# 2.8 Exercise supplement

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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.This module contains the exercise supplement for the chapter "Basic Properties of Real Numbers".

## Symbols and notations ( [link] )

For the following problems, simplify the expressions.

$12+7\left(4+3\right)$

61

$9\left(4-2\right)+6\left(8+2\right)-3\left(1+4\right)$

$6\left[1+8\left(7+2\right)\right]$

438

$26÷2-10$

$\frac{\left(4+17+1\right)+4}{14-1}$

2

$51÷3÷7$

$\left(4+5\right)\left(4+6\right)-\left(4+7\right)$

79

$8\left(2\cdot 12÷13\right)+2\cdot 5\cdot 11-\left[1+4\left(1+2\right)\right]$

$\frac{3}{4}+\frac{1}{12}\left(\frac{3}{4}-\frac{1}{2}\right)$

$\frac{37}{48}$

$48-3\left[\frac{1+17}{6}\right]$

$\frac{29+11}{6-1}$

8

$\frac{\frac{88}{11}+\frac{99}{9}+1}{\frac{54}{9}-\frac{22}{11}}$

$\frac{8\cdot 6}{2}+\frac{9\cdot 9}{3}-\frac{10\cdot 4}{5}$

43

For the following problems, write the appropriate relation symbol $\left(=,<,>\right)$ in place of the $\ast$ .

$22\ast 6$

$9\left[4+3\left(8\right)\right]\ast 6\left[1+8\left(5\right)\right]$

$252>246$

$3\left(1.06+2.11\right)\ast 4\left(11.01-9.06\right)$

$2\ast 0$

$2>0$

For the following problems, state whether the letters or symbols are the same or different.

>and ≮

different

$a=b\text{\hspace{0.17em}}\text{and}\text{\hspace{0.17em}}b=a$

Represent the sum of $c$ and $d$ two different ways.

$c+d;d+c$

For the following problems, use algebraic notataion.

8 plus 9

62 divided by $f$

$\frac{62}{f}\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}62÷f$

8 times $\left(x+4\right)$

6 times $x$ , minus 2

$6x-2$

$x+1$ divided by $x-3$

$y+11$ divided by $y+10$ , minus 12

$\left(y+11\right)÷\left(y+10\right)-12\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\frac{y+11}{y+10}-12$

zero minus $a$ times $b$

## The real number line and the real numbers ( [link] )

Is every natural number a whole number?

yes

Is every rational number a real number?

For the following problems, locate the numbers on a number line by placing a point at their (approximate) position.

2

$3.6$

$-1\frac{3}{8}$

0

$-4\frac{1}{2}$

Draw a number line that extends from 10 to 20. Place a point at all odd integers.

Draw a number line that extends from $-10$ to $10$ . Place a point at all negative odd integers and at all even positive integers.

Draw a number line that extends from $-5$ to $10$ . Place a point at all integers that are greater then or equal to $-2$ but strictly less than 5.

Draw a number line that extends from $-10$ to $10$ . Place a point at all real numbers that are strictly greater than $-8$ but less than or equal to 7.

Draw a number line that extends from $-10$ to $10$ . Place a point at all real numbers between and including $-6$ and 4.

For the following problems, write the appropriate relation symbol $\left(=,<,>\right).$

$\begin{array}{cc}-3& 0\end{array}$

$-3<0$

$\begin{array}{cc}-1& 1\end{array}$

$\begin{array}{cc}-8& -5\end{array}$

$-8<-5$

$\begin{array}{cc}-5& -5\frac{1}{2}\end{array}$

Is there a smallest two digit integer? If so, what is it?

$\text{yes,}\text{\hspace{0.17em}}-99$

Is there a smallest two digit real number? If so, what is it?

For the following problems, what integers can replace $x$ so that the statements are true?

$4\le x\le 7$

$4,\text{\hspace{0.17em}}5,\text{\hspace{0.17em}}6,\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}7$

$-3\le x<1$

$-3

$-2,\text{\hspace{0.17em}}-1,\text{\hspace{0.17em}}0,\text{\hspace{0.17em}}1,\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}2$

The temperature today in Los Angeles was eighty-two degrees. Represent this temperature by real number.

