The document discusses order of operations and using parentheses to indicate the order in which calculations should be performed. It provides examples of calculating the total value of bills using multiplication and addition. Rules for order of operations are given as: 1) perform operations within parentheses first, 2) do multiplication and division from left to right, 3) do addition and subtraction from left to right. Following the proper order of operations is important for obtaining the correct solution.
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, exponents, and grouping symbols. The order of operations is: 1) operations within grouping symbols from innermost to outermost, 2) multiplication and division from left to right, 3) addition and subtraction from left to right. Examples are worked through step-by-step to demonstrate applying the order of operations rules.
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, grouping symbols and exponents. The key steps are to perform operations within grouping symbols from the innermost out, then multiplication and division from left to right, followed by addition and subtraction from left to right. Setting clear rules for order of operations ensures the correct solution is obtained.
The document discusses order of operations and provides examples to illustrate how to correctly evaluate mathematical expressions involving multiple operations. It establishes that the order of operations is: 1) operations within grouping symbols from innermost to outermost, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Examples with step-by-step workings demonstrate applying this order of operations to evaluate expressions involving grouping symbols, multiplication, division, addition and subtraction.
Binary search: illustrated step-by-step walk throughYoshi Watanabe
A step-by-step illustration of Binary Search to help you walk through a series of operations. Illustration is accompanied by actual code with bold line indicating the current operation.
https://github.com/yoshiwatanabe/Algorithms/blob/master/Finding/BinarySearch.cs
Merge sort: illustrated step-by-step walk throughYoshi Watanabe
A step-by-step illustration of Merge sort to help you walk through a series of operations. Illustration is accompanied by actual code with bold line indicating the current operation.
The document shows a series of arithmetic operations with integers. Each line shows the step-by-step working out of the operations, moving from the original expressions on the left to the final solutions on the right. A variety of operations are used, including addition, subtraction, multiplication, division and use of brackets.
This document contains an activity on introductory R commands and operations using basic statistical functions. It includes examples of adding variables, performing calculations, creating graphs and loading built-in datasets. For one activity, commute times are entered and organized using R commands. Standard deviations, means and medians are calculated for price data. Probabilities are found for standard normal distributions.
The document discusses approximate string matching and the k-mismatch problem. It presents an algorithm that uses a state bit-vector s to efficiently find all occurrences of a pattern p in a string x that are within k mismatches. The algorithm constructs s by iterating through x and updating s based on a pre-calculated table t, and reports a match when s[|p|] ≤ k.
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, exponents, and grouping symbols. The order of operations is: 1) operations within grouping symbols from innermost to outermost, 2) multiplication and division from left to right, 3) addition and subtraction from left to right. Examples are worked through step-by-step to demonstrate applying the order of operations rules.
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of evaluating expressions involving multiplication, division, addition, subtraction, grouping symbols and exponents. The key steps are to perform operations within grouping symbols from the innermost out, then multiplication and division from left to right, followed by addition and subtraction from left to right. Setting clear rules for order of operations ensures the correct solution is obtained.
The document discusses order of operations and provides examples to illustrate how to correctly evaluate mathematical expressions involving multiple operations. It establishes that the order of operations is: 1) operations within grouping symbols from innermost to outermost, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Examples with step-by-step workings demonstrate applying this order of operations to evaluate expressions involving grouping symbols, multiplication, division, addition and subtraction.
Binary search: illustrated step-by-step walk throughYoshi Watanabe
A step-by-step illustration of Binary Search to help you walk through a series of operations. Illustration is accompanied by actual code with bold line indicating the current operation.
https://github.com/yoshiwatanabe/Algorithms/blob/master/Finding/BinarySearch.cs
Merge sort: illustrated step-by-step walk throughYoshi Watanabe
A step-by-step illustration of Merge sort to help you walk through a series of operations. Illustration is accompanied by actual code with bold line indicating the current operation.
The document shows a series of arithmetic operations with integers. Each line shows the step-by-step working out of the operations, moving from the original expressions on the left to the final solutions on the right. A variety of operations are used, including addition, subtraction, multiplication, division and use of brackets.
This document contains an activity on introductory R commands and operations using basic statistical functions. It includes examples of adding variables, performing calculations, creating graphs and loading built-in datasets. For one activity, commute times are entered and organized using R commands. Standard deviations, means and medians are calculated for price data. Probabilities are found for standard normal distributions.
The document discusses approximate string matching and the k-mismatch problem. It presents an algorithm that uses a state bit-vector s to efficiently find all occurrences of a pattern p in a string x that are within k mismatches. The algorithm constructs s by iterating through x and updating s based on a pre-calculated table t, and reports a match when s[|p|] ≤ k.
