# 1.1 Real numbers: algebra essentials  (Page 4/35)

 Page 4 / 35

## Differentiating the sets of numbers

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

1. $\sqrt{36}$
2. $\frac{8}{3}$
3. $\sqrt{73}$
4. $-6$
5. $3.2121121112\dots$
N W I Q Q′
a. $\text{\hspace{0.17em}}\sqrt{36}=6$ X X X X
b. $\text{\hspace{0.17em}}\frac{8}{3}=2.\overline{6}$ X
c. $\text{\hspace{0.17em}}\sqrt{73}$ X
d. –6 X X
e. 3.2121121112... X

Classify each number as being a natural number ( N ), whole number ( W ), integer ( I ), rational number ( Q ), and/or irrational number ( Q′ ).

1. $-\frac{35}{7}$
2. $0$
3. $\sqrt{169}$
4. $\sqrt{24}$
5. $4.763763763\dots$
N W I Q Q'
a. $\text{\hspace{0.17em}}-\frac{35}{7}$ X X
b. 0 X X X
c. $\text{\hspace{0.17em}}\sqrt{169}$ X X X X
d. $\text{\hspace{0.17em}}\sqrt{24}$ X
e. 4.763763763... X

## Performing calculations using the order of operations

When we multiply a number by itself, we square it or raise it to a power of 2. For example, $\text{\hspace{0.17em}}{4}^{2}=4\cdot 4=16.\text{\hspace{0.17em}}$ We can raise any number to any power. In general, the exponential notation     $\text{\hspace{0.17em}}{a}^{n}\text{\hspace{0.17em}}$ means that the number or variable $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is used as a factor $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ times.

In this notation, $\text{\hspace{0.17em}}{a}^{n}\text{\hspace{0.17em}}$ is read as the n th power of $\text{\hspace{0.17em}}a,\text{\hspace{0.17em}}$ where $\text{\hspace{0.17em}}a\text{\hspace{0.17em}}$ is called the base    and $\text{\hspace{0.17em}}n\text{\hspace{0.17em}}$ is called the exponent     . A term in exponential notation may be part of a mathematical expression, which is a combination of numbers and operations. For example, $\text{\hspace{0.17em}}24+6\cdot \frac{2}{3}-{4}^{2}\text{\hspace{0.17em}}$ is a mathematical expression.

To evaluate a mathematical expression, we perform the various operations. However, we do not perform them in any random order. We use the order of operations    . This is a sequence of rules for evaluating such expressions.

Recall that in mathematics we use parentheses ( ), brackets [ ], and braces { } to group numbers and expressions so that anything appearing within the symbols is treated as a unit. Additionally, fraction bars, radicals, and absolute value bars are treated as grouping symbols. When evaluating a mathematical expression, begin by simplifying expressions within grouping symbols.

The next step is to address any exponents or radicals. Afterward, perform multiplication and division from left to right and finally addition and subtraction from left to right.

Let’s take a look at the expression provided.

$24+6\cdot \frac{2}{3}-{4}^{2}$

There are no grouping symbols, so we move on to exponents or radicals. The number 4 is raised to a power of 2, so simplify $\text{\hspace{0.17em}}{4}^{2}\text{\hspace{0.17em}}$ as 16.

$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-{4}^{2}\hfill \\ 24+6\cdot \frac{2}{3}-16\hfill \end{array}\hfill \end{array}$

Next, perform multiplication or division, left to right.

$\begin{array}{l}\hfill \\ \begin{array}{l}24+6\cdot \frac{2}{3}-16\hfill \\ 24+4-16\hfill \end{array}\hfill \end{array}$

Lastly, perform addition or subtraction, left to right.

Therefore, $\text{\hspace{0.17em}}24+6\cdot \frac{2}{3}-{4}^{2}=12.$

For some complicated expressions, several passes through the order of operations will be needed. For instance, there may be a radical expression inside parentheses that must be simplified before the parentheses are evaluated. Following the order of operations ensures that anyone simplifying the same mathematical expression will get the same result.

## Order of operations

Operations in mathematical expressions must be evaluated in a systematic order, which can be simplified using the acronym PEMDAS :

P (arentheses)
E (xponents)
M (ultiplication) and D (ivision)
A (ddition) and S (ubtraction)

Given a mathematical expression, simplify it using the order of operations.

1. Simplify any expressions within grouping symbols.
2. Simplify any expressions containing exponents or radicals.
3. Perform any multiplication and division in order, from left to right.
4. Perform any addition and subtraction in order, from left to right.

bsc F. y algebra and trigonometry pepper 2
given that x= 3/5 find sin 3x
4
DB
remove any signs and collect terms of -2(8a-3b-c)
-16a+6b+2c
Will
Joeval
(x2-2x+8)-4(x2-3x+5)
sorry
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
x²-2x+9-4x²+12x-20 -3x²+10x+11
Miranda
(X2-2X+8)-4(X2-3X+5)=0 ?
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
The anwser is imaginary number if you want to know The anwser of the expression you must arrange The expression and use quadratic formula To find the answer
master
Y
master
master
Soo sorry (5±Root11* i)/3
master
Mukhtar
explain and give four example of hyperbolic function
What is the correct rational algebraic expression of the given "a fraction whose denominator is 10 more than the numerator y?
y/y+10
Mr
Find nth derivative of eax sin (bx + c).
Find area common to the parabola y2 = 4ax and x2 = 4ay.
Anurag
A rectangular garden is 25ft wide. if its area is 1125ft, what is the length of the garden
to find the length I divide the area by the wide wich means 1125ft/25ft=45
Miranda
thanks
Jhovie
What do you call a relation where each element in the domain is related to only one value in the range by some rules?
A banana.
Yaona
given 4cot thither +3=0and 0°<thither <180° use a sketch to determine the value of the following a)cos thither
what are you up to?
nothing up todat yet
Miranda
hi
jai
hello
jai
Miranda Drice
jai
aap konsi country se ho
jai
which language is that
Miranda
I am living in india
jai
good
Miranda
what is the formula for calculating algebraic
I think the formula for calculating algebraic is the statement of the equality of two expression stimulate by a set of addition, multiplication, soustraction, division, raising to a power and extraction of Root. U believe by having those in the equation you will be in measure to calculate it
Miranda
state and prove Cayley hamilton therom
hello
Propessor
hi
Miranda
the Cayley hamilton Theorem state if A is a square matrix and if f(x) is its characterics polynomial then f(x)=0 in another ways evey square matrix is a root of its chatacteristics polynomial.
Miranda
hi
jai
hi Miranda
jai
thanks
Propessor
welcome
jai
What is algebra
algebra is a branch of the mathematics to calculate expressions follow.
Miranda
Miranda Drice would you mind teaching me mathematics? I think you are really good at math. I'm not good at it. In fact I hate it. 😅😅😅
Jeffrey
lolll who told you I'm good at it
Miranda
something seems to wispher me to my ear that u are good at it. lol
Jeffrey
lolllll if you say so
Miranda
but seriously, Im really bad at math. And I hate it. But you see, I downloaded this app two months ago hoping to master it.
Jeffrey
which grade are you in though
Miranda
oh woww I understand
Miranda
Jeffrey
Jeffrey
Miranda
how come you finished in college and you don't like math though
Miranda
gotta practice, holmie
Steve
if you never use it you won't be able to appreciate it
Steve
I don't know why. But Im trying to like it.
Jeffrey
yes steve. you're right
Jeffrey
so you better
Miranda
what is the solution of the given equation?
which equation
Miranda
I dont know. lol
Jeffrey
Miranda
Jeffrey