This document discusses the rules for multiplying signed numbers. It states that to multiply two signed numbers, you multiply their absolute values and use the sign rules. The sign of the product is positive if the factors have the same sign or if there is an even number of negative factors, and negative if the factors have opposite signs or if there is an odd number of negative factors. Examples are provided to illustrate multiplying signed numbers and determining the sign of the product using the even-odd rule. Algebraic notation for multiplication is also discussed.
8 multiplication division of signed numbers, order of operationselem-alg-sample
This document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses that in algebra, multiplication is implied without an explicit operator between terms. The sign of a product of many numbers is determined by the even-odd rule - an even number of negative factors yields a positive product; an odd number yields a negative product. Examples are provided to illustrate the rules.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
The document discusses the process of long division for both numbers and polynomials. It demonstrates long division of numbers step-by-step using the example of 78 divided by 2. It then explains that long division of polynomials follows the same process, setting up the division with the numerator polynomial inside and denominator polynomial outside. An example problem divides the polynomial 2x^2 - 3x + 20 by the polynomial x - 4 using the long division process.
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
Properties of addition and multiplicationShiara Agosto
This document discusses properties of addition and multiplication. It explains the commutative, associative, identity, and distributive properties and provides examples of how each applies to addition and multiplication. Practice problems are included for students to identify which properties are being used and to solve expressions using the properties.
Commutative And Associative PropertiesEunice Myers
The document discusses the commutative and associative properties of real numbers. The commutative property states that the order of numbers does not matter in addition and multiplication, but it does matter in subtraction and division. The associative property states that the grouping of numbers does not matter in addition and multiplication, but it does matter in subtraction and division. Both properties only apply to addition and multiplication, not subtraction and division.
The document discusses three mathematical properties: the associative property, which allows changing the grouping of operations without changing the result; the commutative property, which allows changing the order of operations without changing the result; and the distributive property, which distributes multiplication over addition or subtraction. It provides examples of how to use each property to solve problems involving areas, discounts, and total costs.
The document discusses three properties of addition:
1) The associative property - The grouping of addends does not change the sum
2) The commutative property - The order of addends does not change the sum
3) The identity property - Adding 0 to a number does not change the number
It provides examples of applying each property to demonstrate that the sum is unchanged regardless of grouping or order of addends.
8 multiplication division of signed numbers, order of operationselem-alg-sample
This document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses that in algebra, multiplication is implied without an explicit operator between terms. The sign of a product of many numbers is determined by the even-odd rule - an even number of negative factors yields a positive product; an odd number yields a negative product. Examples are provided to illustrate the rules.
1 s3 multiplication and division of signed numbersmath123a
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use rules to determine the sign of the product: two numbers with the same sign yield a positive product, while two numbers with opposite signs yield a negative product. It also discusses that in algebra, operations are often implied rather than written out. The even-odd rule determines the sign of products with multiple factors: an even number of negative factors yields a positive product, while an odd number yields a negative product.
The document discusses the process of long division for both numbers and polynomials. It demonstrates long division of numbers step-by-step using the example of 78 divided by 2. It then explains that long division of polynomials follows the same process, setting up the division with the numerator polynomial inside and denominator polynomial outside. An example problem divides the polynomial 2x^2 - 3x + 20 by the polynomial x - 4 using the long division process.
1) The document provides an overview of properties and operations of real numbers including identifying different types of real numbers like integers, rational numbers, and irrational numbers.
2) It discusses ordering real numbers and using symbols like <, >, ≤, ≥ to compare them. Properties of addition, multiplication and other operations are also covered.
3) Examples are provided to illustrate concepts like using properties of real numbers to evaluate expressions and convert between units like miles and kilometers.
Properties of addition and multiplicationShiara Agosto
This document discusses properties of addition and multiplication. It explains the commutative, associative, identity, and distributive properties and provides examples of how each applies to addition and multiplication. Practice problems are included for students to identify which properties are being used and to solve expressions using the properties.
Commutative And Associative PropertiesEunice Myers
The document discusses the commutative and associative properties of real numbers. The commutative property states that the order of numbers does not matter in addition and multiplication, but it does matter in subtraction and division. The associative property states that the grouping of numbers does not matter in addition and multiplication, but it does matter in subtraction and division. Both properties only apply to addition and multiplication, not subtraction and division.
The document discusses three mathematical properties: the associative property, which allows changing the grouping of operations without changing the result; the commutative property, which allows changing the order of operations without changing the result; and the distributive property, which distributes multiplication over addition or subtraction. It provides examples of how to use each property to solve problems involving areas, discounts, and total costs.
