9.1 Square root expressions  (Page 2/2)

 Page 2 / 2

The number $\sqrt{50}$ is between what two whole numbers?

Since ${7}^{2}=49,\text{\hspace{0.17em}}\sqrt{49}=7.$

Since ${8}^{2}=64,\text{\hspace{0.17em}}\sqrt{64}=8.$ Thus,

$7<\sqrt{50}<8$

Thus, $\sqrt{50}$ is a number between 7 and 8.

Practice set b

Write the principal and secondary square roots of each number.

100

$\sqrt{100}=10$ and $-\sqrt{100}=-10$

121

$\sqrt{121}=11$ and $-\sqrt{121}=-11$

35

$\sqrt{35}$ and $-\sqrt{35}$

Use a calculator to obtain a decimal approximation for the two square roots of 35. Round to two decimal places.

$5.92\text{\hspace{0.17em}and\hspace{0.17em}}-5.92$

Meaningful expressions

Since we know that the square of any real number is a positive number or zero, we can see that expressions such as $\sqrt{-16}$ do not describe real numbers. There is no real number that can be squared that will produce −16. For $\sqrt{x}$ to be a real number, we must have $x\ge 0.$ In our study of algebra, we will assume that all variables and all expressions in radicands represent nonnegative numbers (numbers greater than or equal to zero).

Sample set c

Write the proper restrictions that must be placed on the variable so that each expression represents a real number.

For $\sqrt{x-3}$ to be a real number, we must have

$\begin{array}{lllll}x-3\ge 0\hfill & \hfill & \text{or}\hfill & \hfill & x\ge 3\hfill \end{array}$

For $\sqrt{2m+7}$ to be a real number, we must have

$\begin{array}{lllllllll}2m+7\ge 0\hfill & \hfill & \text{or}\hfill & \hfill & 2m\ge -7\hfill & \hfill & \text{or}\hfill & \hfill & m\ge \frac{-7}{2}\hfill \end{array}$

Practice set c

Write the proper restrictions that must be placed on the variable so that each expression represents a real number.

$\sqrt{x+5}$

$x\ge -5$

$\sqrt{y-8}$

$y\ge 8$

$\sqrt{3a+2}$

$a\ge -\frac{2}{3}$

$\sqrt{5m-6}$

$m\ge \frac{6}{5}$

Simplifying square roots

When variables occur in the radicand, we can often simplify the expression by removing the radical sign. We can do so by keeping in mind that the radicand is the square of some other expression. We can simplify a radical by seeking an expression whose square is the radicand. The following observations will help us find the square root of a variable quantity.

Since ${\left({x}^{3}\right)}^{2}={x}^{3}\text{\hspace{0.17em}}·{\text{\hspace{0.17em}}}^{2}={x}^{6},{x}^{3}$ is a square root of ${x}^{6}.$ Also

Since ${\left({x}^{4}\right)}^{2}={x}^{4·2}={x}^{8},{x}^{4}$ is a square root of ${x}^{8}.$ Also

Since ${\left({x}^{6}\right)}^{2}={x}^{6·2}={x}^{12},{x}^{6}$ is a square root of ${x}^{12}.$ Also

These examples suggest the following rule:

If a variable has an even exponent, its square root can be found by dividing that exponent by 2.

The examples of Sample Set B illustrate the use of this rule.

Sample set d

Simplify each expression by removing the radical sign. Assume each variable is nonnegative .

$\begin{array}{lll}\sqrt{{a}^{2}}.\hfill & \hfill & \text{We\hspace{0.17em}seek\hspace{0.17em}an\hspace{0.17em}expression\hspace{0.17em}whose\hspace{0.17em}square\hspace{0.17em}is\hspace{0.17em}}{a}^{2}.\text{\hspace{0.17em}Since}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(a\right)}^{2}={a}^{2},\hfill \\ \sqrt{{a}^{2}}=a\hfill & \hfill & \text{Notice\hspace{0.17em}that\hspace{0.17em}}2÷2=1.\hfill \end{array}$

$\begin{array}{lll}\sqrt{{y}^{8}}.\hfill & \hfill & \text{We\hspace{0.17em}seek\hspace{0.17em}an\hspace{0.17em}expression\hspace{0.17em}whose\hspace{0.17em}square\hspace{0.17em}is\hspace{0.17em}}{y}^{8}.\text{\hspace{0.17em}Since}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left({y}^{4}\right)}^{2}={y}^{8},\hfill \\ \sqrt{{y}^{8}}={y}^{4}\hfill & \hfill & \text{Notice\hspace{0.17em}that\hspace{0.17em}}8÷2=4.\hfill \end{array}$

$\begin{array}{lll}\sqrt{25{m}^{2}{n}^{6}}.\hfill & \hfill & \text{We\hspace{0.17em}seek\hspace{0.17em}an\hspace{0.17em}expression\hspace{0.17em}whose\hspace{0.17em}square\hspace{0.17em}is\hspace{0.17em}}25{m}^{2}{n}^{6}.\text{\hspace{0.17em}Since}\text{\hspace{0.17em}}\text{\hspace{0.17em}}{\left(5m{n}^{3}\right)}^{2}=25{m}^{2}{n}^{6},\hfill \\ \sqrt{25{m}^{2}{n}^{6}}=5m{n}^{3}\hfill & \hfill & \text{Notice\hspace{0.17em}that\hspace{0.17em}}2÷2=1\text{\hspace{0.17em}and\hspace{0.17em}}6÷2=3.\hfill \end{array}$