This document summarizes the rules for multiplying integers. It provides examples of multiplying positive and negative integers and the resulting products. The four rules are:
1) Multiplying two positive integers equals a positive product.
2) Multiplying a positive and negative integer equals a negative product.
3) Multiplying a negative and positive integer equals a negative product.
4) Multiplying two negative integers equals a positive product.
1 s2 addition and subtraction of signed numbersmath123a
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
This document provides an introduction to quadratic equations including:
- The definition of a quadratic equation as an equation of the form ax^2 + bx + c = 0 where a ≠ 0.
- Methods for solving quadratic equations including factorization, completing the square, and the quadratic formula.
- Examples of solving quadratic equations by each method and deriving the quadratic formula.
- Applications of quadratic equations such as finding the time it takes a ball to hit the ground after being thrown up and determining the width of metal needed to cut a frame with a given area.
1. The document provides information about exercises 51-88 on finding the definite integral to calculate the area between a curve and the x-axis over an interval [a,b].
2. Exercises 55-62 involve sketching graphs of functions and finding their average values over given intervals.
3. Exercises 63-70 involve using the integral definition as a limit of Riemann sums to evaluate definite integrals.
This document provides examples of solving mathematical expressions by following the proper order of operations. It demonstrates working through expressions step-by-step, starting with operations in parentheses and then moving from left to right with exponents, multiplication, division, addition and subtraction. A few sample expressions are provided along with the full work shown to arrive at the solution.
Prefix Sum Algorithm | Prefix Sum Array Implementation | EP2Kanahaiya Gupta
Prefix sum algorithm is mainly used for range query and the complexity of prefix sum algorithm is O(n).
This video explains the working of prefix sum algorithm.
This is the second part of the video and please watch the first part (why you must learn prefix sum algorithm) before watching this.
✅ Why you must learn prefix sum algorithm part one link : https://youtu.be/scD312I7kkE
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#JAVAAID #HackerRankSolutions #HackerRankTutorials #implementation #prefixsum #HackerRank #JavaAidTutorials #Programming #DataStructures #algorithms #coding #competitiveprogramming #JavaAidTutorials #Java #codinginterview #problemsolving #KanahaiyaGupta #hackerrankchallenges
1. The document discusses the history and concepts of integrals in mathematics.
2. Key figures mentioned include Bernhard Riemann, who developed the formal definition of integrals, and Henri Lebesgue, who developed the theory of integration.
3. Integral substitution and partial integration are introduced as methods for evaluating integrals. Examples are provided to demonstrate these methods.
BSC_Computer Science_Discrete Mathematics_Unit-IRai University
This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
- The nth derivative of a function y with respect to x is denoted as d^n y/dx^n.
- Standard formulas are given for finding the nth derivative of functions such as x^m, e^ax, a^x, 1/(ax+b), (ax+b)^m, log(ax+b), sin(ax+b), and cos(ax+b).
- Examples demonstrate calculating specific high-order derivatives such as the 10th derivative of x
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of calculating the total value of different combinations of bills. It explains that operations inside parentheses should be performed first, followed by multiplication and division from left to right, and then addition and subtraction from left to right. This established order of operations ensures the correct solution is obtained. The document also includes an example problem set for readers to practice applying the proper order of operations without performing steps that are excluded based on the established rules.
The document discusses order of operations and exponents. It explains that when evaluating arithmetic expressions, operations should be performed in a specific order: 1) operations within grouping symbols, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Exponents are defined as the number of times a base is used as a factor, with the exponent written as a superscript. Examples are provided to demonstrate applying order of operations correctly.
This document discusses adding and subtracting integers. It provides the following rules:
1) When adding integers with the same sign, add their values and use the same sign for the sum.
2) When adding integers with opposite signs, change them to positive, subtract the smaller from the larger, and use the sign of the larger number for the sum.
3) When subtracting integers, add the opposite integer.
It then provides examples of applying these rules and an independent practice section with problems to solve.
The document provides information about order of operations in math. It explains that order of operations is important to get the correct answer when a math problem contains multiple operations. It presents the mnemonic "PEMDAS" (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) as the standard order of operations. Several examples of applying order of operations to evaluate expressions are shown. The document is intended to teach students the proper order for solving expressions with multiple operations.
1) Completing the square allows quadratic expressions to be written in the form x + a^2 + b. This is useful for solving quadratic equations and finding the turning point of parabolas.