The document discusses three properties of addition:
1) The associative property - The grouping of addends does not change the sum
2) The commutative property - The order of addends does not change the sum
3) The identity property - Adding 0 to a number does not change the number
It provides examples of applying each property to demonstrate that the sum is unchanged regardless of grouping or order of addends.
Properties of addition and multiplicationShiara Agosto
This document discusses properties of addition and multiplication. It explains the commutative, associative, identity, and distributive properties for both addition and multiplication. Examples are provided to illustrate each property. Practice problems are included for students to identify which properties are being used. The document also shows how to solve equations by applying properties of operations.
Math 7 lesson 11 properties of real numbersAriel Gilbuena
At the end of the lesson, the learner should be able to:
recall the different properties of real numbers
write equivalent statements involving variables using the properties of real numbers
Ben harvested 73 eggplant and 94 pieces of okra, for a total of 73 + 94 = 167 pieces of vegetables harvested. The associative property of addition states that changing the grouping of the addends does not change the sum, as shown in examples such as (4 + 3) + 5 = 4 + (3 + 5).
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
The document discusses properties of real numbers including:
- The commutative property of addition and multiplication, where changing the order of terms does not change the result.
- The identity properties of addition and multiplication, where adding/multiplying a number and the identity element (0 for addition, 1 for multiplication) does not change the number.
- The inverse properties of addition and multiplication, where adding/multiplying a number and its inverse/opposite results in the identity element.
1) Rules for adding and subtracting integers include keeping the sign the same when adding like signs, and using the sign of the larger number when subtracting or adding opposite signs.
2) When multiplying integers, the sign of the product is determined by the number of negative factors. If even, the product is positive, and if odd, the product is negative.
3) Integers are closed under addition, subtraction, and multiplication, and follow properties like commutativity and associativity for these operations.
The document discusses the properties of real numbers. It outlines six axioms of equality that define the equal sign and substitution in real numbers. These include reflexive, symmetric, transitive, addition, multiplication, and replacement properties. It then outlines six field axioms that define the algebraic structure of real numbers under addition and multiplication, including closure, associative, commutative, distributive, identity, and inverse axioms.
The document provides instructions and examples for performing arithmetic operations on integers using the Rathmell model. It defines the Rathmell triangle and how to manipulate signed chips to represent addition, subtraction, multiplication, and division of integers. Rules are given for each operation, depending on whether the numbers have like or unlike signs. Examples are worked out step-by-step to demonstrate applying the rules correctly.
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
Real numbers follow rules of equality and substitution. If two numbers are equal, then they are equal regardless of any addition, subtraction, multiplication, or division operations performed on them. Equality is also reflexive, symmetric, and transitive - a number equals itself; if a equals b then b equals a; and if a equals b and b equals c, then a equals c.
- The document is a lesson on identifying and graphing linear functions. It provides examples of determining if a graph or set of ordered pairs represents a linear function based on whether a constant change in the x-value results in a constant change in the y-value.
- It also gives examples of writing functions in standard form (y=mx+b) to identify if they are linear and how to graph linear functions by choosing x-values and finding the corresponding y-values.
- Applications word problems are presented where the domain and range may be restricted based on real-world constraints.
This document contains NCERT solutions for Class 8 Maths Chapter 1 on Rational Numbers. It includes solutions to 11 questions from Exercise 1.1 that involve finding additive and multiplicative inverses of rational numbers, identifying properties used in multiplication, determining if a number is its own reciprocal, and filling in blanks about rational numbers and their reciprocals. The questions cover key concepts about rational numbers such as their properties and operations involving addition, subtraction, multiplication and division.
Multiply matrices by following rules: matrices can be multiplied if the number of columns in the first equals the number of rows in the second. The product is an m x p matrix if the first is m x n and the second is n x p. Square matrices can be multiplied by themselves to find powers; the identity matrix I has 1s on the main diagonal and 0s elsewhere. Businesses can use matrix multiplication to calculate total revenues, costs, and profits from sales data.
This document provides information on factoring polynomials. It defines factors as numbers that are multiplied together to get a product. Factoring is rewriting an expression as a product of its factors. The document discusses factoring whole numbers, polynomials, and quadratic trinomials. It presents different methods for factoring quadratic trinomials with leading coefficients of 1, including factoring by decomposition, using temporary factors, and the window pane method. Examples are provided to illustrate each factoring technique.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
Rational numbers are closed under addition, subtraction, and multiplication, but not division. Addition, subtraction and multiplication of rational numbers are commutative, but division is not. Addition of rational numbers is associative, but subtraction is not.
The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
22 multiplication and division of signed numbersalg1testreview
To multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product:
- Two numbers with the same sign yield a positive product
- Two numbers with opposite signs yield a negative product
In algebra, if there is no indicated operation between quantities, it represents multiplication. For example, xy means x * y. However, if there is a + or - between parentheses and a quantity, it represents combining terms rather than multiplication.
