# 1.1 Real numbers: algebra essentials  (Page 5/35)

 Page 5 / 35

## 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

sin theta=3/4.prove that sec square theta barabar 1 + tan square theta by cosec square theta minus cos square theta
Umesh Reply
I want to know trigonometry but I can't understand it anyone who can help
Siyabonga Reply
Yh
Idowu
which part of trig?
Nyemba
functions
Siyabonga
trigonometry
Ganapathi
differentiation doubhts
Ganapathi
hi
Ganapathi
hello
Brittany
Prove that 4sin50-3tan 50=1
Sudip Reply
f(x)= 1 x    f(x)=1x  is shifted down 4 units and to the right 3 units.
Sebit Reply
f (x) = −3x + 5 and g (x) = x − 5 /−3
Sebit
what are real numbers
Marty Reply
I want to know partial fraction Decomposition.
Adama Reply
classes of function in mathematics
Yazidu Reply
divide y2_8y2+5y2/y2
Sumanth Reply
wish i knew calculus to understand what's going on 🙂
Dashawn Reply
@dashawn ... in simple terms, a derivative is the tangent line of the function. which gives the rate of change at that instant. to calculate. given f(x)==ax^n. then f'(x)=n*ax^n-1 . hope that help.
Christopher
thanks bro
Dashawn
maybe when i start calculus in a few months i won't be that lost 😎
Dashawn
what's the derivative of 4x^6
Axmed Reply
24x^5
James
10x
Axmed
24X^5
Taieb
Thanks for this helpfull app
Axmed Reply
secA+tanA=2√5,sinA=?
richa Reply
tan2a+tan2a=√3
Rahulkumar
classes of function
Yazidu
if sinx°=sin@, then @ is - ?
NAVJIT Reply
the value of tan15°•tan20°•tan70°•tan75° -
NAVJIT
0.037 than find sin and tan?
Jon Reply
cos24/25 then find sin and tan
Deepak Reply

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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