\frac{{a}^{m}}{{a}^{n}}\normalsize ={a}^{n-m}$ .	${x}^{\color{red}{11-7}}$
Simplify.	${x}^{4}$

try it

When dividing terms that also contain coefficients, divide the coefficients and then divide variable powers with the same base by subtracting the exponents.

Example

Simplify. $\displaystyle \frac{12{{x}^{4}}}{2x}$

Show Solution

Separate into numerical and variable factors.

(\frac{12}{2}) (\frac{x^{4}}{x})

$\displaystyle \left( \frac{12}{2} \right)\left( \frac{{{x}^{4}}}{x} \right)$

Since the bases of the exponents are the same, you can apply the Quotient Rule. Divide the coefficients and subtract the exponents of matching variables.

$\displaystyle 6\left( {{x}^{4-1}} \right)$

Answer
$\frac{12{{x}^{4}}}{2x}=6{{x}^{3}}$

Watch the following video for more examples of how to simplify quotients that contain exponents. Pay attention to the last example where we demonstrate the difference between subtracting terms with exponents, and subtracting exponents to simplify a quotient.

Simplify Quotients Raised to a Power

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

Now let’s look at what happens if you raise a quotient to a power. Remember that quotient means divide. Suppose you have $\displaystyle \frac{3}{4}$ and raise it to the [latex]3^rd[/latex]power.

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

You can see that raising the quotient to the power of $3$ can also be written as the numerator $(3)$ to the power of $3$ , and the denominator $(4)$ to the power of $3$ .

Similarly, if you are using variables, the quotient raised to a power is equal to the numerator raised to the power over the denominator raised to power.

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

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 $m$ is a counting number, then

${\left(\Large\frac{a}{b}\normalsize\right)}^{m}=\Large\frac{{a}^{m}}{{b}^{m}}$
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:

example

Simplify:

1. ${\left(\Large\frac{5}{8}\normalsize\right)}^{2}$
2. ${\left(\Large\frac{x}{3}\normalsize\right)}^{4}$
3. ${\left(\Large\frac{y}{m}\normalsize\right)}^{3}$

Show Solution

Solution

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

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

3.
	$\Large(\frac{y}{m})^3$
Raise the numerator and denominator to the third power.	$\Large\frac{y^{3}}{m^{3}}$

try it

Example

Simplify. $\displaystyle {{\left( \frac{2{x}^{2}y}{x} \right)}^{3}}$

Show Solution

Apply the power to each factor individually.

$\displaystyle \frac{{{2}^{3}{\left({x}^{2}\right)}^{3}{y}^{3}}}{{{x}^{3}}}$

Separate into numerical and variable factors.

$\displaystyle {{2}^{3}}\cdot \frac{{{x}^{3\cdot2}}}{{{x}^{3}}}\cdot \frac{{{y}^{3}}}{1}$

Simplify by taking $2$ to the third power and applying the Power and Quotient Rules for exponents—multiply and subtract the exponents of matching variables.

$\displaystyle 8\cdot {{x}^{(6-3)}}\cdot {{y}^{3}}$

Simplify.

$\displaystyle 8{{x}^{3}}{{y}^{3}}$

Answer

$\displaystyle {{\left( \frac{2{x}^{2}y}{x} \right)}^{3}}=8{{x}^{3}}{{y}^{3}}$

For more examples of how to simplify a quotient raised to a power, watch the following video.

In the following video you will be shown examples of simplifying quotients that are raised to a power.

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UNIT 4