# 9.4 Division of square root expressions

 Page 1 / 1
This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The distinction between the principal square root of the number x and the secondary square root of the number x is made by explanation and by example. The simplification of the radical expressions that both involve and do not involve fractions is shown in many detailed examples; this is followed by an explanation of how and why radicals are eliminated from the denominator of a radical expression. Real-life applications of radical equations have been included, such as problems involving daily output, daily sales, electronic resonance frequency, and kinetic energy.Objectives of this module: be able to use the division property of square roots, the method of rationalizing the denominator, and conjugates to divide square roots.

## Overview

• The Division Property of Square Roots
• Rationalizing the Denominator
• Conjugates and Rationalizing the Denominator

## The division property of square roots

In our work with simplifying square root expressions, we noted that

$\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$

Since this is an equation, we may write it as

$\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}$

To divide two square root expressions, we use the division property of square roots.

## The division property $\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}$

$\frac{\sqrt{x}}{\sqrt{y}}=\sqrt{\frac{x}{y}}$

The quotient of the square roots is the square root of the quotient.

## Rationalizing the denominator

As we can see by observing the right side of the equation governing the division of square roots, the process may produce a fraction in the radicand. This means, of course, that the square root expression is not in simplified form. It is sometimes more useful to rationalize the denominator of a square root expression before actually performing the division.

## Sample set a

Simplify the square root expressions.

$\sqrt{\frac{3}{7}}.$

This radical expression is not in simplified form since there is a fraction under the radical sign. We can eliminate this problem using the division property of square roots.

$\sqrt{\frac{3}{7}}=\frac{\sqrt{3}}{\sqrt{7}}=\frac{\sqrt{3}}{\sqrt{7}}·\frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{3}\sqrt{7}}{7}=\frac{\sqrt{21}}{7}$

$\frac{\sqrt{5}}{\sqrt{3}}.$

A direct application of the rule produces $\sqrt{\frac{5}{3}},$ which must be simplified. Let us rationalize the denominator before we perform the division.

$\frac{\sqrt{5}}{\sqrt{3}}=\frac{\sqrt{5}}{\sqrt{3}}·\frac{\sqrt{3}}{\sqrt{3}}=\frac{\sqrt{5}\sqrt{3}}{3}=\frac{\sqrt{15}}{3}$

$\frac{\sqrt{21}}{\sqrt{7}}=\sqrt{\frac{21}{7}}=\sqrt{3}.$

The rule produces the quotient quickly. We could also rationalize the denominator first and produce the same result.

$\frac{\sqrt{21}}{\sqrt{7}}=\frac{\sqrt{21}}{7}·\frac{\sqrt{7}}{\sqrt{7}}=\frac{\sqrt{21·7}}{7}=\frac{\sqrt{3·7·7}}{7}=\frac{\sqrt{3·{7}^{2}}}{7}=\frac{7\sqrt{3}}{7}=\sqrt{3}$

$\frac{\sqrt{80{x}^{9}}}{\sqrt{5{x}^{4}}}=\sqrt{\frac{80{x}^{9}}{5{x}^{4}}}=\sqrt{16{x}^{5}}=\sqrt{16}\sqrt{{x}^{4}x}=4{x}^{2}\sqrt{x}$

$\frac{\sqrt{50{a}^{3}{b}^{7}}}{\sqrt{5a{b}^{5}}}=\sqrt{\frac{50{a}^{3}{b}^{7}}{5a{b}^{5}}}=\sqrt{10{a}^{2}{b}^{2}}=ab\sqrt{10}$

$\frac{\sqrt{5a}}{\sqrt{b}}.$

Some observation shows that a direct division of the radicands will produce a fraction. This suggests that we rationalize the denominator first.