2) To solve the equation x^2 + 4x - 7 = 0 by completing the square:
a) Write the expression in the form x + a^2 + b as x + 2^2 - 11
b) Set this equal to 0 and solve for x, obtaining the solutions x = -2 ± √11.
3) Completing the square and writing quadratic expressions in the form x + a^2 + b allows them to be more easily solved and for properties like the
The document contains information about rational numbers including integers, fractions, and decimals. It provides examples of adding and subtracting rational numbers on a number line. Key points include:
- Rational numbers include integers, fractions, and decimals.
- Zero is a whole number but not a positive integer.
- Examples are given of comparing rational numbers and performing addition and subtraction on a number line.
- Properties of addition like commutativity and inverses are illustrated.
1) The document discusses properties of real numbers including integers, rational numbers, decimals, and fractions. It covers the four fundamental operations on integers - addition, subtraction, multiplication, and division.
2) Key properties of integer addition and subtraction are discussed, including closure, commutativity, associativity, and additive identity. Addition is commutative and associative, while subtraction is not commutative or associative.
3) Examples are provided to illustrate performing the four operations on integers and evaluating expressions involving integers. Rules for multiplying and dividing positive and negative integers are also explained.
The document discusses rules for evaluating sums and products of positive and negative integers:
1) When adding positive integers, add the magnitudes and keep the positive sign. When adding negative integers, add the magnitudes and keep the negative sign.
2) When adding a positive and negative integer, subtract the magnitudes and keep the sign of the integer with the largest magnitude.
3) When multiplying integers, the sign of the product is positive if there is an even number of negative factors and negative if there is an odd number of negative factors.
The document discusses integers and rational numbers including:
- All integers are rational numbers because they can be written as fractions.
- Negative numbers are expressed with a negative sign such as -77 feet below sea level.
- When multiplying integers, a positive result occurs with an even number of negatives and a negative result with an odd number.
- Coordinates are plotted on a coordinate plane and basic operations are performed with integers and rational numbers.
This document provides an introduction to integers through five parts:
Part I defines key integer vocabulary like positive and negative numbers. It discusses integer properties like opposites and compares/orders integers on number lines. Real world applications like temperature, sea level, and money are explored.
Part II covers integer addition rules - signs the same means keep the sign, signs different means subtract the numbers and keep the larger absolute value sign. Number lines demonstrate adding integers visually.
Part III explains that subtracting a negative number is the same as adding a positive number through changing operation and number signs. More examples solidify this rule.
Part IV proves this subtraction rule is true by using the same checking method as regular subtraction equations
This document provides an animated demonstration of the order of operations in mathematics (BIDMAS). It shows examples of calculations with different arrangements of addition, subtraction, multiplication and division and whether the results are correct according to the standard order of operations or not. The standard order is brackets, indices, division, multiplication, addition and subtraction from left to right. Several practice questions are provided for students to calculate following this order of operations.
This document provides an animated demonstration of the order of operations in mathematics (BIDMAS). It shows examples of calculations with different arrangements of addition, subtraction, multiplication and division and whether the results are correct according to the standard order of operations or not. The standard order is brackets, indices, division, multiplication, addition and subtraction from left to right. Several practice questions are provided for students to calculate following this order of operations.
The document discusses the order of operations in mathematics. It states that brackets, exponents, multiplication, division, addition and subtraction should be followed in that order when evaluating expressions. It provides examples such as 2 + 3 x 4 = 14 by first multiplying 3 x 4 before adding. The document also discusses order of operations with fractions, noting that fractions act as brackets. It provides practice problems and solutions to reinforce these concepts.
This document summarizes the rules for multiplying integers. It provides examples of multiplying positive and negative integers and the resulting products. The four rules are:
1) Multiplying two positive integers equals a positive product.
2) Multiplying a positive and negative integer equals a negative product.
3) Multiplying a negative and positive integer equals a negative product.
4) Multiplying two negative integers equals a positive product.
1 s2 addition and subtraction of signed numbersmath123a
The document discusses addition and subtraction of signed numbers. It states that adding signed numbers involves removing parentheses and combining the numbers. Examples are provided to demonstrate this process. For subtraction, the concept of opposite numbers is introduced, where the opposite of a positive number x is -x, and the opposite of a negative number -x is x. The process of finding opposites is demonstrated using examples.
This document provides an introduction to quadratic equations including:
- The definition of a quadratic equation as an equation of the form ax^2 + bx + c = 0 where a ≠ 0.
- Methods for solving quadratic equations including factorization, completing the square, and the quadratic formula.
- Examples of solving quadratic equations by each method and deriving the quadratic formula.
