# 5.1 Real numbers: algebra essentials  (Page 9/35)

 Page 9 / 35

Simplify each algebraic expression.

1. $\frac{2}{3}y-2\left(\frac{4}{3}y+z\right)$
2. $\frac{5}{t}-2-\frac{3}{t}+1$
3. $4p\left(q-1\right)+q\left(1-p\right)$
4. $9r-\left(s+2r\right)+\left(6-s\right)$
1. $\text{\hspace{0.17em}}\frac{2}{t}-1;$
2. $\text{\hspace{0.17em}}3pq-4p+q;$
3. $\text{\hspace{0.17em}}7r-2s+6$

## Simplifying a formula

A rectangle with length $\text{\hspace{0.17em}}L\text{\hspace{0.17em}}$ and width $\text{\hspace{0.17em}}W\text{\hspace{0.17em}}$ has a perimeter $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ given by $\text{\hspace{0.17em}}P=L+W+L+W.\text{\hspace{0.17em}}$ Simplify this expression.

If the amount $\text{\hspace{0.17em}}P\text{\hspace{0.17em}}$ is deposited into an account paying simple interest $\text{\hspace{0.17em}}r\text{\hspace{0.17em}}$ for time $\text{\hspace{0.17em}}t,$ the total value of the deposit $\text{\hspace{0.17em}}A\text{\hspace{0.17em}}$ is given by $\text{\hspace{0.17em}}A=P+Prt.\text{\hspace{0.17em}}$ Simplify the expression. (This formula will be explored in more detail later in the course.)

$A=P\left(1+rt\right)$

Access these online resources for additional instruction and practice with real numbers.

## Key concepts

• Rational numbers may be written as fractions or terminating or repeating decimals. See [link] and [link] .
• Determine whether a number is rational or irrational by writing it as a decimal. See [link] .
• The rational numbers and irrational numbers make up the set of real numbers. See [link] . A number can be classified as natural, whole, integer, rational, or irrational. See [link] .
• The order of operations is used to evaluate expressions. See [link] .
• The real numbers under the operations of addition and multiplication obey basic rules, known as the properties of real numbers. These are the commutative properties, the associative properties, the distributive property, the identity properties, and the inverse properties. See [link] .
• Algebraic expressions are composed of constants and variables that are combined using addition, subtraction, multiplication, and division. See [link] . They take on a numerical value when evaluated by replacing variables with constants. See [link] , [link] , and [link]
• Formulas are equations in which one quantity is represented in terms of other quantities. They may be simplified or evaluated as any mathematical expression. See [link] and [link] .

## Verbal

Is $\text{\hspace{0.17em}}\sqrt{2}\text{\hspace{0.17em}}$ an example of a rational terminating, rational repeating, or irrational number? Tell why it fits that category.

irrational number. The square root of two does not terminate, and it does not repeat a pattern. It cannot be written as a quotient of two integers, so it is irrational.

What is the order of operations? What acronym is used to describe the order of operations, and what does it stand for?

What do the Associative Properties allow us to do when following the order of operations? Explain your answer.

The Associative Properties state that the sum or product of multiple numbers can be grouped differently without affecting the result. This is because the same operation is performed (either addition or subtraction), so the terms can be re-ordered.

## Numeric

For the following exercises, simplify the given expression.

$10+2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\left(5-3\right)$

$6÷2-\left(81÷{3}^{2}\right)$

$-6$

$18+{\left(6-8\right)}^{3}$

$-2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}{\left[16÷{\left(8-4\right)}^{2}\right]}^{2}$

$-2$

$4-6+2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}7$

$3\left(5-8\right)$

$-9$

$4+6-10÷2$

$12÷\left(36÷9\right)+6$

9

${\left(4+5\right)}^{2}÷3$

$3-12\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2+19$

-2

$2+8\text{\hspace{0.17em}}×\text{\hspace{0.17em}}7÷4$

$5+\left(6+4\right)-11$

4

$9-18÷{3}^{2}$

$14\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3÷7-6$

0

$9-\left(3+11\right)\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2$

$6+2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2-1$

9

$64÷\left(8+4\text{\hspace{0.17em}}×\text{\hspace{0.17em}}2\right)$

$9+4\left({2}^{2}\right)$

25

${\left(12÷3\text{\hspace{0.17em}}×\text{\hspace{0.17em}}3\right)}^{2}$

$25÷{5}^{2}-7$

$-6$

$\left(15-7\right)\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\left(3-7\right)$

$2\text{\hspace{0.17em}}×\text{\hspace{0.17em}}4-9\left(-1\right)$

17

${4}^{2}-25\text{\hspace{0.17em}}×\text{\hspace{0.17em}}\frac{1}{5}$

$12\left(3-1\right)÷6$

4

## Algebraic

For the following exercises, solve for the variable.

$8\left(x+3\right)=64$

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
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
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salma
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
hii
Uday
hi
salma
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×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
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No. 7x -4y is simplified from 4x + (3y + 3x) -7y
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
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Yasmin
what is the function of carbon nanotubes?
Cesar
I'm interested in nanotube
Uday
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
Hello
Uday
I'm interested in Nanotube
Uday
this technology will not going on for the long time , so I'm thinking about femtotechnology 10^-15
Prasenjit
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
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