3.6: Summary of Key Concepts
- Page ID
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\(\newcommand{\avec}{\mathbf a}\) \(\newcommand{\bvec}{\mathbf b}\) \(\newcommand{\cvec}{\mathbf c}\) \(\newcommand{\dvec}{\mathbf d}\) \(\newcommand{\dtil}{\widetilde{\mathbf d}}\) \(\newcommand{\evec}{\mathbf e}\) \(\newcommand{\fvec}{\mathbf f}\) \(\newcommand{\nvec}{\mathbf n}\) \(\newcommand{\pvec}{\mathbf p}\) \(\newcommand{\qvec}{\mathbf q}\) \(\newcommand{\svec}{\mathbf s}\) \(\newcommand{\tvec}{\mathbf t}\) \(\newcommand{\uvec}{\mathbf u}\) \(\newcommand{\vvec}{\mathbf v}\) \(\newcommand{\wvec}{\mathbf w}\) \(\newcommand{\xvec}{\mathbf x}\) \(\newcommand{\yvec}{\mathbf y}\) \(\newcommand{\zvec}{\mathbf z}\) \(\newcommand{\rvec}{\mathbf r}\) \(\newcommand{\mvec}{\mathbf m}\) \(\newcommand{\zerovec}{\mathbf 0}\) \(\newcommand{\onevec}{\mathbf 1}\) \(\newcommand{\real}{\mathbb R}\) \(\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}\) \(\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}\) \(\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}\) \(\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}\) \(\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}\) \(\newcommand{\laspan}[1]{\text{Span}\{#1\}}\) \(\newcommand{\bcal}{\cal B}\) \(\newcommand{\ccal}{\cal C}\) \(\newcommand{\scal}{\cal S}\) \(\newcommand{\wcal}{\cal W}\) \(\newcommand{\ecal}{\cal E}\) \(\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}\) \(\newcommand{\gray}[1]{\color{gray}{#1}}\) \(\newcommand{\lgray}[1]{\color{lightgray}{#1}}\) \(\newcommand{\rank}{\operatorname{rank}}\) \(\newcommand{\row}{\text{Row}}\) \(\newcommand{\col}{\text{Col}}\) \(\renewcommand{\row}{\text{Row}}\) \(\newcommand{\nul}{\text{Nul}}\) \(\newcommand{\var}{\text{Var}}\) \(\newcommand{\corr}{\text{corr}}\) \(\newcommand{\len}[1]{\left|#1\right|}\) \(\newcommand{\bbar}{\overline{\bvec}}\) \(\newcommand{\bhat}{\widehat{\bvec}}\) \(\newcommand{\bperp}{\bvec^\perp}\) \(\newcommand{\xhat}{\widehat{\xvec}}\) \(\newcommand{\vhat}{\widehat{\vvec}}\) \(\newcommand{\uhat}{\widehat{\uvec}}\) \(\newcommand{\what}{\widehat{\wvec}}\) \(\newcommand{\Sighat}{\widehat{\Sigma}}\) \(\newcommand{\lt}{<}\) \(\newcommand{\gt}{>}\) \(\newcommand{\amp}{&}\) \(\definecolor{fillinmathshade}{gray}{0.9}\)Summary of Key Concepts
Exponential Notation
Exponential notation is a description of repeated multiplication.
Exponent
An exponent records the number of identical factors repeated in a multiplication.
In a number such as \(7^3\).
Base
7 is called the base.
Exponent
3 is called the exponent, or power.
Power
\(7^3\) is read "seven to the third power," or "seven cubed."
Squared, Cubed
A number raised to the second power is often called squared. A number raised to the third power is often called cubed.
Root
In mathematics, the word root is used to indicate that, through repeated multiplication, one number is the source of another number.
The Radical Sign \(\sqrt{\ \ }\)
The symbol \(\sqrt{\ \ }\) is called a radical sign and indicates the square root of a number. The symbol \(\sqrt[n]{\ \ }\) represents the \(n\)th root.
Radical, Index, Radicand
An expression such as \(\sqrt[4]{16}\) is called a radical and 4 is called the index. The number 16 is called the radicand.
Grouping Symbols
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
- Perform all operations inside grouping symbols, beginning with the innermost set, in the order of 2, 3, and 4 below.
- Perform all exponential and root operations, moving left to right.
- Perform all multiplications and division, moving left to right.
- Perform all additions and subtractions, moving left to right.
One Number as the Factor of Another
A first number is a factor of a second number if the first number divides into the second number a whole number of times.
Prime Number
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 first prime number and the only even prime number.
Composite Number
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
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
The prime factorization of 45 is \(3 \cdot 3 \cdot 5\). The numbers that occur in this factorization of 45 are each prime.
Determining the Prime Factorization of a Whole Number
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 \(2 \cdot 2 \cdot 3 \cdot 11 = 2^2 \cdot 3 \cdot 11\).
Common Factor
A factor that occurs in each number of a group of numbers is called a common factor. 3 is a common factor to the group 18, 6, and 45
Greatest Common Factor (GCF)
The largest common factor of a group of whole numbers is called the greatest common factor. For example, to find the greatest common factor of 12 and 20,
Write the prime factorization of each number.
\(\text{array} {l} {12 = 2 \cdot 2 \cdot 3 = 2^2 \cdot 3} \\ {60 = 2 \cdot 2 \cdot 3 \cdot 5 = 2^2 \cdot 3 \cdot 5} \end{array}\)
Write each base that is common to each of the numbers:
2 and 3
The smallest exponent appearing on 2 is 2.
The smallest exponent appearing on 3 is 1.
The GCF of 12 and 60 is the product of the numbers \(2^2\) and 3.
\(2^2 \cdot 3 = 4 \cdot 3 = 12\)
Thus, 12 is the largest number that divides both 12 and 60 without a remainder.
Finding the GCF
There is a simple method, based on prime factorization, that determines the GCF of a group of whole numbers.
Multiple
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.
Common Multiples
Multiples that are common to a group of whole numbers are called common multiples. Some common multiples of 6 and 9 are 18, 36, and 54.
The LCM
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
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.
Write the prime factorization of each number
\(\begin{array} {l} {28 = 2 \cdot 2 \cdot 7 = 2^2 \cdot 7} \\ {72 = 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 = 2^3 \cdot 3^2} \end{array}\)
Write each base that appears in each of the prime factorizations, 2, 3, and 7.
To each of the bases listed in step 2, attach the largest exponent that appears on it in the prime factorization.
\(2^3\), \(3^2\), and 7
The LCM is the product of the numbers found in step 3.
\(2^3 \cdot 3^2 \cdot 7 = 8 \cdot 9 \cdot 7 = 504\)
Thus, 504 is the smallest number that both 28 and 72 will divide into without a remainder.
The Difference Between the GCF and the LCM
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.