# 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

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

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

$4{b}^{8}$

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

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

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

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

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

$81{a}^{4}$

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

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

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

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

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

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

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

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

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

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

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

${x}^{11}$

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

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

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

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

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

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

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

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

${x}^{n+m}$

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

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

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

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

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

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

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

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

${u}^{w-k}$

show that the set of all natural number form semi group under the composition of addition
what is the meaning
Dominic
explain and give four Example hyperbolic function
_3_2_1
felecia
⅗ ⅔½
felecia
_½+⅔-¾
felecia
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
x*x=2
felecia
2+2x=
felecia
×/×+9+6/1
Debbie
Q2 x+(x+2)+(x+4)=60 3x+6=60 3x+6-6=60-6 3x=54 3x/3=54/3 x=18 :. The numbers are 18,20 and 22
Naagmenkoma
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Abdullahi
hi mam
Mark
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
16
Makan
x=16
Makan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
What is the expressiin for seven less than four times the number of nickels
How do i figure this problem out.
how do you translate this in Algebraic Expressions
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
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?
how did you get the value of 2000N.What calculations are needed to arrive at it
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