$\frac{\sqrt{5a}}{\sqrt{b}}=\frac{\sqrt{5a}}{\sqrt{b}}·\frac{\sqrt{b}}{\sqrt{b}}=\frac{\sqrt{5a}\sqrt{b}}{b}=\frac{\sqrt{5ab}}{b}$

$\frac{\sqrt{m-6}}{\sqrt{m+2}}=\frac{\sqrt{m-6}}{\sqrt{m+2}}·\frac{\sqrt{m+2}}{\sqrt{m+2}}=\frac{\sqrt{{m}^{2}-4m-12}}{m+2}$

$\frac{\sqrt{{y}^{2}-y-12}}{\sqrt{y+3}}=\sqrt{\frac{{y}^{2}-y-12}{y+3}}=\sqrt{\frac{\left(y+3\right)\left(y-4\right)}{\left(y+3\right)}}=\sqrt{\frac{\overline{)\left(y+3\right)}\left(y-4\right)}{\overline{)\left(y+3\right)}}}=\sqrt{y-4}$

## Practice set a

Simplify the square root expressions.

$\frac{\sqrt{26}}{\sqrt{13}}$

$\sqrt{2}$

$\frac{\sqrt{7}}{\sqrt{3}}$

$\frac{\sqrt{21}}{3}$

$\frac{\sqrt{80{m}^{5}{n}^{8}}}{\sqrt{5{m}^{2}n}}$

$4m{n}^{3}\sqrt{mn}$

$\frac{\sqrt{196{\left(x+7\right)}^{8}}}{\sqrt{2{\left(x+7\right)}^{3}}}$

$7{\left(x+7\right)}^{2}\sqrt{2\left(x+7\right)}$

$\frac{\sqrt{n+4}}{\sqrt{n-5}}$

$\frac{\sqrt{{n}^{2}-n-20}}{n-5}$

$\frac{\sqrt{{a}^{2}-6a+8}}{\sqrt{a-2}}$

$\sqrt{a-4}$

$\frac{\sqrt{{x}^{3}{}^{n}}}{\sqrt{{x}^{n}}}$

${x}^{n}$

$\frac{\sqrt{{a}^{3m-5}}}{\sqrt{{a}^{m-1}}}$

${a}^{m-2}$

## Conjugates and rationalizing the denominator

To perform a division that contains a binomial in the denominator, such as $\frac{3}{4+\sqrt{6}},$ we multiply the numerator and denominator by a conjugate of the denominator.

## Conjugate

A conjugate of the binomial $a+b$ is $a-b$ . Similarly, a conjugate of $a-b$ is $a+b$ .

Notice that when the conjugates $a+b$ and $a-b$ are multiplied together, they produce a difference of two squares.

$\left(a+b\right)\left(a-b\right)={a}^{2}-ab+ab-{b}^{2}={a}^{2}-{b}^{2}$

This principle helps us eliminate square root radicals, as shown in these examples that illustrate finding the product of conjugates.

$\begin{array}{lll}\left(5+\sqrt{2}\right)\left(5-\sqrt{2}\right)\hfill & =\hfill & {5}^{2}-{\left(\sqrt{2}\right)}^{2}\hfill \\ \hfill & =\hfill & 25-2\hfill \\ \hfill & =\hfill & 23\hfill \end{array}$

$\begin{array}{lll}\left(\sqrt{6}-\sqrt{7}\right)\left(\sqrt{6}+\sqrt{7}\right)\hfill & =\hfill & {\left(\sqrt{6}\right)}^{2}-{\left(\sqrt{7}\right)}^{2}\hfill \\ \hfill & =\hfill & 6-7\hfill \\ \hfill & =\hfill & -1\hfill \end{array}$

## Sample set b

Simplify the following expressions.

$\frac{3}{4+\sqrt{6}}$ .

The conjugate of the denominator is $4-\sqrt{6.}$ Multiply the fraction by 1 in the form of $\frac{4-\sqrt{6}}{4-\sqrt{6}}$ . $\begin{array}{lll}\frac{3}{4+\sqrt{6}}·\frac{4-\sqrt{6}}{4-\sqrt{6}}\hfill & =\hfill & \frac{3\left(4-\sqrt{6}\right)}{{4}^{2}-{\left(\sqrt{6}\right)}^{2}}\hfill \\ \hfill & =\hfill & \frac{12-3\sqrt{6}}{16-6}\hfill \\ \hfill & =\hfill & \frac{12-3\sqrt{6}}{10}\hfill \end{array}$