$\begin{array}{lllll}-\sqrt{121{a}^{10}{\left(b-1\right)}^{4}}.\hfill & \hfill & \hfill & \hfill & \text{We\hspace{0.17em}seek\hspace{0.17em}an\hspace{0.17em}expression\hspace{0.17em}whose\hspace{0.17em}square\hspace{0.17em}is\hspace{0.17em}}121{a}^{10}{\left(b-1\right)}^{4}.\text{\hspace{0.17em}}\text{Since}\hfill \\ {\left[11{a}^{5}{\left(b-1\right)}^{2}\right]}^{2}\hfill & =\hfill & 121{a}^{10}{\left(b-1\right)}^{4},\hfill & \hfill & \hfill \\ \sqrt{121{a}^{10}{\left(b-1\right)}^{4}}\hfill & =\hfill & 11{a}^{5}{\left(b-1\right)}^{2}\hfill & \hfill & \hfill \\ \text{Then,}\text{\hspace{0.17em}}-\sqrt{121{a}^{10}{\left(b-1\right)}^{4}}\hfill & =\hfill & -11{a}^{5}{\left(b-1\right)}^{2}\hfill & \hfill & \text{Notice\hspace{0.17em}that\hspace{0.17em}}10÷2=5\text{\hspace{0.17em}and\hspace{0.17em}}4÷2=2.\hfill \end{array}$

Practice set d

Simplify each expression by removing the radical sign. Assume each variable is nonnegative.

$\sqrt{{y}^{8}}$

${y}^{4}$

$\sqrt{16{a}^{4}}$

$4{a}^{2}$

$\sqrt{49{x}^{4}{y}^{6}}$

$7{x}^{2}{y}^{3}$

$-\sqrt{100{x}^{8}{y}^{12}{z}^{2}}$

$-10{x}^{4}{y}^{6}z$

$-\sqrt{36{\left(a+5\right)}^{4}}$

$-6{\left(a+5\right)}^{2}$

$\sqrt{225{w}^{4}{\left({z}^{2}-1\right)}^{2}}$

$15{w}^{2}\left({z}^{2}-1\right)$

$\sqrt{0.25{y}^{6}{z}^{14}}$

$0.5{y}^{3}{z}^{7}$

$\sqrt{{x}^{2n}},$ where $n$ is a natural number.

${x}^{n}$

$\sqrt{{x}^{4n}},$ where $n$ is a natural number.

${x}^{2n}$

Exercises

How many square roots does every positive real number have?

two

The symbol $\sqrt{\begin{array}{cc}& \end{array}}$ represents which square root of a number?

The symbol – $\sqrt{\begin{array}{cc}& \end{array}}$ represents which square root of a number?

secondary

For the following problems, find the two square roots of the given number.

64

81

9 and $-9$

25

121

11 and $-11$

144

225

15 and $-15$

10,000

$\frac{1}{16}$

$\frac{1}{4}\text{\hspace{0.17em}}\text{and}-\frac{1}{4}$

$\frac{1}{49}$

$\frac{25}{36}$

$\frac{5}{6}\text{\hspace{0.17em}and\hspace{0.17em}}-\frac{5}{6}$

$\frac{121}{225}$

$0.04$

$0.2\text{\hspace{0.17em}}\text{and}-0.2$

$0.16$

$1.21$

$1.1\text{\hspace{0.17em}}\text{and}-1.1$

For the following problems, evaluate each expression. If the expression does not represent a real number, write "not a real number."

$\sqrt{49}$

$\sqrt{64}$

8

$-\sqrt{36}$

$-\sqrt{100}$

$-10$

$-\sqrt{169}$

$-\sqrt{\frac{36}{81}}$

$-\frac{2}{3}$

$-\sqrt{\frac{121}{169}}$

$\sqrt{-225}$

not a real number

$\sqrt{-36}$

$-\sqrt{-1}$

not a real number

$-\sqrt{-5}$

$-\left(-\sqrt{9}\right)$

3

$-\left(-\sqrt{0.81}\right)$

For the following problems, write the proper restrictions that must be placed on the variable so that the expression represents a real number.

$\sqrt{y+10}$

$y\ge -10$

$\sqrt{x+4}$

$\sqrt{a-16}$

$a\ge 16$

$\sqrt{h-11}$

$\sqrt{2k-1}$

$k\ge \frac{1}{2}$

$\sqrt{7x+8}$

$\sqrt{-2x-8}$

$x\le -4$

$\sqrt{-5y+15}$

For the following problems, simplify each expression by removing the radical sign.