23 multiplication and division of signed numbersalg-ready-review
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses how multiplication is implied in algebra without an explicit operation symbol between terms.
This document discusses key concepts in the real number system including:
- Rational numbers that can be expressed as ratios of integers, and irrational numbers that cannot.
- Integers, including positive, negative and whole numbers.
- Properties of addition like commutativity, associativity and closure.
- Properties of multiplication like commutativity, associativity and distributivity.
- Absolute value and rules for performing operations on signed numbers like addition, subtraction, multiplication and division.
An algebraic expression is a combination of letters and numbers linked by operation signs: addition, subtraction, multiplication, division and exponentiation. Algebraic expressions allow us, for example, to find areas and volumes. Some examples given are the circumference of a circle (2πr), the area of a square (s=l2), and the volume of a cube (V=a3). The document then provides examples and explanations of algebraic addition, subtraction, multiplication, division, and factorization.
Properties of addition and multiplicationShiara Agosto
This document discusses properties of addition and multiplication. It explains the commutative, associative, identity, and distributive properties for both addition and multiplication. Examples are provided to illustrate each property. Practice problems are included for students to identify which properties are being used. The document also shows how to solve equations by applying properties of operations.
Math 7 lesson 11 properties of real numbersAriel Gilbuena
At the end of the lesson, the learner should be able to:
recall the different properties of real numbers
write equivalent statements involving variables using the properties of real numbers
Ben harvested 73 eggplant and 94 pieces of okra, for a total of 73 + 94 = 167 pieces of vegetables harvested. The associative property of addition states that changing the grouping of the addends does not change the sum, as shown in examples such as (4 + 3) + 5 = 4 + (3 + 5).
10 more on factoring trinomials and factoring by formulaselem-alg-sample
The document discusses two methods for factoring trinomials of the form ax^2 + bx + c. The first method is short but not always reliable, while the second method takes more steps but always provides a definite answer. This second method, called the reversed FOIL method, involves finding four numbers that satisfy certain properties to factor the trinomial. An example is worked out step-by-step to demonstrate how to use the reversed FOIL method to factor the trinomial 3x^2 + 5x + 2.
The document discusses properties of real numbers including:
- The commutative property of addition and multiplication, where changing the order of terms does not change the result.
- The identity properties of addition and multiplication, where adding/multiplying a number and the identity element (0 for addition, 1 for multiplication) does not change the number.
- The inverse properties of addition and multiplication, where adding/multiplying a number and its inverse/opposite results in the identity element.
1) Rules for adding and subtracting integers include keeping the sign the same when adding like signs, and using the sign of the larger number when subtracting or adding opposite signs.
2) When multiplying integers, the sign of the product is determined by the number of negative factors. If even, the product is positive, and if odd, the product is negative.
3) Integers are closed under addition, subtraction, and multiplication, and follow properties like commutativity and associativity for these operations.
The document discusses the properties of real numbers. It outlines six axioms of equality that define the equal sign and substitution in real numbers. These include reflexive, symmetric, transitive, addition, multiplication, and replacement properties. It then outlines six field axioms that define the algebraic structure of real numbers under addition and multiplication, including closure, associative, commutative, distributive, identity, and inverse axioms.
The document provides instructions and examples for performing arithmetic operations on integers using the Rathmell model. It defines the Rathmell triangle and how to manipulate signed chips to represent addition, subtraction, multiplication, and division of integers. Rules are given for each operation, depending on whether the numbers have like or unlike signs. Examples are worked out step-by-step to demonstrate applying the rules correctly.
The document discusses several properties of real numbers including:
1) The closure properties of addition and multiplication - the sum and product of any two real numbers is a real number.
2) Commutative, associative, and distributive properties of addition and multiplication.
3) Identity properties of addition and multiplication with 0 and 1 being the identity elements.
4) Inverse properties of addition and multiplication where adding/multiplying a number and its inverse results in the identity element.
Real numbers follow rules of equality and substitution. If two numbers are equal, then they are equal regardless of any addition, subtraction, multiplication, or division operations performed on them. Equality is also reflexive, symmetric, and transitive - a number equals itself; if a equals b then b equals a; and if a equals b and b equals c, then a equals c.
- The document is a lesson on identifying and graphing linear functions. It provides examples of determining if a graph or set of ordered pairs represents a linear function based on whether a constant change in the x-value results in a constant change in the y-value.
- It also gives examples of writing functions in standard form (y=mx+b) to identify if they are linear and how to graph linear functions by choosing x-values and finding the corresponding y-values.
- Applications word problems are presented where the domain and range may be restricted based on real-world constraints.