$\frac{\sqrt{2x}}{\sqrt{3}-\sqrt{5x}}.$

The conjugate of the denominator is $\sqrt{3}+\sqrt{5x.}$ Multiply the fraction by 1 in the form of $\frac{\sqrt{3}+\sqrt{5x}}{\sqrt{3}+\sqrt{5x}}.$

$\begin{array}{lll}\frac{\sqrt{2x}}{\sqrt{3}-\sqrt{5x}}·\frac{\sqrt{3}+\sqrt{5x}}{\sqrt{3}+\sqrt{5x}}\hfill & =\hfill & \frac{\sqrt{2x}\left(\sqrt{3}+\sqrt{5x}\right)}{{\left(\sqrt{3}\right)}^{2}-{\left(\sqrt{5x}\right)}^{2}}\hfill \\ \hfill & =\hfill & \frac{\sqrt{2x}\sqrt{3}+\sqrt{2x}\sqrt{5x}}{3-5x}\hfill \\ \hfill & =\hfill & \frac{\sqrt{6x}+\sqrt{10{x}^{2}}}{3-5x}\hfill \\ \hfill & =\hfill & \frac{\sqrt{6x}+x\sqrt{10}}{3-5x}\hfill \end{array}$

## Practice set b

Simplify the following expressions.

$\frac{5}{9+\sqrt{7}}$

$\frac{45-5\sqrt{7}}{74}$

$\frac{-2}{1-\sqrt{3x}}$

$\frac{-2-2\sqrt{3x}}{1-3x}$

$\frac{\sqrt{8}}{\sqrt{3x}+\sqrt{2x}}$

$\frac{2\sqrt{6x}-4\sqrt{x}}{x}$

$\frac{\sqrt{2m}}{m-\sqrt{3m}}$

$\frac{\sqrt{2m}+\sqrt{6}}{m-3}$

## Exercises

For the following problems, simplify each expressions.

