# 2.3 Properties of the real numbers  (Page 2/2)

 Page 2 / 2

$\begin{array}{cc}\left(9y\right)4=9\left(y4\right)& \text{Both}\text{\hspace{0.17em}}\text{represent}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{same}\text{\hspace{0.17em}}\text{product}\text{.}\end{array}$

## Practice set b

Fill in the $\left(\begin{array}{cc}& \end{array}\right)$ to make each statement true. Use the associative properties.

$\left(9+2\right)+5=9+\left(\begin{array}{cc}& \end{array}\right)$

$2+5$

$x+\left(5+y\right)=\left(\begin{array}{cc}& \end{array}\right)+y$

$x+5$

$\left(11a\right)6=11\left(\begin{array}{cc}& \end{array}\right)$

$a\cdot 6$

$\left[\left(7m-2\right)\left(m+3\right)\right]\left(m+4\right)=\left(7m-2\right)\left[\left(\begin{array}{cc}& \end{array}\right)\left(\begin{array}{cc}& \end{array}\right)\right]$

$\left(m+3\right)\left(m+4\right)$

## Sample set c

Simplify (rearrange into a simpler form): $5x6b8ac4$ .

According to the commutative property of multiplication, we can make a series of consecutive switches and get all the numbers together and all the letters together.

$\begin{array}{ll}5\cdot 6\cdot 8\cdot 4\cdot x\cdot b\cdot a\cdot c\hfill & \hfill \\ 960xbac\hfill & \text{Multiply}\text{\hspace{0.17em}}\text{the}\text{\hspace{0.17em}}\text{numbers}\text{.}\hfill \\ 960abcx\hfill & \text{By}\text{\hspace{0.17em}}\text{convention,}\text{\hspace{0.17em}}\text{we}\text{\hspace{0.17em}}\text{will,}\text{\hspace{0.17em}}\text{when}\text{\hspace{0.17em}}\text{possible,}\text{\hspace{0.17em}}\text{write}\text{\hspace{0.17em}}\text{all}\text{\hspace{0.17em}}\text{letters}\text{\hspace{0.17em}}\text{in}\text{\hspace{0.17em}}\text{alphabetical}\text{\hspace{0.17em}}\text{order}\text{.}\hfill \end{array}$

## Practice set c

Simplify each of the following quantities.

$3a7y9d$

$189ady$

$6b8acz4\cdot 5$

$960abcz$

$4p6qr3\left(a+b\right)$

$72pqr\left(a+b\right)$

## The distributive properties

When we were first introduced to multiplication we saw that it was developed as a description for repeated addition.

$4+4+4=3\cdot 4$

Notice that there are three 4’s, that is, 4 appears 3 times . Hence, 3 times 4.
We know that algebra is generalized arithmetic. We can now make an important generalization.

When a number $a$ is added repeatedly $n$ times, we have
$\underset{a\text{\hspace{0.17em}}\text{appears}\text{\hspace{0.17em}}n\text{\hspace{0.17em}}\text{times}}{\underbrace{a+a+a+\cdots +a}}$
Then, using multiplication as a description for repeated addition, we can replace
$\begin{array}{ccc}\underset{n\text{\hspace{0.17em}}\text{times}}{\underbrace{a+a+a+\cdots +a}}& \text{with}& na\end{array}$

For example:

$x+x+x+x$ can be written as $4x$ since $x$ is repeatedly added 4 times.

$x+x+x+x=4x$

$r+r$ can be written as $2r$ since $r$ is repeatedly added 2 times.

$r+r=2r$

The distributive property involves both multiplication and addition. Let’s rewrite $4\left(a+b\right).$ We proceed by reading $4\left(a+b\right)$ as a multiplication: 4 times the quantity $\left(a+b\right)$ . This directs us to write

$\begin{array}{lll}4\left(a+b\right)\hfill & =\hfill & \left(a+b\right)+\left(a+b\right)+\left(a+b\right)+\left(a+b\right)\hfill \\ \hfill & =\hfill & a+b+a+b+a+b+a+b\hfill \end{array}$

Now we use the commutative property of addition to collect all the $a\text{'}s$ together and all the $b\text{'}s$ together.

