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

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## 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.

An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation
12, 17, 22.... 25th term
12, 17, 22.... 25th term
Akash
College algebra is really hard?
Absolutely, for me. My problems with math started in First grade...involving a nun Sister Anastasia, bad vision, talking & getting expelled from Catholic school. When it comes to math I just can't focus and all I can hear is our family silverware banging and clanging on the pink Formica table.
Carole
I'm 13 and I understand it great
AJ
I am 1 year old but I can do it! 1+1=2 proof very hard for me though.
Atone
hi
Not really they are just easy concepts which can be understood if you have great basics. I am 14 I understood them easily.
Vedant
find the 15th term of the geometric sequince whose first is 18 and last term of 387
I know this work
salma
The given of f(x=x-2. then what is the value of this f(3) 5f(x+1)
hmm well what is the answer
Abhi
how do they get the third part x = (32)5/4
make 5/4 into a mixed number, make that a decimal, and then multiply 32 by the decimal 5/4 turns out to be
AJ
can someone help me with some logarithmic and exponential equations.
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
hmm
Abhi
is it a question of log
Abhi
🤔.
Abhi
I rally confuse this number And equations too I need exactly help
salma
But this is not salma it's Faiza live in lousvile Ky I garbage this so I am going collage with JCTC that the of the collage thank you my friends
salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
hi
Ayuba
Hello
opoku
hi
Ali
greetings from Iran
Ali
salut. from Algeria
Bach
hi
Nharnhar
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×