$\frac{\sqrt{28}}{\sqrt{2}}$

$\sqrt{14}$

$\frac{\sqrt{200}}{\sqrt{10}}$

$\frac{\sqrt{28}}{\sqrt{7}}$

2

$\frac{\sqrt{96}}{\sqrt{24}}$

$\frac{\sqrt{180}}{\sqrt{5}}$

6

$\frac{\sqrt{336}}{\sqrt{21}}$

$\frac{\sqrt{162}}{\sqrt{18}}$

3

$\sqrt{\frac{25}{9}}$

$\sqrt{\frac{36}{35}}$

$\frac{6\sqrt{35}}{35}$

$\sqrt{\frac{225}{16}}$

$\sqrt{\frac{49}{225}}$

$\frac{7}{15}$

$\sqrt{\frac{3}{5}}$

$\sqrt{\frac{3}{7}}$

$\frac{\sqrt{21}}{7}$

$\sqrt{\frac{1}{2}}$

$\sqrt{\frac{5}{2}}$

$\frac{\sqrt{10}}{2}$

$\sqrt{\frac{11}{25}}$

$\sqrt{\frac{15}{36}}$

$\frac{\sqrt{15}}{6}$

$\sqrt{\frac{5}{16}}$

$\sqrt{\frac{7}{25}}$

$\frac{\sqrt{7}}{5}$

$\sqrt{\frac{32}{49}}$

$\sqrt{\frac{50}{81}}$

$\frac{5\sqrt{2}}{9}$

$\frac{\sqrt{125{x}^{5}}}{\sqrt{5{x}^{3}}}$

$\frac{\sqrt{72{m}^{7}}}{\sqrt{2{m}^{3}}}$

$6{m}^{2}$

$\frac{\sqrt{162{a}^{11}}}{\sqrt{2{a}^{5}}}$

$\frac{\sqrt{75{y}^{10}}}{\sqrt{3{y}^{4}}}$

$5{y}^{3}$

$\frac{\sqrt{48{x}^{9}}}{\sqrt{3{x}^{2}}}$

$\frac{\sqrt{125{a}^{14}}}{\sqrt{5{a}^{5}}}$

$5{a}^{4}\sqrt{a}$

$\frac{\sqrt{27{a}^{10}}}{\sqrt{3{a}^{5}}}$

$\frac{\sqrt{108{x}^{21}}}{\sqrt{3{x}^{4}}}$

$6{x}^{8}\sqrt{x}$

$\frac{\sqrt{48{x}^{6}{y}^{7}}}{\sqrt{3xy}}$

$\frac{\sqrt{45{a}^{3}{b}^{8}{c}^{2}}}{\sqrt{5a{b}^{2}c}}$

$3a{b}^{3}\sqrt{c}$

$\frac{\sqrt{66{m}^{12}{n}^{15}}}{\sqrt{11m{n}^{8}}}$

$\frac{\sqrt{30{p}^{5}{q}^{14}}}{\sqrt{5{q}^{7}}}$

${p}^{2}{q}^{3}\sqrt{6pq}$

$\frac{\sqrt{b}}{\sqrt{5}}$

$\frac{\sqrt{5x}}{\sqrt{2}}$

$\frac{\sqrt{10x}}{2}$

$\frac{\sqrt{2{a}^{3}b}}{\sqrt{14a}}$

$\frac{\sqrt{3{m}^{4}{n}^{3}}}{\sqrt{6m{n}^{5}}}$

$\frac{m\sqrt{2m}}{2n}$

$\frac{\sqrt{5{\left(p-q\right)}^{6}{\left(r+s\right)}^{4}}}{\sqrt{25{\left(r+s\right)}^{3}}}$

$\frac{\sqrt{m\left(m-6\right)-{m}^{2}+6m}}{\sqrt{3m-7}}$

0

$\frac{\sqrt{r+1}}{\sqrt{r-1}}$

$\frac{\sqrt{s+3}}{\sqrt{s-3}}$

$\frac{\sqrt{{s}^{2}-9}}{s-3}$

$\frac{\sqrt{{a}^{2}+3a+2}}{\sqrt{a+1}}$

$\frac{\sqrt{{x}^{2}-10x+24}}{\sqrt{x-4}}$

$\sqrt{x-6}$

$\frac{\sqrt{{x}^{2}-2x-8}}{\sqrt{x+2}}$

$\frac{\sqrt{{x}^{2}-4x+3}}{\sqrt{x-3}}$

$\sqrt{x-1}$

$\frac{\sqrt{2{x}^{2}-x-1}}{\sqrt{x-1}}$

$\frac{-5}{4+\sqrt{5}}$

$\frac{-20+5\sqrt{5}}{11}$

$\frac{1}{1+\sqrt{x}}$

$\frac{2}{1-\sqrt{a}}$

$\frac{2\left(1+\sqrt{a}\right)}{1-a}$

$\frac{-6}{\sqrt{5}-1}$

$\frac{-6}{\sqrt{7}+2}$

$-2\left(\sqrt{7}-2\right)$

$\frac{3}{\sqrt{3}-\sqrt{2}}$

$\frac{4}{\sqrt{6}+\sqrt{2}}$

$\sqrt{6}-\sqrt{2}$

$\frac{\sqrt{5}}{\sqrt{8}-\sqrt{6}}$

$\frac{\sqrt{12}}{\sqrt{12}-\sqrt{8}}$

$3+\sqrt{6}$

$\frac{\sqrt{7x}}{2-\sqrt{5x}}$

$\frac{\sqrt{6y}}{1+\sqrt{3y}}$

$\frac{\sqrt{6y}-3y\sqrt{2}}{1-3y}$

$\frac{\sqrt{2}}{\sqrt{3}-\sqrt{2}}$

$\frac{\sqrt{a}}{\sqrt{a}+\sqrt{b}}$

$\frac{a-\sqrt{ab}}{a-b}$

$\frac{\sqrt{{8}^{3}{b}^{5}}}{4-\sqrt{2ab}}$

$\frac{\sqrt{7x}}{\sqrt{5x}+\sqrt{x}}$

$\frac{\sqrt{35}-\sqrt{7}}{4}$

$\frac{\sqrt{3y}}{\sqrt{2y}-\sqrt{y}}$

## Exercises for review

( [link] ) Simplify ${x}^{8}{y}^{7}\left(\frac{{x}^{4}{y}^{8}}{{x}^{3}{y}^{4}}\right).$

${x}^{9}{y}^{11}$

( [link] ) Solve the compound inequality $-8\le 7-5x\le -23.$

( [link] ) Construct the graph of $y=\frac{2}{3}x-4.$

( [link] ) The symbol $\sqrt{x}$ represents which square root of the number $x,\text{\hspace{0.17em}}x\ge 0$ ?