This document contains NCERT solutions for Class 8 Maths Chapter 1 on Rational Numbers. It includes solutions to 11 questions from Exercise 1.1 that involve finding additive and multiplicative inverses of rational numbers, identifying properties used in multiplication, determining if a number is its own reciprocal, and filling in blanks about rational numbers and their reciprocals. The questions cover key concepts about rational numbers such as their properties and operations involving addition, subtraction, multiplication and division.
Multiply matrices by following rules: matrices can be multiplied if the number of columns in the first equals the number of rows in the second. The product is an m x p matrix if the first is m x n and the second is n x p. Square matrices can be multiplied by themselves to find powers; the identity matrix I has 1s on the main diagonal and 0s elsewhere. Businesses can use matrix multiplication to calculate total revenues, costs, and profits from sales data.
This document provides information on factoring polynomials. It defines factors as numbers that are multiplied together to get a product. Factoring is rewriting an expression as a product of its factors. The document discusses factoring whole numbers, polynomials, and quadratic trinomials. It presents different methods for factoring quadratic trinomials with leading coefficients of 1, including factoring by decomposition, using temporary factors, and the window pane method. Examples are provided to illustrate each factoring technique.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0.
Rational numbers are closed under addition, subtraction, and multiplication, but not division. Addition, subtraction and multiplication of rational numbers are commutative, but division is not. Addition of rational numbers is associative, but subtraction is not.
The document provides information about graphing and transforming quadratic functions:
- It discusses graphing quadratic functions by making tables of x- and y-values and plotting points. Examples show graphing f(x) = x^2 - 4x + 3 and g(x) = -x^2 + 6x - 8.
- Transformations of quadratic functions are described as translating the graph left/right or up/down, reflecting across an axis, or stretching/compressing vertically or horizontally. Examples demonstrate translating, reflecting, and compressing the graph of f(x) = x^2.
- The vertex form of a quadratic function f(x) = a(x-h)^2 +
22 multiplication and division of signed numbersalg1testreview
To multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product:
- Two numbers with the same sign yield a positive product
- Two numbers with opposite signs yield a negative product
In algebra, if there is no indicated operation between quantities, it represents multiplication. For example, xy means x * y. However, if there is a + or - between parentheses and a quantity, it represents combining terms rather than multiplication.
23 multiplication and division of signed numbersalg-ready-review
The document discusses rules for multiplying signed numbers. It states that to multiply two signed numbers, multiply their absolute values and use the following rules for the sign of the product: two numbers with the same sign yield a positive product; two numbers with opposite signs yield a negative product. It also discusses how multiplication is implied in algebra without an explicit operation symbol between terms.
This document discusses key concepts in the real number system including:
- Rational numbers that can be expressed as ratios of integers, and irrational numbers that cannot.
- Integers, including positive, negative and whole numbers.
- Properties of addition like commutativity, associativity and closure.
- Properties of multiplication like commutativity, associativity and distributivity.
- Absolute value and rules for performing operations on signed numbers like addition, subtraction, multiplication and division.
An algebraic expression is a combination of letters and numbers linked by operation signs: addition, subtraction, multiplication, division and exponentiation. Algebraic expressions allow us, for example, to find areas and volumes. Some examples given are the circumference of a circle (2πr), the area of a square (s=l2), and the volume of a cube (V=a3). The document then provides examples and explanations of algebraic addition, subtraction, multiplication, division, and factorization.
This document defines and explains key concepts related to real numbers and algebraic expressions. It introduces sets and subsets of real numbers like integers, rational numbers, and irrational numbers. It describes properties of real numbers including addition, multiplication, order, and absolute value. It also covers representing real numbers on a number line, algebraic expressions, and properties of negatives.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q is not equal to 0. Rational numbers are closed under addition, subtraction, and multiplication, but not division. They are commutative for addition and multiplication, but not for subtraction or division. Addition is associative for rational numbers, but subtraction is not.
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
Rational numbers are numbers that can be written as fractions p/q, where p and q are integers and q ≠ 0. Some key properties of rational numbers are:
1) Rational numbers are closed under addition, subtraction, and multiplication but not division. For any two rational numbers a and b, a + b, a - b, and a × b are rational, but a ÷ b may not be rational if b = 0.
2) Addition and multiplication of rational numbers are commutative but subtraction and division are not.
3) Addition and multiplication of rational numbers are associative but subtraction and division are not.
Death Valley is a low-lying desert region in southeastern California that was named in 1849 after 30 people attempting a shortcut to the California goldfields only 18 survived. Much of the valley is below sea level, with the lowest point being Badwater at 282 feet below sea level. Mount McKinley in Alaska, also called Denali, is the tallest mountain in North America at 20,320 feet above sea level and glows in early morning sunlight. The total distance between the top of Mount McKinley and the bottom of Death Valley is 20,602 feet.
