The document discusses natural numbers and factors. It defines natural numbers as whole numbers used to count items. A factor of a natural number x is another natural number y that x can be divided by evenly. The document then provides an example of listing the factors and multiples of 12. It also discusses prime numbers as numbers with only two factors, 1 and the number itself. The document concludes with definitions of factoring a number and exponents.
We know that a number that can be written as \frac{p}{q}, where p and q are integers and q \neq 0, is known as RATIONAL NUMBERS. Thus, the set of the rational numbers contains all integers and fractions. The set of rational numbers is denoted by Q. Therefore, N \subseteq W \subseteq Z \subseteq Q.
Fundamentals of AlgebraChu v. NguyenIntegral ExponentsDustiBuckner14
Fundamentals of Algebra
Chu v. Nguyen
Integral Exponents
Exponents
If n is a positive integer (a whole number, i.e., a number without decimal part) and x is a number, then
The number x is called the base and n is called the exponent.
The most common ways of referring to are “ x to the nth power,”
“ x to the nth,” or “the nth power of x.”
Integral Exponents (cont.)
For any non-zero number x and a positive integer n
and
Note: is not defined
and
Rules Concerning Integral Exponents
Following are five rules in which m and n are positive integers:
Rule 1: ; for example,
Rule 2: ; for example
or
Rules Concerning Integral Exponents (Cont.)
Rule 3: ; for example
or
Rule 4: ; for example
or
Rule 5: ; for example
or
Basic Rules for Operating with Fractions
Since dividing by zero is not defined, we assume that the denominator
is not zero.
Following are the eight basic rules for operating with fractions.
Rule 1: ; for example
Rule 2: ; for example
Rule 3: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 4: ; for example
Rule 5: ; for example
Rule 6: ; for example
Basic Rules for Operating with Fractions (cont.)
Rule 7: ; for example
Rule 8: ; for example
Notes: a*b +a*x may be expressed as a(b + x)
a*b + 1 may be written as a(b + ), and
m*x – y may be expressed as m(x - )
Square Root
Generally, for a>0 , there is exactly one positive number x such that
, we say that x is the root of a, written as
for
When n = 2, we say that x is the square root of “a” and is denoted by
or or
For example:
or
Practices
Carrying out the following operations:
24 ; 2-2 ; 2322, ; 252-5 ; and (2x3)5
; ; ; and
3.
4.
5.
n
m
n
m
x
x
x
+
=
n
x
1
0
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2
(
2
2
2
=
=
=
-
+
-
x
x
x
x
1
0
=
x
0
0
n
n
n
x
x
x
x
=
=
+
0
0
n
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2 ...
2024.06.01 Introducing a competency framework for languag learning materials ...Sandy Millin
http://sandymillin.wordpress.com/iateflwebinar2024
Published classroom materials form the basis of syllabuses, drive teacher professional development, and have a potentially huge influence on learners, teachers and education systems. All teachers also create their own materials, whether a few sentences on a blackboard, a highly-structured fully-realised online course, or anything in between. Despite this, the knowledge and skills needed to create effective language learning materials are rarely part of teacher training, and are mostly learnt by trial and error.
Knowledge and skills frameworks, generally called competency frameworks, for ELT teachers, trainers and managers have existed for a few years now. However, until I created one for my MA dissertation, there wasn’t one drawing together what we need to know and do to be able to effectively produce language learning materials.
This webinar will introduce you to my framework, highlighting the key competencies I identified from my research. It will also show how anybody involved in language teaching (any language, not just English!), teacher training, managing schools or developing language learning materials can benefit from using the framework.
Students, digital devices and success - Andreas Schleicher - 27 May 2024..pptxEduSkills OECD
Andreas Schleicher presents at the OECD webinar ‘Digital devices in schools: detrimental distraction or secret to success?’ on 27 May 2024. The presentation was based on findings from PISA 2022 results and the webinar helped launch the PISA in Focus ‘Managing screen time: How to protect and equip students against distraction’ https://www.oecd-ilibrary.org/education/managing-screen-time_7c225af4-en and the OECD Education Policy Perspective ‘Students, digital devices and success’ can be found here - https://oe.cd/il/5yV
The Roman Empire A Historical Colossus.pdfkaushalkr1407
The Roman Empire, a vast and enduring power, stands as one of history's most remarkable civilizations, leaving an indelible imprint on the world. It emerged from the Roman Republic, transitioning into an imperial powerhouse under the leadership of Augustus Caesar in 27 BCE. This transformation marked the beginning of an era defined by unprecedented territorial expansion, architectural marvels, and profound cultural influence.