( [link] ) Simplify $\sqrt{{a}^{2}+8a+16}$ .

$a+4$

mcq creativity involves
Sanjika
mCQ creativity involves
Sanjika
ology means a study of
is a greek word means soul
SharaAmor
science
Tina
a branch of knowledge
Kanchan
psych means soul and ology means to study.
Wajid
Can someone help me with tha latent functions of different social institutions
Usman
***interestinglydifferenttopics.blogspot.com/2020/10/reflexes-psychological-debate.html Do check this out and please correct me if I'm wrong anywhere in this
Factors that affect learning?
I am struggling with survivors guilt and complicated grief after traumatic death of spouse. how to cope.
Try to talk with someone you trust and join people who makes you feel worthy. try to help someone. Helping someone makes you feel better
sherlock
I think you must seek professional help.
I am extremely sorry for your loss...I know that the pain of your loss is overwhelming and that you are experiencing all kinds of difficult and unexpected emotions. please don't get me wrong but the best way to cope is to move on..i know its hard..but that's the only way..(continued.)
Princess
stop burdening yourself with thoughts that say you could have done something..you couldn't..your just a human being and you cant always be a superman who saves the day..please dont feel guilty when you have no reason to..(cont.)
Princess
learn to love yourself again..learn to live ..for yourself,for the people who love you and need you in their lives.live for people who you once loved...Think about how you can make your life meaningful once again.It might seem impossible but trust me its necessary ..
Princess
Let your feelings of pain out ..you dont have to suffer ..talk to people whom you trust and love. take care and have a beautiful life ahead.
Princess
First and foremost -- Wow Rowdy! You have been through a highly difficult situation in your life, hands down. As a wife, I simply cannot phathom nor imagine slightly the feelings and thoughts you bare and I'm sure as these words do not help and hit rewind; my heart and prayers are going out to you-
Stephanie
today. I will say as well, everything you are feeling and going through, please know that these feelings and thoughts are all allowed, it is okay for you to find yourself feeling or thinking things that you would never find yourself wanting especially given the situation. I'm sure you question --
Stephanie
yourself, which in turn could make you question who you are right now and what this means for you as a person now and in the future. This is all okay, but there will have to be an understanding as to how to grow from these feelings and thoughts in a more humane manner l as these feelings and ---
Stephanie
thoughts could take a turn for a much more negative outcome as you find yourself trapped in a darkness that, if strong enough, trials and tribulations take years of complex development and experience, less strong minded individuals sometimes either never find the light, or end up being stuck in ----
Stephanie
a subconscious netherworld as they are consumed given the manipulation that was not understood to overcome. It results in losing all sense of heart, faith, soul and intelligence so that you don't grow, which in your case, growth is crucial. This is what your spouse will do their best in trying -----
Stephanie
to guide you through; keep an your senses wide open so that you can learn when she is trying to communicate.
Stephanie
Angela
Thanks angela
Ali
Ali
what is this
Maryam
Geo
Geo
actually I don't know
Maryam
I badly need some for my research :<
Geo
But I need that😥
Mohammed
***courses.lumenlearning.com /boundless-sociology/chapter/theories-of-socialization/
Jobin
branches of psychology
neuropsychology
jeddy
behaviorism
Mira
Legal psychology
Miljana
Developmental psychology, Sports psychology , Clinical psychology , Marketing psychology , counseling psychology , Biological psychology, Educational psychology , Positive psychology etc.
child.psychology
Nss
social, occupational psychology
jeddy
what does it mean when the temporal lobe is anterior to occipital lobe
the cortex refers to what part of the brain
John
the cortex refers to what part of the brain
John
cerebral cortex
jeddy
behind?
jeddy
in what time of situation is training most useful?
Nyakuei
i think its not about suitation, its about the thing tatz need training.
gokul
