# 5.7 Simplify and use square roots  (Page 3/8)

 Page 3 / 8

Suppose your calculator has a $\text{10-digit}$ display. Using it to find the square root of $5$ will give $2.236067977.$ This is the approximate square root of $5.$ When we report the answer, we should use the “approximately equal to” sign instead of an equal sign.

$\sqrt{5}\approx 2.236067978$

You will seldom use this many digits for applications in algebra. So, if you wanted to round $\sqrt{5}$ to two decimal places, you would write

$\sqrt{5}\approx 2.24$

How do we know these values are approximations and not the exact values? Look at what happens when we square them.

$\begin{array}{ccc}\hfill {2.236067978}^{2}& =& 5.000000002\hfill \\ \hfill {2.24}^{2}& =& 5.0176\hfill \end{array}$

The squares are close, but not exactly equal, to $5.$

Round $\sqrt{17}$ to two decimal places using a calculator.

## Solution

 $\sqrt{17}$ Use the calculator square root key. $4.123105626$ Round to two decimal places. $4.12$ $\sqrt{17}\approx 4.12$

Round $\sqrt{11}$ to two decimal places.

≈ 3.32

Round $\sqrt{13}$ to two decimal places.

≈ 3.61

## Simplify variable expressions with square roots

Expressions with square root that we have looked at so far have not had any variables. What happens when we have to find a square root of a variable expression?

Consider $\sqrt{9{x}^{2}},$ where $x\ge 0.$ Can you think of an expression whose square is $9{x}^{2}?$

$\begin{array}{ccc}\hfill {\left(?\right)}^{2}& =& 9{x}^{2}\hfill \\ \hfill {\left(3x\right)}^{2}& =& 9{x}^{2}\phantom{\rule{2em}{0ex}}\text{so}\phantom{\rule{0.2em}{0ex}}\sqrt{9{x}^{2}}=3x\hfill \end{array}$

When we use a variable in a square root expression, for our work, we will assume that the variable represents a non-negative number. In every example and exercise that follows, each variable in a square root expression is greater than or equal to zero.

Simplify: $\sqrt{{x}^{2}}.$

## Solution

Think about what we would have to square to get ${x}^{2}$ . Algebraically, ${\left(?\right)}^{2}={x}^{2}$

 $\sqrt{{x}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(x\right)}^{2}$ $x$

Simplify: $\sqrt{{y}^{2}}.$

y

Simplify: $\sqrt{{m}^{2}}.$

m

Simplify: $\sqrt{16{x}^{2}}.$

## Solution

 $\sqrt{16{x}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(4x\right)}^{2}=16{x}^{2}$ $4x$

Simplify: $\sqrt{64{x}^{2}}.$

8 x

Simplify: $\sqrt{169{y}^{2}}.$

13 y

Simplify: $-\sqrt{81{y}^{2}}.$

## Solution

 $-\sqrt{81{y}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(9y\right)}^{2}=81{y}^{2}$ $-9y$

Simplify: $-\sqrt{121{y}^{2}}.$

−11 y

Simplify: $-\sqrt{100{p}^{2}}.$

−10 p

Simplify: $\sqrt{36{x}^{2}{y}^{2}}.$

## Solution

 $\sqrt{36{x}^{2}{y}^{2}}$ $\text{Since}\phantom{\rule{0.2em}{0ex}}{\left(6xy\right)}^{2}=36{x}^{2}{y}^{2}$ $6xy$

Simplify: $\sqrt{100{a}^{2}{b}^{2}}.$

10 ab

Simplify: $\sqrt{225{m}^{2}{n}^{2}}.$

15 mn

## Use square roots in applications

As you progress through your college courses, you’ll encounter several applications of square roots. Once again, if we use our strategy for applications, it will give us a plan for finding the answer!

## Use a strategy for applications with square roots.

1. Identify what you are asked to find.
2. Write a phrase that gives the information to find it.
3. Translate the phrase to an expression.
4. Simplify the expression.
5. Write a complete sentence that answers the question.

## Square roots and area

We have solved applications with area before. If we were given the length of the sides of a square, we could find its area by squaring the length of its sides. Now we can find the length of the sides of a square if we are given the area, by finding the square root of the area.

If the area of the square is $A$ square units, the length of a side is $\sqrt{A}$ units. See [link] .

Area (square units) Length of side (units)
$9$ $\sqrt{9}=3$
$144$ $\sqrt{144}=12$
$A$ $\sqrt{A}$

Mike and Lychelle want to make a square patio. They have enough concrete for an area of $200$ square feet. To the nearest tenth of a foot, how long can a side of their square patio be?

## Solution

We know the area of the square is $200$ square feet and want to find the length of the side. If the area of the square is $A$ square units, the length of a side is $\sqrt{A}$ units.

 What are you asked to find? The length of each side of a square patio Write a phrase. The length of a side Translate to an expression. $\sqrt{A}$ Evaluate $\sqrt{A}$ when $A=200$ . $\sqrt{200}$ Use your calculator. $14.142135...$ Round to one decimal place. $\text{14.1 feet}$ Write a sentence. Each side of the patio should be $14.1$ feet.

find the 15th term of the geometric sequince whose first is 18 and last term of 387
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
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?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
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
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.
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
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
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?