# 10.4 Divide monomials  (Page 2/5)

 Page 2 / 5

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}\frac{{x}^{8}}{{x}^{15}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{{12}^{11}}{{12}^{21}}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{{x}^{7}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{{12}^{10}}$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}\frac{{m}^{17}}{{m}^{26}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{{7}^{8}}{{7}^{14}}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{{m}^{9}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{{7}^{6}}$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}\frac{{a}^{5}}{{a}^{9}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{{x}^{11}}{{x}^{7}}$

## Solution

 ⓐ Since 9>5, there are more $a$ 's in the denominator and so we will end up with factors in the denominator. $\frac{{a}^{5}}{{a}^{9}}$ Use the Quotient Property for $n>m,\frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}}.$ Simplify. $\frac{1}{{a}^{4}}$
 ⓑ Notice there are more factors of $x$ in the numerator, since 11>7. So we will end up with factors in the numerator. $\frac{{x}^{11}}{{x}^{7}}$ Use the Quotient Property for $m>n,\frac{{a}^{m}}{{a}^{n}}={a}^{n-m}.$ Simplify. ${x}^{4}$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}\frac{{b}^{19}}{{b}^{11}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{{z}^{5}}{{z}^{11}}$

1. $\phantom{\rule{0.2em}{0ex}}{b}^{8}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{1}{{z}^{6}}$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}\frac{{p}^{9}}{{p}^{17}}$
2. $\phantom{\rule{0.2em}{0ex}}\frac{{w}^{13}}{{w}^{9}}$

1. $\phantom{\rule{0.2em}{0ex}}\frac{1}{{p}^{8}}$
2. $\phantom{\rule{0.2em}{0ex}}{w}^{4}$

## 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 $\frac{{a}^{m}}{{a}^{m}}.$ From earlier work with fractions, we know that

$\frac{2}{2}=1\phantom{\rule{2em}{0ex}}\frac{17}{17}=1\phantom{\rule{2em}{0ex}}\frac{-43}{-43}=1$

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

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

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

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

Now we will simplify $\frac{{a}^{m}}{{a}^{m}}$ in two ways to lead us to the definition of the zero exponent    .

## 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.

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{12}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}{y}^{0}$

## Solution

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

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

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{17}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}{m}^{0}$

1. 1
2. 1

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{k}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}{29}^{0}$

1. 1
2. 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 non-zero expression raised to the zero power is one.

Simplify: ${\left(7z\right)}^{0}.$

## Solution

 ${\left(7z\right)}^{0}$ Use the definition of the zero exponent. 1

Simplify: ${\left(-4y\right)}^{0}.$

1

Simplify: ${\left(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}x\right)}^{0}.$

1

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(-3{x}^{2}y\right)}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}-3{x}^{2}{y}^{0}$

## Solution

 ⓐ The product is raised to the zero power. ${\left(-3{x}^{2}y\right)}^{0}$ Use the definition of the zero exponent. $1$
 ⓑ Notice that only the variable $y$ is being raised to the zero power. ${\left(-3{x}^{2}y\right)}^{0}$ Use the definition of the zero exponent. $-3{x}^{2}\cdot 1$ Simplify. $-3{x}^{2}$

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}{\left(7{x}^{2}y\right)}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}7{x}^{2}{y}^{0}$

1. 1
2. 7 x 2 1

Simplify:

1. $\phantom{\rule{0.2em}{0ex}}-23{x}^{2}{y}^{0}$
2. $\phantom{\rule{0.2em}{0ex}}{\left(-23{x}^{2}y\right)}^{0}$

1. −23 x 2
2. 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.

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

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

We see that ${\left(\frac{x}{y}\right)}^{3}$ is $\frac{{x}^{3}}{{y}^{3}}.$

$\begin{array}{ccccc}\text{We write:}\hfill & & & & {\left(\frac{x}{y}\right)}^{3}\hfill \\ & & & & \frac{{x}^{3}}{{y}^{3}}\hfill \end{array}$

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

${\left(\frac{a}{b}\right)}^{m}=\phantom{\rule{0.2em}{0ex}}\frac{{a}^{m}}{{b}^{m}}$

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

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