The empire's roots lie in the city of Rome, founded, according to legend, by Romulus in 753 BCE. Over centuries, Rome evolved from a small settlement to a formidable republic, characterized by a complex political system with elected officials and checks on power. However, internal strife, class conflicts, and military ambitions paved the way for the end of the Republic. Julius Caesar’s dictatorship and subsequent assassination in 44 BCE created a power vacuum, leading to a civil war. Octavian, later Augustus, emerged victorious, heralding the Roman Empire’s birth.
Under Augustus, the empire experienced the Pax Romana, a 200-year period of relative peace and stability. Augustus reformed the military, established efficient administrative systems, and initiated grand construction projects. The empire's borders expanded, encompassing territories from Britain to Egypt and from Spain to the Euphrates. Roman legions, renowned for their discipline and engineering prowess, secured and maintained these vast territories, building roads, fortifications, and cities that facilitated control and integration.
The Roman Empire’s society was hierarchical, with a rigid class system. At the top were the patricians, wealthy elites who held significant political power. Below them were the plebeians, free citizens with limited political influence, and the vast numbers of slaves who formed the backbone of the economy. The family unit was central, governed by the paterfamilias, the male head who held absolute authority.
Culturally, the Romans were eclectic, absorbing and adapting elements from the civilizations they encountered, particularly the Greeks. Roman art, literature, and philosophy reflected this synthesis, creating a rich cultural tapestry. Latin, the Roman language, became the lingua franca of the Western world, influencing numerous modern languages.
Roman architecture and engineering achievements were monumental. They perfected the arch, vault, and dome, constructing enduring structures like the Colosseum, Pantheon, and aqueducts. These engineering marvels not only showcased Roman ingenuity but also served practical purposes, from public entertainment to water supply.
Unit 8 - Information and Communication Technology (Paper I).pdfThiyagu K
This slides describes the basic concepts of ICT, basics of Email, Emerging Technology and Digital Initiatives in Education. This presentations aligns with the UGC Paper I syllabus.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
Operation “Blue Star” is the only event in the history of Independent India where the state went into war with its own people. Even after about 40 years it is not clear if it was culmination of states anger over people of the region, a political game of power or start of dictatorial chapter in the democratic setup.
The people of Punjab felt alienated from main stream due to denial of their just demands during a long democratic struggle since independence. As it happen all over the word, it led to militant struggle with great loss of lives of military, police and civilian personnel. Killing of Indira Gandhi and massacre of innocent Sikhs in Delhi and other India cities was also associated with this movement.
2. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
The Basics
3. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
The Basics
4. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
The Basics
15 can be divided by 3
5. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
The Basics
15 can be divided by 3
6. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12:
The Basics
15 can be divided by 3
7. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1,
The Basics
15 can be divided by 3
8. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2,
The Basics
15 can be divided by 3
9. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
15 can be divided by 3
10. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
15 can be divided by 3
11. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
12. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The Basics
13. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12:
The Basics
14. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12,
The Basics
15. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
The Basics
16. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
The Basics
Your Turn: List the factors and the multiples of 18.
17. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
The Basics
A prime number is a natural number that can not be divided
evenly into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
Your Turn: List the factors and the multiples of 18.
18. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
The Basics
A prime number is a natural number that can not be divided
evenly into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
The numbers 2, 3, 5, 7, (not 9), 11, 13,… are prime numbers.
Your Turn: List the factors and the multiples of 18.
19. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
The Basics
A prime number is a natural number that can not be divided
evenly into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
The numbers 2, 3, 5, 7, (not 9), 11, 13,… are prime numbers.