$\begin{array}{lll}4\left(a+b\right)\hfill & =\hfill & \underset{4a\text{'}s}{\underbrace{a+a+a+a}}+\underset{4b\text{'}s}{\underbrace{b+b+b+b}}\hfill \end{array}$

Now, using multiplication as a description for repeated addition, we have

$\begin{array}{lll}4\left(a+b\right)\hfill & =\hfill & 4a+4b\hfill \end{array}$

We have distributed the 4 over the sum to both $a$ and $b$ .

## The distributive property

$\begin{array}{cc}a\left(b+c\right)=a\cdot b+a\cdot c& \left(b+c\right)\end{array}a=a\cdot b+a\cdot c$

The distributive property is useful when we cannot or do not wish to perform operations inside parentheses.

## Sample set d

Use the distributive property to rewrite each of the following quantities.

## Practice set d

What property of real numbers justifies
$a\left(b+c\right)=\left(b+c\right)a?$

the commutative property of multiplication

Use the distributive property to rewrite each of the following quantities.

$3\left(2+1\right)$

$6+3$

$\left(x+6\right)7$

$7x+42$

$4\left(a+y\right)$

$4a+4y$

$\left(9+2\right)a$

$9a+2a$

$a\left(x+5\right)$

$ax+5a$

$1\left(x+y\right)$

$x+y$

## The identity properties

The number 0 is called the additive identity since when it is added to any real number, it preserves the identity of that number. Zero is the only additive identity.
For example, $6+0=6$ .

## Multiplicative identity

The number 1 is called the multiplicative identity since when it multiplies any real number, it preserves the identity of that number. One is the only multiplicative identity.
For example $6\cdot 1=6$ .

We summarize the identity properties as follows.

$\begin{array}{cc}\begin{array}{l}\text{ADDITIVE}\text{\hspace{0.17em}}\text{IDENTITY}\\ \text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{PROPERTY}\end{array}& \begin{array}{l}\text{MULTIPLICATIVE}\text{\hspace{0.17em}}\text{IDENTITY}\\ \text{​}\text{​}\text{​}\text{​}\text{​}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{PROPERTY}\end{array}\\ \text{If}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{real}\text{\hspace{0.17em}}\text{number,\hspace{0.17em}then}& \text{If}\text{\hspace{0.17em}}a\text{\hspace{0.17em}}\text{is}\text{\hspace{0.17em}}\text{a}\text{\hspace{0.17em}}\text{real}\text{\hspace{0.17em}}\text{number,}\text{\hspace{0.17em}}\text{then}\\ a+0=a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}0+a=a& a\cdot 1=a\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}\text{\hspace{0.17em}}1\cdot a=a\end{array}$

## The inverse properties

When two numbers are added together and the result is the additive identity, 0, the numbers are called additive inverses of each other. For example, when 3 is added to $-3$ the result is 0, that is, $3+\left(-3\right)=0$ . The numbers 3 and $-3$ are additive inverses of each other.

## Multiplicative inverses

When two numbers are multiplied together and the result is the multiplicative identity, 1, the numbers are called multiplicative inverses of each other. For example, when 6 and $\frac{1}{6}$ are multiplied together, the result is 1, that is, $6\cdot \frac{1}{6}=1$ . The numbers 6 and $\frac{1}{6}$ are multiplicative inverses of each other.

We summarize the inverse properties as follows.

## The inverse properties

1. If $a$ is any real number, then there is a unique real number $-a$ , such that
$\begin{array}{ccc}a+\left(-a\right)=0& \text{and}& -a+a=0\end{array}$
The numbers $a$ and $-a$ are called additive inverses of each other.
2. If $a$ is any nonzero real number, then there is a unique real number $\frac{1}{a}$ such that
$\begin{array}{ccc}a\cdot \frac{1}{a}=1& \text{and}& \frac{1}{a}\end{array}\cdot a=1$
The numbers $a$ and $\frac{1}{a}$ are called multiplicative inverses of each other.