training is possibly warranted for all situations.
rhoda
I would think that training should be an ungoing thing. That's before, during and after. just in case there's a reoccurrence
Angela
any time
Aaronz
what is Karma
I think the summation of good and bad deeds you do in your life. Accidentally or otherwise.
Yes. I totally agree.
ana
And if it is bad then the root must have been too.
ana
Yes , In Hinduism we also believe that our parent's Karma also applies to us ! Don't know about that , but it kept people in check .
Adding to Swarada point. karma can be good or bad. both will have consequences. karma is biasedness you have shown in your life. like a king could not have two rules for normal man and his son, that comes under karma.
Sujeet
Oh perfectly said!
Ok. It is also said that when one does something... bad or good..it will come back to you 10 folds of the same.
ana
yea that too ! I remember my mother always said if u stole someone's 10 rupees ur 100 will be stolen ( excuse my English ! I'm a non - English speaking person)
karma is nothing but what one usually mean when they say- what you sow, so shall you reap.
Alisha
certainly
Crazy
more important.. this concept was told by lord Krishna as part of spirituality, to focus on your good deed ,the purpose you are born for and your responsibilities. Not for hoping that bad will happen with people who made you feel bad. just giving example ;)
Sujeet
karma is the boomerang of your actions. what u put out in the universe u get back
David
Who is the founder of psychology
Wilhelm wundt
Amanda
Wilhelm Wundt
Sneha
William wundth
Ihrar
Wilhelm Wundt
Megha
and William James
Amanda
Wilhelm Wundt
Wilhelm Wundt is the founder of structuralism
Angela
Yes and also the "Father of Psychology" I think the correct word is Father not Founder.
oh ok. noted
Angela
i believe it goes beyond him I believe it is Ancient but un formalized
David
David
Oh thanks! But u summarised it well!
what is up my fellow psych peeps!!
what are some academic research on developmental psychology
Chezy
Sawrada Founder of Psychology is W.Wundt but Father of Psychology is Sigmund Freud...
Wajid
what are some good schools to study psychology?
George Mason, Penn State, UVA, university of Michigan
Jasmine
Hi I'm currently doing my experimental psychology research can you suggest a topic or research topic for my research thank you and please see 😊
Hello Jerico. I could help you out with some ideas if you're interested!
Yasmine
can u give me some ideas too?
Geo
For sure. It depends if you're trying to do an experiment (using statistics probably) or a case study what are you aiming for?
Yasmine
Hello Jerico. Try to opt for something simple and easy to do. I wouls say perhaps study the effects of color on concentration? You could get a group of people and have them take a memory test (you can find some online) in a white room for example. Then have them take another memory test
Yasmine
similar to the first one but in a different room and then you check the responses if the performance is better
Yasmine
but you have to know that a lot of factors will interfere in the performance such as mental state etc so make sure to choose a certain specific population. that's very important
Yasmine
you could be a bit more specific about it , to isolate the variables that can mess it up like age, sex, etc... you gotta take that into account
Yasmine
so that the results get somewhat accurate
Yasmine
that is very detailed information I really like it thank you very much 😊 your so very kind
Jerico
You're very welcome
Yasmine
what are your interests in general?
Yasmine
Althogh I'm persuing botany and engineering.... but I also want to study human behaviour and psychology.... I would like if anyone can share their knowledge with me !!!!
Crazy
I'm interested on effect of electroconvulsive therapy on people having a depression
Jerico
What is Electroconvulsive therapy?
Pixeled
Astral Projection
Mansi
Electroconvulsive therapy (ECT) is a medical treatment that is most commonly used in patients with severe major depression.
Jerico
What are the expected dutties functions and responsibilities of a head of an agency based on the principles, Practice of industrial psychology?
what are the psychological characteristics expected from an employee wether regular or occupying a position?
Paul
Got questions? Join the online conversation and get instant answers!