Even numbers are not prime so prime numbers must be odd
(except for 2).
Your Turn: List the factors and the multiples of 18.
20. Whole numbers 1, 2, 3, 4,.. are called natural numbers because
they are used to count whole items like chickens and cows.
If x and y are two natural numbers and x can be divided by y,
we say y is a factor of x. We also say that x is a multiple of y.
Hence 3 is a factor of 15 and that 15 is a multiple of 3.
A number x is always the multiple and a factor of itself.
Example A. List the factors and the multiples of 12.
The factors of 12: 1, 2, 3, 4, 6, and 12.
The multiples of 12: 12=1x12, 24=2x12, 36=3x12, 48, 60, etc…
The Basics
A prime number is a natural number that can not be divided
evenly into smaller numbers (except as 1’s), that is to say,
a prime number only has 1 and itself as factors.
The numbers 2, 3, 5, 7, (not 9), 11, 13,… are prime numbers.
Even numbers are not prime so prime numbers must be odd
(except for 2). The number 1 is not a prime number.
Your Turn: List the factors and the multiples of 18.
21. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*...
The Basics
22. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6
The Basics
23. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3.
The Basics
24. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers.
The Basics
25. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely,
The Basics
26. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
The Basics
27. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
The Basics
28. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
The Basics
29. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
The Basics
30. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
We write x*x*x…*x as xN where N is the number of x’s
multiplying to itself.
The Basics
31. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
We write x*x*x…*x as xN where N is the number of x’s
multiplying to itself. N is called the exponent or the power of xN.
The Basics
32. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 =
43 =
We write x*x*x…*x as xN where N is the number of x’s
multiplying to itself. N is called the exponent or the power of xN.
The Basics
33. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 =
We write x*x*x…*x as xN where N is the number of x’s
multiplying to itself. N is called the exponent or the power of xN.
The Basics
34. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4
We write x*x*x…*x as xN where N is the number of x’s
multiplying to itself. N is called the exponent or the power of xN.
The Basics
35. To factor a number x means to break down x as a product,
i.e. to write x as a*b*c*... e.g. 12 can be factored as
12 = 2*6 = 3*4 = 2*2*3. We say the factorization is complete
if all the factors are prime numbers. Hence 12=2*2*3 is factored
completely, 12=2*6 is not complete because 6 is not prime.
Just as in chemistry that every chemical is a group of bonded
elements, every number is the product of a group of prime
numbers.
Exponents
To simplify writing repetitive multiplication, we write 22 for 2*2,
we write 23 for 2*2*2, we write 24 for 2*2*2*2 and so on.
Example B. Calculate.
32 = 3*3 = 9 (note that 3*2 = 6)
43 = 4*4*4 = 64 (note that 4*3 = 12)
We write x*x*x…*x as xN where N is the number of x’s
multiplying to itself. N is called the exponent or the power of xN.
The Basics
Your Turn:
Calculate 24, 34, 44,
and 2*4, 3*4, 4*4
36. Numbers that are factored completely may be written
using the exponential notation.
The Basics
37. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 =
The Basics
38. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3
The Basics
39. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
The Basics
40. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 =
The Basics
41. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25
The Basics
42. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5
The Basics
43. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
44. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order.
45. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
46. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
47. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
48. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
49. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
50. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
2 22 2
51. Numbers that are factored completely may be written
using the exponential notation.
Hence, factored completely, 12 = 2*2*3 = 22*3,
200 = 8*25 = 2*2*2*5*5 = 23*52.
The Basics
Each number has a unique form when its completely factored
into prime factors and arranged in order. This is analogous to
chemistry where each chemical has a unique chemical
composition of basic elements such as H20 for water.
Larger numbers may be factored using a vertical format.
Example C. Factor 144 completely. (Vertical format)
144
12 12
3 4 3 4
2 22 2
Gather all the prime numbers at
the end of the branches we have
144 = 3*3*2*2*2*2 = 3224.
Note that we obtain the same answer
regardless how we factor at each step.
Your Turn: Factor 72 completely.
52. Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷.
53. Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts.
54. Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
55. Associative Law for Addition and Multiplication
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
56. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
57. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3
1 + (2 + 3)
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
58. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6,
1 + (2 + 3)
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
59. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6,
1 + (2 + 3) = 1 + 5 = 6.
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
60. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as
1 + (2 + 3) = 1 + 5 = 6.
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
So if you want to
total the money in
your pockets,
it does not matter
which two pockets
you add first.
61. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as
1 + (2 + 3) = 1 + 5 = 6.
Subtraction and division are not associative.
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
So if you want to
total the money in
your pockets,
it does not matter
which two pockets
you add first.
62. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as
1 + (2 + 3) = 1 + 5 = 6.
Subtraction and division are not associative.
For example, ( 3 – 2 ) – 1
3 – ( 2 – 1 )
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
So if you want to
total the money in
your pockets,
it does not matter
which two pockets
you add first.
63. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as
1 + (2 + 3) = 1 + 5 = 6.
Subtraction and division are not associative.
For example, ( 3 – 2 ) – 1 = 0
3 – ( 2 – 1 )
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
So if you want to
total the money in
your pockets,
it does not matter
which two pockets
you add first.
64. Associative Law for Addition and Multiplication
( a + b ) + c = a + ( b + c )
( a * b ) * c = a * ( b * c )
Note: We are to perform the operations inside the “( )” first.
For example, (1 + 2) + 3 = 3 + 3 = 6, which is the same as
1 + (2 + 3) = 1 + 5 = 6.
Subtraction and division are not associative.
For example, ( 3 – 2 ) – 1 = 0 is different from
3 – ( 2 – 1 ) = 2.
Basic Laws
Of the four arithmetic operations +, – , *, and ÷,
+, and * behave nicer than – or ÷. We often take advantage
of the this nice “behavior” of the addition and multiplication
operations to get short cuts. However, be sure you don’t
mistakenly apply these short cuts to – and ÷.
So if you want to
total the money in
your pockets,
it does not matter
which two pockets
you add first.
66. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
Basic Laws
67. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Basic Laws
68. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
Basic Laws
69. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
Basic Laws
70. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
Basic Laws
71. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Basic Laws
72. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Hence to take advantages of this when adding a list numbers,
it's easier to add the ones that add to multiples of 10 first.
Basic Laws
73. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Hence to take advantages of this when adding a list numbers,
it's easier to add the ones that add to multiples of 10 first.
Example D.
14 + 3 + 16 + 8 + 35 + 15
Basic Laws
74. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Hence to take advantages of this when adding a list numbers,
it's easier to add the ones that add to multiples of 10 first.
Example D.
14 + 3 + 16 + 8 + 35 + 15
= 30
Basic Laws
75. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Hence to take advantages of this when adding a list numbers,
it's easier to add the ones that add to multiples of 10 first.
Example D.
14 + 3 + 16 + 8 + 35 + 15
= 30 + 50
Basic Laws
76. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Hence to take advantages of this when adding a list numbers,
it's easier to add the ones that add to multiples of 10 first.
Example D.
14 + 3 + 16 + 8 + 35 + 15
= 30 + 11 + 50
Basic Laws
77. Commutative Law for Addition and Multiplication
a + b = b + a
a * b = b * a
For example, 3*4 = 4*3 = 12.
Subtraction and division don't satisfy commutative laws.
For example: 2 1 1 2.
From the above laws, we get the following important facts.
When adding, the order of addition doesn’t matter.
Hence to take advantages of this when adding a list numbers,
it's easier to add the ones that add to multiples of 10 first.
Example D.
14 + 3 + 16 + 8 + 35 + 15
= 30 + 11 + 50 = 91
Basic Laws
79. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
When multiplying many numbers, always multiply them in
pairs first.
80. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
81. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
82. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
83. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
84. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
85. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Because the order of the additions (or multiplications) may be
scrambled, this offers a strategy to check our answers–
do it in two different ways.
86. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Because the order of the additions (or multiplications) may be
scrambled, this offers a strategy to check our answers–
do it in two different ways.
If both yield the same answer, it’s likely the answer is correct.