## Expanding quantities

When we perform operations such as $6\left(a+3\right)=6a+18$ , we say we are expanding the quantity $6\left(a+3\right)$ .

## Exercises

Use the commutative property of addition and multiplication to write expressions for an equal number for the following problems. You need not perform any calculations.

$x+3$

$3+x$

$5+y$

$10x$

$10x$

$18z$

$r6$

$6r$

$ax$

$xc$

$cx$

$7\left(2+b\right)$

$6\left(s+1\right)$

$\left(s+1\right)6$

$\left(8+a\right)\left(x+6\right)$

$\left(x+16\right)\left(a+7\right)$

$\left(a+7\right)\left(x+16\right)$

$\left(x+y\right)\left(x-y\right)$

$0.06m$

$m\left(0.06\right)$

$5\left(6h+1\right)$

$\left(6h+1\right)5$

$m\left(a+2b\right)$

$k\left(10a-b\right)$

$\left(10a-b\right)k$

$\left(21c\right)\left(0.008\right)$

$\left(-16\right)\left(4\right)$

$\left(4\right)\left(-16\right)$

$\left(5\right)\left(b-6\right)$

$\square \text{\hspace{0.17em}}\cdot ○$

$○\text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}\square$

Simplify using the commutative property of multiplication for the following problems. You need not use the distributive property.

$9x2y$

$18xy$

$5a6b$

$2a3b4c$

$24abc$

$5x10y5z$

$1u3r2z5m1n$

$30mnruz$

$6d4e1f2\left(g+2h\right)$

$\left(\frac{1}{2}\right)d\left(\frac{1}{4}\right)e\left(\frac{1}{2}\right)a$

$\frac{1}{16}ade$

$3\left(a+6\right)2\left(a-9\right)6b$

$1\left(x+2y\right)\left(6+z\right)9\left(3x+5y\right)$

$9\left(x+2y\right)\left(6+z\right)\left(3x+5y\right)$

For the following problems, use the distributive property to expand the quantities.

$2\left(y+9\right)$

$b\left(r+5\right)$

$br+5b$

$m\left(u+a\right)$

$k\left(j+1\right)$

$jk+k$

$x\left(2y+5\right)$

$z\left(x+9w\right)$

$xz+9wz$

$\left(1+\text{\hspace{0.17em}}d\right)e$

$\left(8+\text{\hspace{0.17em}}2f\right)g$

$8g+2fg$

$c\left(2a+\text{\hspace{0.17em}}10b\right)$

$15x\left(2y+\text{\hspace{0.17em}}3z\right)$

$30xy+45xz$

$8y\left(12a+b\right)$

$z\left(x+y+m\right)$

$xz+yz+mz$

$\left(a+6\right)\left(x+y\right)$

$\left(x+10\right)\left(a+b+c\right)$

$ax+bx+cx+10a+10b+10c$

$1\left(x+y\right)$

$1\left(a+16\right)$

$a+16$

$0.48\left(0.34a+0.61\right)$

$21.5\left(16.2a+3.8b+0.7c\right)$

$348.3a+81.7b+15.05c$

$2{z}_{t}\left({L}_{m}+8k\right)$

$2{L}_{m}{z}_{t}+16k{z}_{t}$

## Exercises for review

( [link] ) Find the value of $4\cdot 2+5\left(2\cdot 4-6÷3\right)-2\cdot 5$ .

( [link] ) Is the statement $3\left(5\cdot 3-3\cdot 5\right)+6\cdot 2-3\cdot 4<0$ true or false?

false

( [link] ) Draw a number line that extends from $-2$ to 2 and place points at all integers between and including $-2$ and 3.

( [link] ) Replace the $\ast$ with the appropriate relation symbol $\left(<,>\right).-7\ast -3$ .

$<$

( [link] ) What whole numbers can replace $x$ so that the statement $-2\le x<2$ is true?

find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
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
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
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
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?
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
Got questions? Join the online conversation and get instant answers!