Makalah

Power , Root , and Factors

Disusun guna memenuhi tugas mata kuliah Bahasa Inggris Keilmuan

Dosen pengampu : Aulia Hanifah Q.M.Pd

Disusun oleh:

1. FIKA RIZKY ( 21310010 )

2. FADILA SICILIA MAHARDIKA ( 21310025 )

Program Studi Pendidikan Matematika

Fakultas Keguruan dan Ilmu Pendidikan
UNIVERSITAS MUHAMMADIYAH METRO
2021/2022
 Foreword
Praise and gratitude to Allah SWT who has given the author the opportunity to complete this
paper entitled power,Roots, and Factors.

Materials for power, roots, and factors were prepared to fulfill the assignment from Mrs.
Aulia Hanifah Q.M.Pd in the scientific English course at Muhammadiyah Metro University. In
addition, the author also hopes that this paper can add insight for readers about the
power,roots,and factors.

The author would like to thank Mrs. Aulia Hanifah Q.M.Pd as an English scientific lecturer.
With the task that has been given, it can add insight and knowledge related to the field that the
author is engaged in. The author would also like to thank all those who have helped in the
preparation of this paper.

This paper was written based on various sources of information and various media related to
this material. With the creation of this paper, hopefully it can be useful for all of us and add to
our insight into the power,roots,and factors.

TABLE OF CONTENTENTS
PREFACE .......................................................................................................................................

FOREWORD ...................................................................................................................................

CHAPTER I INTRODUCTION ...........................................................................................................

1.1 Background Of The Paper ........................................................................................................

1.2 Purpose Of The Paper ..............................................................................................................

1.3 Problem Formulation ...............................................................................................................

CHAPTER II THEORY AND DISCUSION ............................................................................................

2.1 Definition power ......................................................................................................................

2.2 Definition roots ........................................................................................................................

2.3 definition factors .....................................................................................................................

CHAPTER III CONCLUSION ..............................................................................................................

3.1 Conclusion ...............................................................................................................................

3.2 Suggest ....................................................................................................................................

BIBLIOGRAPHY ...............................................................................................................................

CHAPTER 1
 INTRODUCTION

A. Background
Mathematics is one of the subjects studied from basic education to secondary education and even
higher education. In mathematics lessons, we find a lot of formulas that are difficult for some students
to understand, even though mathematics is a systematic lesson, or between formulas that are related to
each other.
In mathematics, there are several areas of study that we can know, one of the powers, roots and
factors. This material is restated to help you recall so that you become more familiar with this concept.
In this paper it appears that the concepts of powers and roots are often used. Thus, experience with this
material is a wasted work.

B. Problem Formulation
1. What is the meaning of rank?
2. What is the meaning of root?
3. What is the meaning of factor?

C. Purpose
1. To know the meaning of rank
2. To know the meaning of the root
3. To know the meaning of factor
 CHAPTER 2
DISCUSSION

2.1 DEFINITION OF POWER

Power is a mathematical operation for repeated multiplication of a number by as many powers as . The
exponent of a number is a number that is written lower and is located slightly up.

A. POWER

1. Positive Round POWER

a. Definition of Positive Integer
If a is a real number and n is a positive integer, then an (read "a to the power of n") is the product of n
factors, each of which is a. So, positive integer exponents are generally expressed in the form
an = a×a×a×...×a as many as n factors
with :
a = base number (base)
n = power or exponent
an = number to the power of
b. PROPERTIES OF APPOINTMENT OPERATION
1). The Nature of Multiplying Powered Numbers
For a R and m, n are positive integers, then:
am×an = am + n

Proof :
am × an =a ×a×a×...×a×a×a×a×...×a

=a×a×a×...×a×a×a×a×...×a = am + n
(proven )
2). Properties of Division of Powers
For a R, a ≠ 0 and m,n are positive integers that satisfy m>n.

am : an = am – n

Proof :
am : an =
= a×a×a×...×a = am – n (proven )

3). The Power of Multiplication of Numbers

For a R and m, n are positive integers, then:
( am )n = am . n

Proof :

( am )n = am×am×am×...×am

=( a ×a×...×a)×(a×a×...×a)×...×(a×a×...×a) = = am . n
(proven)
4). The Power of Multiplication of Numbers
For a, b R and n, positive integers apply:
( a . b )n = an . bn

Proof :
( a . b )n =ab ×ab×ab×...×ab = ( a×a×a×...×a) × (b×b×b×...×b) = an . bn (proven)

5). Power Properties of Number Division

For a,b R, b ≠ 0 and n positive integers, then:
n
=

Proof :

n
=

= = (proven)

2. Negative and Zero Integer Powers

a. Power of Zero
For a R dan a ≠ 0 then
a0 = 1

Proof :
a0 = an – n

= (the property of dividing numbers to exponents)

=1

So, a0 = 1

b. Negative Power Number

For a R and a ≠ 0

a-n =

This definition comes from the following form.