- Applications of quadratic equations such as finding the time it takes a ball to hit the ground after being thrown up and determining the width of metal needed to cut a frame with a given area.
1. The document provides information about exercises 51-88 on finding the definite integral to calculate the area between a curve and the x-axis over an interval [a,b].
2. Exercises 55-62 involve sketching graphs of functions and finding their average values over given intervals.
3. Exercises 63-70 involve using the integral definition as a limit of Riemann sums to evaluate definite integrals.
This document provides examples of solving mathematical expressions by following the proper order of operations. It demonstrates working through expressions step-by-step, starting with operations in parentheses and then moving from left to right with exponents, multiplication, division, addition and subtraction. A few sample expressions are provided along with the full work shown to arrive at the solution.
Prefix Sum Algorithm | Prefix Sum Array Implementation | EP2Kanahaiya Gupta
Prefix sum algorithm is mainly used for range query and the complexity of prefix sum algorithm is O(n).
This video explains the working of prefix sum algorithm.
This is the second part of the video and please watch the first part (why you must learn prefix sum algorithm) before watching this.
✅ Why you must learn prefix sum algorithm part one link : https://youtu.be/scD312I7kkE
Subscribe for more and hit the bell icon to get video updates:
https://www.youtube.com/channel/UCx1hbK753l3WhwXP5r93eYA?sub_confirmation=1
Like us on Facebook: https://www.facebook.com/HackerRankSolutionTutorials
Share this video with a YouTuber friend: https://youtu.be/pVS3yhlzrlQ
✚ Join our community ►
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Telegram link: https://t.me/hackerranksolutions
✅ Recommended playlists ►
All hackerrank solutions: https://www.youtube.com/watch?v=oz_yowFTrgs&list=PLSIpQf0NbcCltzNFrOJkQ4J4AAjW3TSmA
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#JAVAAID #HackerRankSolutions #HackerRankTutorials #implementation #prefixsum #HackerRank #JavaAidTutorials #Programming #DataStructures #algorithms #coding #competitiveprogramming #JavaAidTutorials #Java #codinginterview #problemsolving #KanahaiyaGupta #hackerrankchallenges
1. The document discusses the history and concepts of integrals in mathematics.
2. Key figures mentioned include Bernhard Riemann, who developed the formal definition of integrals, and Henri Lebesgue, who developed the theory of integration.
3. Integral substitution and partial integration are introduced as methods for evaluating integrals. Examples are provided to demonstrate these methods.
BSC_Computer Science_Discrete Mathematics_Unit-IRai University
This document discusses successive differentiation and provides examples of finding the nth derivative of common functions such as polynomials, exponentials, logarithms, trigonometric functions, and rational functions. Some key points covered include:
- The nth derivative of a function y with respect to x is denoted as d^n y/dx^n.
- Standard formulas are given for finding the nth derivative of functions such as x^m, e^ax, a^x, 1/(ax+b), (ax+b)^m, log(ax+b), sin(ax+b), and cos(ax+b).
- Examples demonstrate calculating specific high-order derivatives such as the 10th derivative of x
The document discusses order of operations and how to correctly evaluate mathematical expressions. It provides examples of calculating the total value of different combinations of bills. It explains that operations inside parentheses should be performed first, followed by multiplication and division from left to right, and then addition and subtraction from left to right. This established order of operations ensures the correct solution is obtained. The document also includes an example problem set for readers to practice applying the proper order of operations without performing steps that are excluded based on the established rules.
The document discusses order of operations and exponents. It explains that when evaluating arithmetic expressions, operations should be performed in a specific order: 1) operations within grouping symbols, 2) multiplication and division from left to right, and 3) addition and subtraction from left to right. Exponents are defined as the number of times a base is used as a factor, with the exponent written as a superscript. Examples are provided to demonstrate applying order of operations correctly.
This document discusses adding and subtracting integers. It provides the following rules:
1) When adding integers with the same sign, add their values and use the same sign for the sum.
2) When adding integers with opposite signs, change them to positive, subtract the smaller from the larger, and use the sign of the larger number for the sum.
3) When subtracting integers, add the opposite integer.
It then provides examples of applying these rules and an independent practice section with problems to solve.
The document provides information about order of operations in math. It explains that order of operations is important to get the correct answer when a math problem contains multiple operations. It presents the mnemonic "PEMDAS" (Parentheses, Exponents, Multiplication, Division, Addition, Subtraction) as the standard order of operations. Several examples of applying order of operations to evaluate expressions are shown. The document is intended to teach students the proper order for solving expressions with multiple operations.
1) