# 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

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

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

$4{b}^{8}$

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

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

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

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

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

$81{a}^{4}$

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

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

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

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

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

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

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

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

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

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

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

${x}^{11}$

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

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

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

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

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

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

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

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

${x}^{n+m}$

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

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

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

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

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

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

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

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

${u}^{w-k}$

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s.
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so some one know about replacing silicon atom with phosphorous in semiconductors device?
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or in general
Ebrahim
in general
s.
Graphene has a hexagonal structure
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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.
how did you get the value of 2000N.What calculations are needed to arrive at it
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