Let am : am – ( m + n ) = a-n

am : am + n = =

Then, a-n =
2.2 UNDERSTANDING ROOTS
The root form is a number whose result is neither a rational number nor an irrational number, and is
used as another form to express a number to a power.

B. ROOT FORM

1. Concept of Irrational Numbers

Irrational numbers are defined as numbers that cannot be expressed in comparison with a,b B
and b 0.

While rational numbers are numbers that can be expressed in the form of comparisons with a,b
B dan b 0.
Examples of irrational numbers:
a. π = 3,141592...
b. e = 2,718281...
c. = 1,414213...
d. = 2,6457...

Examples of rational numbers:

a. = 0,171717...
b. = 3,0000...
c. = 4,0000...

d. = 1,6666... =

It should be noted that irrational numbers are generally found in the form of roots, but not all forms of
roots are irrational numbers.
2. Root Shape
There are 3 parts in the number form of the root that need to be known, namely the symbol of the form
of the root, the radicand, and the index. In general, the root form is written in the form:

( read "root to the n power of a")

With: is called the root form (radical)

is called the root form symbol
n is called the index (root power)
a is called positive real for n natural numbers and n for odd numbers, a is negative real
numbers.
Root forms are divided into 2 types:
1. Namesake root
A root form is said to be the common root if its index (root power) is the same.
Example:
a. , , has index 2
b. , has index 3

2. similar root
A root form is said to be like roots if its index and radii are the same.
Example:
√3 2 ,2√3 2 ,5√3 2 has index 3, the radii is 2
Like exponents, roots also have certain properties, which are as follows:

a × √b = √ a ×b
For a, b real numbers with n corresponding natural numbers apply:
1. √n n n

2.

3.
√n b b √
√n a = n a

p √a ± q √a = ( p ± q ) √a
n n n

The properties of the root form above explain that the product of the two common root forms with
index n is equal to the product of the radicand of each root form with index n. This also applies to the
operation of dividing the form of the namesake root. For addition and subtraction with similar root
forms, what is added or subtracted is the coefficient of each root form, then multiplied by the root form.
3. Unreal rank
Numbers to the power of zero , negative integers , and fractions are also called untrue exponents.
The numbers with positive integer powers are also called real numbers. For any value a with a≠ 0 , m
integers , n natural numbers , and n≥ 2 applies:
1
a. √n a = a n
m
b. √n am = a n
1 m
The numbers a n and a n are called numbers with unreal powers.
4. the properties of the operation of the unreal rank
For a, b ∈ R where a, b ≠ 0 and p,q are rational numbers, the following properties of the untrue
power operation apply.
1. ap× aq = ap + q
2. ap : aq = ap - q
3. (ap)q = ap . q
4. ( a . b )p = ap . bp

()
p p
a a
5. = p ,b≠0
b b
1
6. a-p = p , a ≠ 0
a
operations on unreal exponents explain that basically the operations that apply are the same as
operations on real exponents.
It should be noted here that the exponents used are the powers of zero , negative integers , and
fractional numbers.
c. RATIONALIZING THE DENOMINER BEING A ROOT
In a number operation, there are times when the number has a denominator in the form of a root, such
as:
The forms of these numbers can be simplified by rationalizing the denominators of the fractions. The
activity of rationalizing essentially changes the form of the root in the denominator into the form of a
rational number, which in the end the number can be expressed in a simpler form.
A form of a fraction containing a number in the form of a root is said to be simple if it is fulfilled:
1. Every number has its roots in simple form, and
2. There is no root form in the denominator if the number is a fraction .
In this section , you will learn how to rationalize various forms of fractions to make them simpler .
a
1. Fraction of the Form
√b

a
The root form with b ≠ 0 can be rationalized by multiplying the fraction by √ bso that:
√b
a a √b = a b
= × √
√b √b √b b
a
2. Fraction of the Form
b− √ c
a a
To simplify the fraction form or is to multiply the fraction by the compound form of
b+ √ c b− √ c
the denominator. The compound form of b+ √ c is b−√ c . otherwise, the compound form of b−√ c
is b+ √ c .
a a b− √ c a(b−√ c )
= × =
b+ √ c b+ √ c b− √ c 2
b −c
a a b+ √ c a(b + √ c)
= × =
b+ √ c b− √ c b+ √ c 2
b −c
a
3. Fraction of the form
√ b−¿ √ c
a a
And to simplify the denominator of the fraction form or , that is, by multiplying
√ b+¿ √ c √ b−¿ √ c
by the compound form of the denominator.
The compound form of√ b + √ c adalah √ b - √ c . otherwise the compound form of √ b - √ c is
√ b + √ c so that:
× √ √ = √ √
a a b− c a( b− c)
=
√ b+¿ √ c √ b+¿ √ c √ b−¿ √ c b−c
a
=
a
×
√ b+ √c = a( √b+ √ c)
√ b−¿ √ c √ b−¿ √ c √ b+¿ √ c b−c
4. Simplifying the root form √ ( a+ b )−2 √ a−b
The form √ ( a+ b )−2 √ a−bcan be converted into the form ( √ a ± √ b )with the condition that
a,b∈ R and a > b.
 Proof :
2
( √ a ± √ b) = a ± 2 √ a . √ b + b = ( a + b ) ± 2 √ ab
√ a ± √ b = √ ( a+ b ) ±2 √ ab
so , √ ( a+ b ) ±2 √ ab = √ a ± √ b

