# 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.

can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?