This document discusses several mathematical concepts including sets, real numbers, inequalities, and absolute value. It provides examples and properties for each concept. Sets can be defined by listing elements or with a common characteristic. Real numbers include natural numbers, integers, rationals, and irrationals. Properties of real numbers include closure under addition and multiplication. Inequalities can be solved using the same methods as equations while maintaining the inequality sign. Absolute value gives the distance of a number from zero and has properties related to products and sums.
Rational numbers can be used to solve equations that cannot be solved using only natural numbers, whole numbers, or integers. Rational numbers are numbers that can be expressed as fractions p/q where p and q are integers and q ≠ 0. Rational numbers are closed under addition, subtraction, and multiplication but not division. Addition and multiplication of rational numbers are commutative, but subtraction and division are not. Addition is associative for rational numbers, but subtraction is not.
Integers are the set of whole numbers and their opposites, including positive and negative numbers from infinity to infinity. Each integer has an equal and opposite integer at the same distance from zero on the number line. Integers are closed under addition and multiplication, but not division, and the product of integers can be either positive or negative depending on the signs of the integers. The integers can be constructed by defining equivalence classes of pairs of natural numbers, where (a,b) represents subtracting b from a. Basic arithmetic operations of integers like addition, subtraction, multiplication and division follow predictable rules based on the signs of the integers.
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers:
1) When adding integers with the same sign, add their absolute values and use the common sign. When adding integers with opposite signs, take the absolute difference and use the sign of the larger number.
2) To subtract an integer, add its opposite and then follow the addition rules.
3) When multiplying an even number of negatives, the result is positive. With an odd number of negatives, the result is negative.
The document provides explanations and examples for adding, subtracting, multiplying, and dividing integers.
It begins by explaining the rules for adding integers with the same sign and integers with different signs, providing examples such as -6 + -2 = -8. It then explains that subtracting integers uses the rule of "adding the opposite" and provides examples like 7 - (-6) = 13.
The document also covers multiplying and dividing integers, noting that an even number of negatives yields a positive result and an odd number yields a negative result. It provides examples such as -2(-2)(-2)= 16 and 2 (-5)= -10.
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
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This document provides instructions and examples for adding, subtracting, multiplying, and dividing positive and negative numbers. It begins with placing values on a number line and defining positive and negative numbers. It then covers rules for adding and subtracting, such as keeping the sign of the larger number if the signs are different. Examples are provided for applying these rules. The document finishes with rules for multiplying and dividing, such as the product of like signs being positive and different signs being negative. More examples are given to reinforce these concepts.
This document defines and explains various sets of numbers including natural numbers, integers, rational numbers, irrational numbers, and real numbers. It provides properties and examples of operations like addition, subtraction, multiplication, and division on real numbers. Key points covered include:
- The definitions of natural numbers, integers, rational numbers, irrational numbers, and real numbers as sets.
- Properties of addition, subtraction, multiplication, and division for real numbers like commutativity, associativity, identity elements, and opposites.
- Absolute value and inequalities involving absolute value.
Similar to 3 multiplication and division of signed numbers 125s (20)
The document discusses the concept of slope as it relates to functions. It introduces function notation and defines a function's output f(x). It explains that the slope of the line connecting two points (x, f(x)) and (x+h, f(x+h)) on a function graph is given by the difference quotient formula: m = (f(x+h) - f(x))/h. An example calculates the slope of the cord between points on the graph of f(x) = x^2 - 2x + 2.
The document discusses notation and algebra of functions. It defines a function as a procedure that assigns a unique output to each valid input. Functions are typically represented by mathematical formulas using notation like f(x) = x^2 - 2x + 3, where f(x) is the name of the function, x is the input, and the formula defines the output. The input box (parentheses) holds the input to be evaluated by the formula. New functions can be formed using addition, subtraction, multiplication, and division of existing functions. Examples are provided to demonstrate simplifying expressions involving function notation and evaluating functions for given inputs.
4 graphs of equations conic sections-circlesTzenma
There are two types of x-y formulas for graphing: functions and non-functions. Functions have y as a single-valued function of x, while non-functions cannot separate y and x. Many graphs of second-degree equations (Ax2 + By2 + Cx + Dy = E) are conic sections, including circles, ellipses, parabolas, and hyperbolas. These conic section shapes result from slicing a cone at different angles. Circles consist of all points at a fixed distance from a center point.
The document discusses quadratic functions and parabolas. It begins by defining quadratic functions as functions of the form y = ax2 + bx + c, where a ≠ 0. It then provides an example of graphing the quadratic function y = x2 - 4x - 12. To do this, it finds the vertex by setting x = -b/2a, and uses the vertex and other points like the y-intercept to sketch the parabolic shape. It also discusses general properties of parabolas, such as being symmetric around a center line and having a highest/lowest point called the vertex that sits on this line.