2.3 UNDERSTANDING FACTORS

A factor is a number that is divisible by a number.
1. Factor a Number
A factor of a number is a number that is evenly divisible by the number. To determine the factor of a
number can be reached by looking for pairs of numbers which when multiplied the result is the number
that is looking for the factor. For more details see the following example!
Example
1. 4 : 1 = 4, remainder 0
4: 2 = 2, remainder 0
4: 4 = 1, remainder 0
So, the factors of 4 are 1, 2, and 4.
2. 6 : 1 = 6, remainder 0
6: 2 = 3, remainder 0
6: 3 = 2, remainder 0
6: 6 = 1, remainder 0
So, the factors of 6 are 1, 2, 3, and 6
3. 16 : 1 = 16, remainder 0
16 : 2 = 8, remainder 0
16 : 4 = 4, remainder 0
16 : 16 = 1, remainder 0
16 : 8 = 2, remainder 0
So, the factors of 16 are 1, 2, 4, 8, and 16

2. Multiples of a Number
Multiples of a number are numbers that are the product of the number by the natural number.
Pay attention to the following multiplications!
• Multiples of 2 = 2 x 1, 2 x 2, 2 x 3, 2 x 4, 2 x 5, ..., 2 x 10, .... = 2, 4, 6, 8, 10, .. ., 20, ....
• Multiples of 3 = 3 x 1, 3 x 2, 3 x 3, 3 x 4, 3 x 5, ..., 3 x 10, .... = 3, 6, 9, 12, 15, . . ., 30, . . .
• Multiples of 4 = 4 x 1, 4 x 2, 4 x 3, 4 x 4, 4 x 5, ..., 4 x 10, .... = 4, 8, 12, 16, 20, . . ., 40, . . .

3. Determine KPK and FPB

After you know the meaning of factors and multiples, and how to find factors and multiples of a number,
in this section you will learn about the least common multiple and the greatest common factor of two or
more numbers.
But before studying it, you should first learn how to determine multiples and common factors of two or
more numbers.

4. Determine the Common Multiple of Two Numbers

Common multiples of two numbers are numbers that are multiples of the two numbers.
Look at the multiples of the following two numbers!
Multiples of 2 = 2, 4, 8, 10, 12, 14, 16, 18, 20, 22, 24, 26,...
Multiples of 4 = 4, 8, 12, 16, 20, 24, 28, 32, 36, 40, ....
Which numbers are in multiples of 2 or multiples of 4?
The same numbers found in multiples of 2 and multiples of 4 are: 4, 8, 12, 16, 20, 24, ....
The numbers 4, 8, 12, 16, 20, 24, ... These are the common multiples of 2 and 4.

5. Determining the LCM and GCF from Two or More Numbers

After we learn how to find common multiples and common factors of two or more numbers, in this
section we will learn about how to determine the GCF and LCM of two numbers.
To find out how, follow the following description carefully.
Determine the LCM of two or more numbers
The least common multiple (LCM) of two or more numbers is the smallest number that is a multiple of
these numbers.
In this section we will discuss one method that is commonly used, namely by using the following steps:
following.
1) determine the multiples of each number;
2) determine the common multiple;
3) Determine the smallest number in the common multiple.

6. Determine the GCF of two numbers

The greatest common factor (GCF) of two or more numbers is the largest number that is a factor of
these numbers. How to find the GCF of two or more numbers? To find the GCF of two or more numbers,
there are several ways.
In this section we will discuss one method that is commonly used, namely by using the following steps:
following.
1) determine the factors of each number;
2) determine the common factor;
3) Determine the largest number in the common factor.
 CHAPTER 3
CLOSING

3.1 CONCLUSION
Based on the description above, the conclusion of the paper entitled "Rank, Root and Factor" is
Ranks, roots, and factors are one method that can improve your abilities so that mathematics can
be considered a fun subject, and easy to understand. The quick way to calculate Powers, Roots,
and Factors has an easy procedure in the process, namely by having base numbers, exponents,
and exponents.

3.2 SUGGEST

Through this fast method of mathematics, it can be used as an alternative way for teaching about
Rank and root material so that we can do rooting without complicated work but make it fun.
 BIBLIOGRAPHY

https://bangka.tribunnews.com/2021/06/09/materi-belajar-matematika-kelas-4-sd-tentang-faktor-
bilangan-dan-kelipatan-lengkap-soal-jawaban?page=2

https://sc.syekhnurjati.ac.id/smartcampus/