# 1.1 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.

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

#### Questions & Answers

write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
the polar co-ordinate of the point (-1, -1)
Sumit Reply
prove the identites sin x ( 1+ tan x )+ cos x ( 1+ cot x )= sec x + cosec x
Rockstar Reply
tanh`(x-iy) =A+iB, find A and B
Pankaj Reply
B=Ai-itan(hx-hiy)
Rukmini
what is the addition of 101011 with 101010
Branded Reply
If those numbers are binary, it's 1010101. If they are base 10, it's 202021.
Jack
extra power 4 minus 5 x cube + 7 x square minus 5 x + 1 equal to zero
archana Reply
the gradient function of a curve is 2x+4 and the curve passes through point (1,4) find the equation of the curve
Kc Reply
1+cos²A/cos²A=2cosec²A-1
Ramesh Reply
test for convergence the series 1+x/2+2!/9x3
success Reply

### Read also:

#### Get the best Algebra and trigonometry course in your pocket!

Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

 By By By By Dravida Mahadeo-J... By