The temperature today in Marbelhead was six degrees below zero. Represent this temperature by real number.

$-6°$

On the number line, how many units between $-3$ and 2?

On the number line, how many units between $-4$ and 0?

4

## Properties of the real numbers ( [link] )

$a+b=b+a$ is an illustration of the property of addition.

$st=ts$ is an illustration of the __________ property of __________.

commutative, multiplication

Use the commutative properties of addition and multiplication to write equivalent expressions for the following problems.

$y+12$

$a+4b$

$4b+a$

$6x$

$2\left(a-1\right)$

$\left(a-1\right)2$

$\left(-8\right)\left(4\right)$

$\left(6\right)\left(-9\right)\left(-2\right)$

$\left(-9\right)\left(6\right)\left(-2\right)\text{\hspace{0.17em}}\text{or\hspace{0.17em}}\left(-9\right)\left(-2\right)\left(6\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\left(6\right)\left(-2\right)\left(-9\right)\text{\hspace{0.17em}}\text{or}\text{\hspace{0.17em}}\left(-2\right)\left(-9\right)\left(6\right)$

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

$△\cdot \diamond$

$\diamond \text{\hspace{0.17em}}\cdot \text{\hspace{0.17em}}△$

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

$8x3y$

$16ab2c$

$32abc$

$4axyc4d4e$

$3\left(x+2\right)5\left(x-1\right)0\left(x+6\right)$

0

$8b\left(a-6\right)9a\left(a-4\right)$

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

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

$3a+12$

$a\left(b+3c\right)$

$2g\left(4h+2k\right)$

$8gh+4gk$

$\left(8m+5n\right)6p$

$3y\left(2x+4z+5w\right)$

$6xy+12yz+15wy$

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

$\left(x+y\right)\left(4a+3b\right)$

$4ax+3bx+4ay+3by$

$10{a}_{z}\left({b}_{z}+c\right)$

For the following problems, write the expressions using exponential notation.

$x$ to the fifth.

${x}^{5}$

$\left(y+2\right)$ cubed.

$\left(a+2b\right)$ squared minus $\left(a+3b\right)$ to the fourth.

${\left(a+2b\right)}^{2}-{\left(a+3b\right)}^{4}$

$x$ cubed plus 2 times $\left(y-x\right)$ to the seventh.

$aaaaaaa$

${a}^{7}$

$2\cdot 2\cdot 2\cdot 2$

$\left(-8\right)\left(-8\right)\left(-8\right)\left(-8\right)xxxyyyyy$

${\left(-8\right)}^{4}{x}^{3}{y}^{5}$

$\left(x-9\right)\left(x-9\right)+\left(3x+1\right)\left(3x+1\right)\left(3x+1\right)$

$2zzyzyyy+7zzyz{\left(a-6\right)}^{2}\left(a-6\right)$

$2{y}^{4}{z}^{3}+7y{z}^{3}{\left(a-6\right)}^{3}$

For the following problems, expand the terms so that no exponents appear.

${x}^{3}$

$3{x}^{3}$

$3xxx$

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

${\left(4b\right)}^{2}$

$4b\text{\hspace{0.17em}}·\text{\hspace{0.17em}}4b$

${\left(6{a}^{2}\right)}^{3}{\left(5c-4\right)}^{2}$

${\left({x}^{3}+7\right)}^{2}{\left({y}^{2}-3\right)}^{3}\left(z+10\right)$

$\left(xxx+7\right)\left(xxx+7\right)\left(yy-3\right)\left(yy-3\right)\left(yy-3\right)\left(z+10\right)$

Choose values for $a$ and $b$ to show that

1. ${\left(a+b\right)}^{2}$ is not always equal to ${a}^{2}+{b}^{2}$ .
2. ${\left(a+b\right)}^{2}$ may be equal to ${a}^{2}+{b}^{2}$ .

Choose value for $x$ to show that

1. ${\left(4x\right)}^{2}$ is not always equal to $4{x}^{2}$ .
2. ${\left(4x\right)}^{2}$ may be equal to $4{x}^{2}$ .