The document discusses first degree (linear) functions. It explains that most real-world mathematical functions can be composed of formulas from three groups: algebraic, trigonometric, and exponential-log. Linear functions of the form f(x)=mx+b are especially important, where m is the slope and b is the y-intercept. The graphs of equations of the form Ax+By=C are straight lines. The slope formula for calculating the slope between two points (x1,y1) and (x2,y2) on a line is given as m=(y2-y1)/(x2-x1).
The document discusses the basic language of functions. A function assigns each input exactly one output. Functions can be defined through written instructions, tables, or mathematical formulas. The domain is the set of all inputs, and the range is the set of all outputs. Functions are widely used in mathematics to model real-world relationships.
The document discusses rational equations word problems involving multiplication-division operations and rate-time-distance problems. It provides an example of people sharing a taxi cost and forms a rational equation to determine the number of people. It also shows how to set up rate, time, and distance relationships using a table for problems involving hiking a trail with different rates of travel for the outward and return journeys.
The document discusses using rational equations to solve word problems involving costs shared among groups of people. It provides an example where a taxi costs $20 to rent for a group of x people, with the cost shared equally. If one person leaves the group, the remaining people each pay $1 more. Setting up the cost equations and subtracting them allows solving for x as 5, the number of original people in the group. A table is shown to organize the calculations for different inputs.
The document discusses ratios and proportions. It defines a ratio as two related quantities stated side by side, and gives an example of a 3:4 ratio of eggs to flour in a recipe. It explains how to write ratios as fractions and set up proportion equations. Proportions are equal ratios, like 3:4 being proportional to 6:8. The document shows how to solve proportion equations by cross-multiplying to obtain a regular equation that can then be solved for the unknown value.
The document discusses methods for simplifying complex fractions. A complex fraction is a fraction with fractions in the numerator or denominator. The first method is to combine the numerator and denominator into single fractions using cross multiplication. The second method is to multiply the lowest common denominator of all terms to both the numerator and denominator. Examples are provided to demonstrate both methods.
The document discusses techniques for combining fractions with opposite denominators. It explains that we can multiply the numerator and denominator by -1 to change the denominator to its opposite. It provides examples of switching fractions to their opposite denominators and combining fractions with opposite denominators by first switching one denominator so they are the same. It also discusses an alternative approach of pulling out a "-" from the denominator and passing it to the numerator when switching denominators, ensuring the leading term is positive for polynomial denominators.
The document discusses addition and subtraction of rational expressions. It states that fractions with the same denominator can be directly added or subtracted, while those with different denominators must first be converted to have a common denominator. The document provides an addition/subtraction rule and examples demonstrating how to perform these operations on rational expressions, including converting fractions to equivalent forms with a specified common denominator.
The document discusses the least common multiple (LCM) and provides examples to illustrate the concept. It describes two methods for finding the LCM - the searching method and the construction method. The searching method involves finding the smallest number that is a multiple of all the given numbers. The construction method builds the minimum that covers all requirements by taking just enough of each specification. An example demonstrates taking the maximum number of years required for each subject across different college requirements.
3 multiplication and division of rational expressions xTzenma
The document discusses multiplication and division of rational expressions. It presents the multiplication rule for rational expressions, which is that the product of two rational expressions is equal to the product of the numerators divided by the product of the denominators. It then gives examples of simplifying products and quotients of rational expressions by factoring and canceling like terms.
The document discusses terms, factors, and cancellation in mathematics expressions. It provides examples of identifying terms and factors in expressions, and using common factors to simplify fractions. Key points include:
- A mathematics expression contains one or more quantities called terms.
- A quantity multiplied to other quantities is a factor.
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The document discusses rational expressions, which are expressions of the form P/Q, where P and Q are polynomials. Polynomials are expressions involving powers of variables with numerical coefficients. Rational expressions include polynomials as a special case where P is viewed as P/1. They can be written in expanded or factored form. The factored form is useful for solving equations, determining the domain of a rational expression, evaluating inputs, and determining the sign of outputs. The domain excludes values that make the denominator equal to 0.
The document provides examples of how to translate word problems into mathematical equations using variables. It introduces using a system of two equations to solve problems involving two unknown quantities, labeled as x and y. An example word problem is provided where a rope is cut into two pieces, and the lengths of the pieces are defined using the variables x and y. The equations are set up and solved to find the length of each piece. The document also discusses organizing multiple sets of data into tables to solve word problems involving multiple entities.