87. Basic Laws
When multiplying, the order of multiplication doesn’t
matter.
Example E.
36 = 3*3*3*3*3*3
= 9 * 9 * 9
= 81 * 9 = 729
When multiplying many numbers, always multiply them in
pairs first. This is useful for raising exponents.
Because the order of the additions (or multiplications) may be
scrambled, this offers a strategy to check our answers–
do it in two different ways.
If both yield the same answer, it’s likely the answer is correct.
If the two outcomes are different, then some thing went wrong,
find it and correct it.
89. Basic Laws
The are two $10–bills and three $5–bills in a box.
The Distributive Law
90. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
The Distributive Law
91. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
The Distributive Law
92. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
The Distributive Law
93. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
The Distributive Law
Recording these with symbols:
4(20+15) = 4(35)
= 140
94. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
II. Unpack the boxes, there are 4x20=$80 from the $10-bills,
and 4x15=$60 from the $5-bills, which total to $140.
The Distributive Law
Recording these with symbols:
4(20+15) = 4(35)
= 140
95. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
II. Unpack the boxes, there are 4x20=$80 from the $10-bills,
and 4x15=$60 from the $5-bills, which total to $140.
The Distributive Law
Recording these with symbols:
4(20+15) = 4(35)
or 4(20+15) = 4(20) + 4(15)
= 140
96. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
II. Unpack the boxes, there are 4x20=$80 from the $10-bills,
and 4x15=$60 from the $5-bills, which total to $140.
The Distributive Law
Recording these with symbols: The point here is that
these two ways of
totaling always yield
the same result,
not which is easier.
4(20+15) = 4(35)
or 4(20+15) = 4(20) + 4(15)
= 140
97. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
II. Unpack the boxes, there are 4x20=$80 from the $10-bills,
and 4x15=$60 from the $5-bills, which total to $140.
The Distributive Law
Recording these with symbols: The point here is that
these two ways of
totaling always yield
the same result,
not which is easier.
4(20+15) = 4(35)
or 4(20+15) = 4(20) + 4(15)
= 140
The fact that these two ways are “identical" and that there are
two forms associated with the two ways is important.
Let’s redo this in symbols.
98. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
II. Unpack the boxes, there are 4x20=$80 from the $10-bills,
and 4x15=$60 from the $5-bills, which total to $140.
The Distributive Law
Recording these with symbols: The point here is that
these two ways of
totaling always yield
the same result,
not which is easier.
4(20+15) = 4(35)
or 4(20+15) = 4(20) + 4(15)
= 140
The fact that these two ways are identical and that
there two forms associated with the two ways is important.
Let’s redo this in symbols.
99. Basic Laws
The are two $10–bills and three $5–bills in a box. So there is
the total of 2x10=20 plus 3x5=15, or $35, in the box which is
recorded as (20+15) = 35.
Suppose we have 4 of these boxes, there are two methods to
find their total:
I. Each box has $35 so 4 boxes have 4x35=$140 in total.
II. Unpack the boxes, there are 4x20=$80 from the $10-bills,
and 4x15=$60 from the $5-bills, which total to $140.
The Distributive Law
Recording these with symbols: The point here is that
these two ways of
totaling always yield
the same result,
not which is easier.
4(20+15) = 4(35)
or 4(20+15) = 4(20) + 4(15)
= 140
The fact that these two ways are identical and that
there two forms associated with the two ways is important.
Let’s redo this in symbols.
101. Basic Laws
There are two $10–bills and three $5–bills in a box.
The Distributive Law
102. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
The Distributive Law
103. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F).
The Distributive Law
104. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F). We open all the boxes and take out all the bills,
there are 4 sets of 2T’s or 8T’s
The Distributive Law
105. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F). We open all the boxes and take out all the bills,
there are 4 sets of 2T’s or 8T’s and 4 sets of 3F’s or 12F’s.
The Distributive Law
106. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F). We open all the boxes and take out all the bills,
there are 4 sets of 2T’s or 8T’s and 4 sets of 3F’s or 12F’s.