(a) any value except zero

(b) only zero

## Rules of exponents ( [link] ) - the power rules for exponents ( [link] )

Simplify the following problems.

${4}^{2}+8$

${6}^{3}+5\left(30\right)$

366

${1}^{8}+{0}^{10}+{3}^{2}\left({4}^{2}+{2}^{3}\right)$

${12}^{2}+0.3{\left(11\right)}^{2}$

$180.3$

$\frac{{3}^{4}+1}{{2}^{2}+{4}^{2}+{3}^{2}}$

$\frac{{6}^{2}+{3}^{2}}{{2}^{2}+1}+\frac{{\left(1+4\right)}^{2}-{2}^{3}-{1}^{4}}{{2}^{5}-{4}^{2}}$

10

${a}^{4}{a}^{3}$

$2{b}^{5}2{b}^{3}$

$4{b}^{8}$

$4{a}^{3}{b}^{2}{c}^{8}\cdot 3a{b}^{2}{c}^{0}$

$\left(6{x}^{4}{y}^{10}\right)\left(x{y}^{3}\right)$

$6{x}^{5}{y}^{13}$

$\left(3xy{z}^{2}\right)\left(2{x}^{2}{y}^{3}\right)\left(4{x}^{2}{y}^{2}{z}^{4}\right)$

${\left(3a\right)}^{4}$

$81{a}^{4}$

${\left(10xy\right)}^{2}$

${\left({x}^{2}{y}^{4}\right)}^{6}$

${x}^{12}{y}^{24}$

${\left({a}^{4}{b}^{7}{c}^{7}{z}^{12}\right)}^{9}$

${\left(\frac{3}{4}{x}^{8}{y}^{6}{z}^{0}{a}^{10}{b}^{15}\right)}^{2}$

$\frac{9}{16}{x}^{16}{y}^{12}{a}^{20}{b}^{30}$

$\frac{{x}^{8}}{{x}^{5}}$

$\frac{14{a}^{4}{b}^{6}{c}^{7}}{2a{b}^{3}{c}^{2}}$

$7{a}^{3}{b}^{3}{c}^{5}$

$\frac{11{x}^{4}}{11{x}^{4}}$

${x}^{4}\cdot \frac{{x}^{10}}{{x}^{3}}$

${x}^{11}$

${a}^{3}{b}^{7}\cdot \frac{{a}^{9}{b}^{6}}{{a}^{5}{b}^{10}}$

$\frac{{\left({x}^{4}{y}^{6}{z}^{10}\right)}^{4}}{{\left(x{y}^{5}{z}^{7}\right)}^{3}}$

${x}^{13}{y}^{9}{z}^{19}$

$\frac{{\left(2x-1\right)}^{13}{\left(2x+5\right)}^{5}}{{\left(2x-1\right)}^{10}\left(2x+5\right)}$

${\left(\frac{3{x}^{2}}{4{y}^{3}}\right)}^{2}$

$\frac{9{x}^{4}}{16{y}^{6}}$

$\frac{{\left(x+y\right)}^{9}{\left(x-y\right)}^{4}}{{\left(x+y\right)}^{3}}$

${x}^{n}\cdot {x}^{m}$

${x}^{n+m}$

${a}^{n+2}{a}^{n+4}$

$6{b}^{2n+7}\cdot 8{b}^{5n+2}$

$48{b}^{7n+9}$

$\frac{18{x}^{4n+9}}{2{x}^{2n+1}}$

${\left({x}^{5t}{y}^{4r}\right)}^{7}$

${x}^{35t}{y}^{28r}$

${\left({a}^{2n}{b}^{3m}{c}^{4p}\right)}^{6r}$

$\frac{{u}^{w}}{{u}^{k}}$

${u}^{w-k}$

How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
what is Nano technology ?
write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
What fields keep nano created devices from performing or assimulating ? Magnetic fields ? Are do they assimilate ?
why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
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
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
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
how can I make nanorobot?
Lily
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
how did you get the value of 2000N.What calculations are needed to arrive at it
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