Download Exponents, Roots, Factorization of Whole Numbers

OpenStax-CNX module: m34877
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Exponents, Roots, Factorization of
Whole Numbers: Summary of Key
Concepts
∗
Wade Ellis
Denny Burzynski
This work is produced by OpenStax-CNX and licensed under the
Creative Commons Attribution License 3.0†
Abstract
This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This
module reviews the key concepts from the chapter "Exponents, Roots, Factorization of Whole Numbers."
1 Summary of Key Concepts
Exponential Notation ( here )
Exponential notation is a description of repeated multiplication.
Exponent ( here )
An exponent records the number of identical factors repeated in a multiplication.
1
2
In a number such as 73 ,
Base ( here3 )
7 is called the base.
Exponent ( here4 )
3 is called the exponent, or power.
Power ( here5 )
73 is read "seven to the third power," or "seven cubed."
∗ Version
1.2: Aug 18, 2010 8:25 pm -0500
† http://creativecommons.org/licenses/by/3.0/
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Whole Numbers: Exponents and Roots"
Whole Numbers: Exponents and Roots"
Whole Numbers: Exponents and Roots"
Whole Numbers: Exponents and Roots"
Whole Numbers: Exponents and Roots"
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Squared, Cubed ( here6 )
A number raised to the second power is often called
called cubed.
squared. A number raised to the third power is often
Root ( here7 )
In mathematics, the word root is used to indicate that, through repeated multiplication, one number is the
source of another number.
The Radical Sign √ ( here8 )
The symbol √ is called a radical sign and indicates the square root of a number. The symbol
√
n
represents the nth root.
Radical, Index, Radicand
( here9 )
√
An expression such as
radicand.
4
16 is called a
radical and 4 is called the index. The number 16 is called the
Grouping Symbols ( here10 )
Grouping symbols are used to indicate that a particular collection of numbers and meaningful operations are
to be grouped together and considered as one number. The grouping symbols commonly used in mathematics
are
Parentheses: ( )
Brackets: [ ]
Braces: { }
Bar:
Order of Operations ( here11 )
1. Perform all operations inside grouping symbols, beginning with the innermost set, in the order of 2, 3,
and 4 below.
2. Perform all exponential and root operations, moving left to right.
3. Perform all multiplications and division, moving left to right.
4. Perform all additions and subtractions, moving left to right.
One Number as the Factor of Another ( here12 )
A rst number is a factor of a second number if the rst number divides into the second number a whole
number of times.
Prime Number ( here13 )
A whole number greater than one whose only factors are itself and 1 is called a prime number. The whole
number 1 is not a prime number. The whole number 2 is the rst prime number and the only even prime
number.
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Composite Number ( here14 )
A whole number greater than one that is composed of factors other than itself and 1 is called a composite
number.
Fundamental Principle of Arithmetic ( here15 )
Except for the order of factors, every whole number other than 1 can be written in one and only one way as
a product of prime numbers.
Prime Factorization ( here16 )
The prime factorization of 45 is 3 · 3 · 5. The numbers that occur in this factorization of 45 are each prime.
Determining the Prime Factorization of a Whole Number ( here17 )
There is a simple method, based on division by prime numbers, that produces the prime factorization of a
whole number. For example, we determine the prime factorization of 132 as follows.
The prime factorization of 132 is 2 · 2 · 3 · 11 = 22 · 3 · 11.
Common Factor ( here18 )
A factor that occurs in each number of a group of numbers is called a
factor to the group 18, 6, and 45
common factor. 3 is a common
The largest common factor of a group of whole numbers is called the
example, to nd the greatest common factor of 12 and 20,
greatest common factor. For
Greatest Common Factor (GCF) ( here19 )
1. Write the prime factorization of each number.
12
=
2 · 2 · 3 = 22 · 3
60
=
2 · 2 · 3 · 5 = 22 · 3 · 5
2. Write each base that is common to each of the numbers: 2 and 3
3. The smallest exponent appearing on 2 is 2.
The smallest exponent appearing on 3 is 1.
4. The GCF of 12 and 60 is the product of the numbers 22 and 3. 22 · 3 = 4 · 3 = 12
Thus, 12 is the largest number that divides both 12 and 60 without a remainder.
Finding the GCF ( here20 )
There is a simple method, based on prime factorization, that determines the GCF of a group of whole
numbers.
Multiple ( here21 )
When a whole number is multiplied by all other whole numbers, with the exception of zero, the resulting
individual products are called multiples of that whole number. Some multiples of 7 are 7, 14, 21, and 28.
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Common Multiples ( here22 )
Multiples that are common to a group of whole numbers are called
multiples of 6 and 9 are 18, 36, and 54.
common multiples. Some common
The LCM ( here23 )
The least common multiple (LCM) of a group of whole numbers is the smallest whole number that each
of the given whole numbers divides into without a remainder. The least common multiple of 9 and 6 is 18.
Finding the LCM ( here24 )
There is a simple method, based on prime factorization, that determines the LCM of a group of whole
numbers. For example, the least common multiple of 28 and 72 is found in the following way.
1. Write the prime factorization of each number
28
=
2 · 2 · 7 = 22 · 7
72
=
2 · 2 · 2 · 3 · 3 = 23 · 32
2. Write each base that appears in each of the prime factorizations, 2, 3, and 7.
3. To each of the bases listed in step 2, attach the largest exponent that appears on it in the prime
factorization. 23 , 32 , and 7
4. The LCM is the product of the numbers found in step 3. 23 · 32 · 7 = 8 · 9 · 7 = 504
Thus, 504 is the smallest number that both 28 and 72 will divide into without a remainder.
The Dierence Between the GCF and the LCM ( here25 )
The GCF of two or more whole numbers is the largest number that divides into each of the given whole
numbers. The LCM of two or more whole numbers is the smallest whole number that each of the given
numbers divides into without a remainder.
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