$\sqrt{{m}^{6}}$

${m}^{3}$

$\sqrt{{k}^{10}}$

$\sqrt{{a}^{8}}$

${a}^{4}$

$\sqrt{{h}^{16}}$

$\sqrt{{x}^{4}{y}^{10}}$

${x}^{2}{y}^{5}$

$\sqrt{{a}^{6}{b}^{20}}$

$\sqrt{{a}^{4}{b}^{6}}$

${a}^{2}{b}^{3}$

$\sqrt{{x}^{8}{y}^{14}}$

$\sqrt{81{a}^{2}{b}^{2}}$

$9ab$

$\sqrt{49{x}^{6}{y}^{4}}$

$\sqrt{100{m}^{8}{n}^{2}}$

$10{m}^{4}n$

$\sqrt{225{p}^{14}{r}^{16}}$

$\sqrt{36{x}^{22}{y}^{44}}$

$6{x}^{11}{y}^{22}$

$\sqrt{169{w}^{4}{z}^{6}{\left(m-1\right)}^{2}}$

$\sqrt{25{x}^{12}{\left(y-1\right)}^{4}}$

$5{x}^{6}{\left(y-1\right)}^{2}$

$\sqrt{64{a}^{10}{\left(a+4\right)}^{14}}$

$\sqrt{9{m}^{6}{n}^{4}{\left(m+n\right)}^{18}}$

$3{m}^{3}{n}^{2}{\left(m+n\right)}^{9}$

$\sqrt{25{m}^{26}{n}^{42}{r}^{66}{s}^{84}}$

$\sqrt{{\left(f-2\right)}^{2}{\left(g+6\right)}^{4}}$

$\left(f-2\right){\left(g+6\right)}^{4}$

$\sqrt{{\left(2c-3\right)}^{6}+{\left(5c+1\right)}^{2}}$

$-\sqrt{64{r}^{4}{s}^{22}}$

$-8{r}^{2}{s}^{11}$

$-\sqrt{121{a}^{6}{\left(a-4\right)}^{8}}$

$-\left[-\sqrt{{\left(w+6\right)}^{2}}\right]$

$w+6$

$-\left[-\sqrt{4{a}^{2}{b}^{2}{\left({c}^{2}+8\right)}^{2}}\right]$

$\sqrt{1.21{h}^{4}{k}^{4}}$

$1.1{h}^{2}{k}^{2}$

$\sqrt{2.25{m}^{6}{p}^{6}}$

$-\sqrt{\frac{169{a}^{2}{b}^{4}{c}^{6}}{196{x}^{4}{y}^{6}{z}^{8}}}$

$-\frac{13a{b}^{2}{c}^{3}}{14{x}^{2}{y}^{3}{z}^{4}}$

$-\left[\sqrt{\frac{81{y}^{4}{\left(z-1\right)}^{2}}{225{x}^{8}{z}^{4}{w}^{6}}}\right]$

Exercised for review

( [link] ) Find the quotient. $\frac{{x}^{2}-1}{4{x}^{2}-1}÷\frac{x-1}{2x+1}.$

$\frac{x+1}{2x-1}$

( [link] ) Find the sum. $\frac{1}{x+1}+\frac{3}{x+1}+\frac{2}{{x}^{2}-1}.$

( [link] ) Solve the equation, if possible: $\frac{1}{x-2}=\frac{3}{{x}^{2}-x-2}-\frac{3}{x+1}.$

No solution $\text{;}\text{\hspace{0.17em}}x=2$ is excluded.

( [link] ) Perform the division: $\frac{15{x}^{3}-5{x}^{2}+10x}{5x}.$

( [link] ) Perform the division: $\frac{{x}^{3}-5{x}^{2}+13x-21}{x-3}.$

${x}^{2}-2x+7$

anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
Introduction about quantum dots in nanotechnology
what does nano mean?
nano basically means 10^(-9). nanometer is a unit to measure length.
Bharti
do you think it's worthwhile in the long term to study the effects and possibilities of nanotechnology on viral treatment?
absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
characteristics of micro business
Abigail
for teaching engĺish at school how nano technology help us
Anassong
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
what is the Synthesis, properties,and applications of carbon nano chemistry
Mostly, they use nano carbon for electronics and for materials to be strengthened.
Virgil
is Bucky paper clear?
CYNTHIA
carbon nanotubes has various application in fuel cells membrane, current research on cancer drug,and in electronics MEMS and NEMS etc
NANO
so some one know about replacing silicon atom with phosphorous in semiconductors device?
Yeah, it is a pain to say the least. You basically have to heat the substarte up to around 1000 degrees celcius then pass phosphene gas over top of it, which is explosive and toxic by the way, under very low pressure.
Harper
Do you know which machine is used to that process?
s.
how to fabricate graphene ink ?
for screen printed electrodes ?
SUYASH
What is lattice structure?
of graphene you mean?
Ebrahim
or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
tahir
On having this app for quite a bit time, Haven't realised there's a chat room in it.
Cied
what is biological synthesis of nanoparticles
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
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
Berger describes sociologists as concerned with
Please keep in mind that it's not allowed to promote any social groups (whatsapp, facebook, etc...), exchange phone numbers, email addresses or ask for personal information on QuizOver's platform.