# 1.1 Real numbers: algebra essentials  (Page 3/35)

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## Using the order of operations

Use the order of operations to evaluate each of the following expressions.

1. ${\left(3\cdot 2\right)}^{2}-4\left(6+2\right)$
2. $\frac{{5}^{2}-4}{7}-\sqrt{11-2}$
3. $6-|5-8|+3\left(4-1\right)$
4. $\frac{14-3\cdot 2}{2\cdot 5-{3}^{2}}$
5. $7\left(5\cdot 3\right)-2\left[\left(6-3\right)-{4}^{2}\right]+1$

1. Note that in the first step, the radical is treated as a grouping symbol, like parentheses. Also, in the third step, the fraction bar is considered a grouping symbol so the numerator is considered to be grouped.

2. In this example, the fraction bar separates the numerator and denominator, which we simplify separately until the last step.

Use the order of operations to evaluate each of the following expressions.

1. $\sqrt{{5}^{2}-{4}^{2}}+7{\left(5-4\right)}^{2}$
2. $1+\frac{7\cdot 5-8\cdot 4}{9-6}$
3. $|1.8-4.3|+0.4\sqrt{15+10}$
4. $\frac{1}{2}\left[5\cdot {3}^{2}-{7}^{2}\right]+\frac{1}{3}\cdot {9}^{2}$
5. $\left[{\left(3-8\right)}^{2}-4\right]-\left(3-8\right)$
1. 10
2. 2
3. 4.5
4. 25
5. 26

## Using properties of real numbers

For some activities we perform, the order of certain operations does not matter, but the order of other operations does. For example, it does not make a difference if we put on the right shoe before the left or vice-versa. However, it does matter whether we put on shoes or socks first. The same thing is true for operations in mathematics.

## Commutative properties

The commutative property of addition    states that numbers may be added in any order without affecting the sum.

$a+b=b+a$

We can better see this relationship when using real numbers.

$\begin{array}{lllll}\left(-2\right)+7=5\hfill & \hfill & \text{and}\hfill & \hfill & 7+\left(-2\right)=5\hfill \end{array}$

Similarly, the commutative property of multiplication    states that numbers may be multiplied in any order without affecting the product.

$a\cdot b=b\cdot a$

Again, consider an example with real numbers.

$\begin{array}{ccccc}\left(-11\right)\cdot \left(-4\right)=44& & \text{and}& & \left(-4\right)\cdot \left(-11\right)=44\end{array}$

It is important to note that neither subtraction nor division is commutative. For example, $\text{\hspace{0.17em}}17-5\text{\hspace{0.17em}}$ is not the same as $\text{\hspace{0.17em}}5-17.\text{\hspace{0.17em}}$ Similarly, $\text{\hspace{0.17em}}20÷5\ne 5÷20.$

## Associative properties

The associative property of multiplication    tells us that it does not matter how we group numbers when multiplying. We can move the grouping symbols to make the calculation easier, and the product remains the same.

$a\left(bc\right)=\left(ab\right)c$

Consider this example.

$\begin{array}{ccccc}\left(3\cdot 4\right)\cdot 5=60& & \text{and}& & 3\cdot \left(4\cdot 5\right)=60\end{array}$

The associative property of addition    tells us that numbers may be grouped differently without affecting the sum.

$a+\left(b+c\right)=\left(a+b\right)+c$

This property can be especially helpful when dealing with negative integers. Consider this example.

$\begin{array}{ccccc}\left[15+\left(-9\right)\right]+23=29& & \text{and}& & 15+\left[\left(-9\right)+23\right]=29\end{array}$

Are subtraction and division associative? Review these examples.

#### Questions & Answers

find to nearest one decimal place of centimeter the length of an arc of circle of radius length 12.5cm and subtending of centeral angle 1.6rad
Martina Reply
factoring polynomial
Noven Reply
what's your topic about?
Shin Reply
find general solution of the Tanx=-1/root3,secx=2/root3
Nani Reply
find general solution of the following equation
Nani
the value of 2 sin square 60 Cos 60
Sanjay Reply
0.75
Lynne
0.75
Inkoom
when can I use sin, cos tan in a giving question
duru Reply
depending on the question
Nicholas
I am a carpenter and I have to cut and assemble a conventional roof line for a new home. The dimensions are: width 30'6" length 40'6". I want a 6 and 12 pitch. The roof is a full hip construction. Give me the L,W and height of rafters for the hip, hip jacks also the length of common jacks.
John
I want to learn the calculations
Koru Reply
where can I get indices
Kojo Reply
I need matrices
Nasasira
hi
Raihany
Hi
Solomon
need help
Raihany
maybe provide us videos
Nasasira
about complex fraction
Raihany
Hello
Cromwell
a
Amie
What do you mean by a
Cromwell
nothing. I accidentally press it
Amie
you guys know any app with matrices?
Khay
Ok
Cromwell
Solve the x? x=18+(24-3)=72
Leizel Reply
x-39=72 x=111
Suraj
Solve the formula for the indicated variable P=b+4a+2c, for b
Deadra Reply
Need help with this question please
Deadra
b=-4ac-2c+P
Denisse
b=p-4a-2c
Suddhen
b= p - 4a - 2c
Snr
p=2(2a+C)+b
Suraj
b=p-2(2a+c)
Tapiwa
P=4a+b+2C
COLEMAN
b=P-4a-2c
COLEMAN
like Deadra, show me the step by step order of operation to alive for b
John
A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
yah
immy

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