The document provides an example of solving a system of linear equations using the substitution method. It begins with the system 2x + y = 7 and x + y = 5. It solves the second equation for x in terms of y, getting x = 5 - y. This expression for x is then substituted into the first equation, giving 10 - 2y + y = 7, which can be solved to find the value of y, and then substituted back into the original equation to find the value of x. The solution is presented as (2, 3). The document then provides two additional examples demonstrating how to set up and solve systems of equations using the substitution method.
The document discusses systems of linear equations. It provides examples to illustrate that we need as many equations as unknowns to solve for the unknown variables. For a system with two unknowns, we need two equations; for three unknowns, we need three equations. The document also gives examples of setting up and solving systems of linear equations to find unknown costs given information about total costs.
The document discusses equations of lines. It separates lines into two cases - horizontal and vertical lines which have a slope of 0 or undefined, and their equations are y=c or x=c; and tilted lines, whose equations can be found using the point-slope formula y-y1=m(x-x1) where m is the slope and (x1,y1) is a point on the line. It provides examples of finding equations of lines given their characteristics like slope and intercept points.
Thinking of getting a dog? Be aware that breeds like Pit Bulls, Rottweilers, and German Shepherds can be loyal and dangerous. Proper training and socialization are crucial to preventing aggressive behaviors. Ensure safety by understanding their needs and always supervising interactions. Stay safe, and enjoy your furry friends!
This presentation was provided by Steph Pollock of The American Psychological Association’s Journals Program, and Damita Snow, of The American Society of Civil Engineers (ASCE), for the initial session of NISO's 2024 Training Series "DEIA in the Scholarly Landscape." Session One: 'Setting Expectations: a DEIA Primer,' was held June 6, 2024.
This slide is special for master students (MIBS & MIFB) in UUM. Also useful for readers who are interested in the topic of contemporary Islamic banking.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
How to Fix the Import Error in the Odoo 17Celine George
An import error occurs when a program fails to import a module or library, disrupting its execution. In languages like Python, this issue arises when the specified module cannot be found or accessed, hindering the program's functionality. Resolving import errors is crucial for maintaining smooth software operation and uninterrupted development processes.
it describes the bony anatomy including the femoral head , acetabulum, labrum . also discusses the capsule , ligaments . muscle that act on the hip joint and the range of motion are outlined. factors affecting hip joint stability and weight transmission through the joint are summarized.
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Macroeconomics- Movie Location
This will be used as part of your Personal Professional Portfolio once graded.
Objective:
Prepare a presentation or a paper using research, basic comparative analysis, data organization and application of economic information. You will make an informed assessment of an economic climate outside of the United States to accomplish an entertainment industry objective.
How to Add Chatter in the odoo 17 ERP ModuleCeline George
In Odoo, the chatter is like a chat tool that helps you work together on records. You can leave notes and track things, making it easier to talk with your team and partners. Inside chatter, all communication history, activity, and changes will be displayed.
A workshop hosted by the South African Journal of Science aimed at postgraduate students and early career researchers with little or no experience in writing and publishing journal articles.
3. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
Multiplication and Division of Signed Numbers
4. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
Multiplication and Division of Signed Numbers
5. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Multiplication and Division of Signed Numbers
6. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Multiplication and Division of Signed Numbers
Two numbers with the same sign multiplied yield positive
products.
7. Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Multiplication and Division of Signed Numbers
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
8. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4)
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
9. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
10. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4)
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
11. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
12. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
In algebra, multiplication operations are indicated in many ways.
13. Multiplication and Division of Signed Numbers
Example A.
a. 5 * (4) = –5 * (–4) = 20
b. –5 * (4) = 5 * (–4) = –20
In algebra, multiplication operations are indicated in many ways.
We use the following rules to identify multiplication operations.
Rule for Multiplication of Signed Numbers
To multiple two signed numbers, we multiply their absolute
values and use the following rules for the sign of the product.
+ * + = – * – = + ;
+ * – = – * + = – ;
Two numbers with the same sign multiplied yield a positive
product.
Two numbers with opposite signs multiplied yield a negative
product.
14. Multiplication and Division of Signed Numbers
● If there is no operation indicated between two quantities,
the operation between them is multiplication.
15. Multiplication and Division of Signed Numbers
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
16. Multiplication and Division of Signed Numbers
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
17. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
18. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15,
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
19. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
20. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25,
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
21. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
22. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
To multiply many signed numbers together,
we always determine the sign of the product first,
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
23. Multiplication and Division of Signed Numbers
However, if there is a “+” or “–” sign between the ( ) and a
quantity, then the operation is to combine.
Hence 3(+5) = (+5)3 =15, but 3 + (5) = (3) + 5 = 8,
and –5(–5) = (–5)(–5) = 25, but (–5) – 5 = –5 – (5) = –10.
● If there is no operation indicated between a set of ( ) and
a quantity, the operation between them is multiplication.