We track this unpacking or expanding procedure, as
4(2T + 3F) = 4*2T + 4*3F = 8T + 12F
The Distributive Law
107. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F). We open all the boxes and take out all the bills,
there are 4 sets of 2T’s or 8T’s and 4 sets of 3F’s or 12F’s.
We track this unpacking or expanding procedure, as
4(2T + 3F) = 4*2T + 4*3F = 8T + 12F
The Distributive Law
Expressing this operation in symbol we have the
Distributive Law : a*(b ± c) = a*b ± a*c
108. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F). We open all the boxes and take out all the bills,
there are 4 sets of 2T’s or 8T’s and 4 sets of 3F’s or 12F’s.
We track this unpacking or expanding procedure, as
4(2T + 3F) = 4*2T + 4*3F = 8T + 12F
The Distributive Law
Expressing this operation in symbol we have the
Distributive Law : a*(b ± c) = a*b ± a*c
The “packed” form is
the factored form.
the factored form.
109. Basic Laws
There are two $10–bills and three $5–bills in a box.
Using the letter T to represent a $10–bill and F for a $5–bill,
we may represent this box as (2T + 3F).
If we have four of these boxes, we may record them as
4(2T + 3F). We open all the boxes and take out all the bills,
there are 4 sets of 2T’s or 8T’s and 4 sets of 3F’s or 12F’s.
We track this unpacking or expanding procedure, as
4(2T + 3F) = 4*2T + 4*3F = 8T + 12F
The Distributive Law
Expressing this operation in symbol we have the
Distributive Law : a*(b ± c) = a*b ± a*c
The “packed” form is
the factored form.
The “unpacked” form is
the expanded form.
the factored form. the expanded form.
110. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket,
Basic Laws
111. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket,
Basic Laws
The basket may be written as (4A + 6B).
112. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets.
Basic Laws
The basket may be written as (4A + 6B).
113. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets.
Basic Laws
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B)
114. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B)
115. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B
116. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B = 20A + 30B
so that we have 20 apples and 30 bananas.
117. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
B. We unpack six fruit boxes and obtain 48 apples and
30 bananas. Record this amount using A and B
(i.e. the expanded form),
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B = 20A + 30B
so that we have 20 apples and 30 bananas.
118. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
B. We unpack six fruit boxes and obtain 48 apples and
30 bananas. Record this amount using A and B
(i.e. the expanded form),
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B = 20A + 30B
so that we have 20 apples and 30 bananas.
The fruits may be recorded as 48A + 30B
119. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
B. We unpack six fruit boxes and obtain 48 apples and
30 bananas. Record this amount using A and B
(i.e. the expanded form), then write it in the factored form
(i.e. as 6 packed boxes). What were in each box?
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B = 20A + 30B
so that we have 20 apples and 30 bananas.
The fruits may be recorded as 48A + 30B
120. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
B. We unpack six fruit boxes and obtain 48 apples and
30 bananas. Record this amount using A and B
(i.e. the expanded form), then write it in the factored form
(i.e. as 6 packed boxes). What were in each box?
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B = 20A + 30B
so that we have 20 apples and 30 bananas.
The fruits may be recorded as 48A + 30B, packed in six
baskets, so each has 48/6 = 8 apples and 30/6 = 5 bananas,
121. Example E. A. A fruit basket contains 4 apples and 6
bananas, using A for “an apple” and B for "a banana“ write the
expression for a fruit basket, then write the expression for
5 such baskets. Expand the expression,
how many apples and bananas do we have?
Basic Laws
B. We unpack six fruit boxes and obtain 48 apples and
30 bananas. Record this amount using A and B
(i.e. the expanded form), then write it in the factored form
(i.e. as 6 packed boxes). What were in each box?
The basket may be written as (4A + 6B).
Five baskets are 5(4A + 6B), distribute the 5 and expand,
5(4A + 6B) = 5*4A + 5*6B = 20A + 30B
so that we have 20 apples and 30 bananas.
The fruits may be recorded as 48A + 30B, packed in six
baskets, so each has 48/6 = 8 apples and 30/6 = 5 bananas,
hence 48A + 30B = 6(8A + 5B) in the factored form.
