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.

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
answer and questions in exercise 11.2 sums
how do u calculate inequality of irrational number?
Alaba
give me an example
Chris
and I will walk you through it
Chris
cos (-z)= cos z .
cos(- z)=cos z
Mustafa
what is a algebra
(x+x)3=?
6x
Obed
what is the identity of 1-cos²5x equal to?
__john __05
Kishu
Hi
Abdel
hi
Ye
hi
Nokwanda
C'est comment
Abdel
Hi
Amanda
hello
SORIE
Hiiii
Chinni
hello
Ranjay
hi
ANSHU
hiiii
Chinni
h r u friends
Chinni
yes
Hassan
so is their any Genius in mathematics here let chat guys and get to know each other's
SORIE
I speak French
Abdel
okay no problem since we gather here and get to know each other
SORIE
hi im stupid at math and just wanna join here
Yaona
lol nahhh none of us here are stupid it's just that we have Fast, Medium, and slow learner bro but we all going to work things out together
SORIE
it's 12
what is the function of sine with respect of cosine , graphically
tangent bruh
Steve
cosx.cos2x.cos4x.cos8x
sinx sin2x is linearly dependent
what is a reciprocal
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
I don't understand how radicals works pls
How look for the general solution of a trig function