Hence x(a + b) = x * (a + b ) and (a + b)x = (a + b) * x.
To multiply many signed numbers together,
we always determine the sign of the product first:
the sign of the product is determined by the Even–Odd Rules,
then multiply just the (absolute values of the) numbers.
● If there is no operation indicated between two quantities,
the operation between them is multiplication, so xy means x * y.
● If there is no operation indicated between two sets of ( )’s,
the operation between them is multiplication.
Hence (x + y)(a + b) = (x + y) * (a + b)
24. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
Multiplication and Division of Signed Numbers
25. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Multiplication and Division of Signed Numbers
26. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1)
Multiplication and Division of Signed Numbers
27. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1)
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
28. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
4 came from 1*2*2*1 (just the numbers)
29. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
30. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2)
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
31. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2)
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
four negative numbers, so the product is positive
32. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2) = 16
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
four negative numbers, so the product is positive
33. Even-Odd Rule for the Sign of a Product
• If there are even number of negative numbers in the
multiplication, the product is positive.
• If there are odd number of negative numbers in the
multiplication, the product is negative.
Example B.
a. –1(–2 ) 2 (–1) = – 4
b. (–2)4 = (–2 )(–2)(–2)(–2) = 16
Fact: A quantity raised to an even power is always positive
i.e. xeven is always positive (except 0).
Multiplication and Division of Signed Numbers
three negative numbers, so the product is negative
four negative numbers, so the product is positive
35. Rule for the Sign of a Quotient
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
36. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
37. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
38. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
39. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
40. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
41. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
42. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4)
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
43. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
44. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
c.
(–6)2
=
–4
45. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
c.
(–6)2
=
36
–4–4
46. Rule for the Sign of a Quotient
Division of signed numbers follows the same sign-rules for
multiplications.
Two numbers with the same sign divided yield a positive
quotient.
Two numbers with opposite signs divided yield a negative
quotient.
Multiplication and Division of Signed Numbers
Example C.
a.
b . –20 / 4 = 20 / (–4) = –5
In algebra, a ÷ b is written as a/b or .
a
b
+
+
=
–
– = +
+
+
=
–
–=
–
20
4
= –20
–4
= 5
c.
(–6)2
=
36
–4 =
–4 –9
47. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
48. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
49. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
50. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
five negative signs
so the product is negative
51. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
4(6)(3)
2(5)(12)
five negative signs
so the product is negative
simplify just the numbers
52. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
4(6)(3)
2(5)(12)
= –
3
5
five negative signs
so the product is negative
simplify just the numbers
53. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
five negative signs
so the product is negative
simplify just the numbers
4(6)(3)
2(5)(12)
= –
3
5
Various form of the Even–Odd Rule extend to algebra and
geometry. It’s the basis of many decisions and conclusions in
mathematics problems.
54. The Even–Odd Rule applies to more length * and / operations
problems.
Multiplication and Division of Signed Numbers
Example D. Simplify.
(– 4)6(–1)(–3)
(–2)(–5)12
= –
five negative signs
so the product is negative
simplify just the numbers
4(6)(3)
2(5)(12)
= –
3
5
Various form of the Even–Odd Rule extend to algebra and
geometry. It’s the basis of many decisions and conclusions in
mathematics problems.
The following is an example of the two types of graphs there
are due to this Even–Odd Rule. (Don’t worry about how they
are produced.)
55. The Even Power Graphs vs. Odd Power Graphs of y = xN
Multiplication and Division of Signed Numbers
56. Make sure that you interpret the operations correctly.
Exercise A. Calculate the following expressions.
1. 3 – 3 2. 3(–3) 3. (3) – 3 4. (–3) – 3
5. –3(–3) 6. –(–3)(–3) 7. (–3) – (–3) 8. –(–3) – (–3)
B.Multiply. Determine the sign first.
9. 2(–3) 10. (–2)(–3) 11. (–1)(–2)(–3)
12. 2(–2)(–3) 13. (–2)(–2)(–2) 14. (–2)(–2)(–2)(–2)
15. (–1)(–2)(–2)(–2)(–2) 16. 2(–1)(3)(–1)(–2)
17. 12
–3
18. –12
–3
19. –24
–8
21. (2)(–6)
–8
C. Simplify. Determine the sign and cancel first.
20. 24
–12
22. (–18)(–6)
–9
23. (–9)(6)
(12)(–3)
24. (15)(–4)
(–8)(–10)
25. (–12)(–9)
(– 27)(15)
26. (–2)(–6)(–1)
(2)(–3)(–2)
27. 3(–5)(–4)
(–2)(–1)(–2)
28. (–2)(3)(–4)5(–6)
(–3)(4)(–5)6(–7)
Multiplication and Division of Signed Numbers