# 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

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

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

$4{b}^{8}$

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

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

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

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

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

$81{a}^{4}$

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

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

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

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

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

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

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

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

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

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

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

${x}^{11}$

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

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

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

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

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

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

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

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

${x}^{n+m}$

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

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

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

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

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

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

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

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

${u}^{w-k}$

what are the products of Nano chemistry?
There are lots of products of nano chemistry... Like nano coatings.....carbon fiber.. And lots of others..
learn
Even nanotechnology is pretty much all about chemistry... Its the chemistry on quantum or atomic level
learn
da
no nanotechnology is also a part of physics and maths it requires angle formulas and some pressure regarding concepts
Bhagvanji
Preparation and Applications of Nanomaterial for Drug Delivery
revolt
da
Application of nanotechnology in medicine
what is variations in raman spectra for nanomaterials
I only see partial conversation and what's the question here!
what about nanotechnology for water purification
please someone correct me if I'm wrong but I think one can use nanoparticles, specially silver nanoparticles for water treatment.
Damian
yes that's correct
Professor
I think
Professor
what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
industrial application...? mmm I think on the medical side as drug carrier, but you should go deeper on your research, I may be wrong
Damian
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
how nano science is used for hydrophobicity
Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
what is differents between GO and RGO?
Mahi
what is simplest way to understand the applications of nano robots used to detect the cancer affected cell of human body.? How this robot is carried to required site of body cell.? what will be the carrier material and how can be detected that correct delivery of drug is done Rafiq
Rafiq
if virus is killing to make ARTIFICIAL DNA OF GRAPHENE FOR KILLED THE VIRUS .THIS IS OUR ASSUMPTION
Anam
analytical skills graphene is prepared to kill any type viruses .
Anam
Any one who tell me about Preparation and application of Nanomaterial for drug Delivery
Hafiz
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
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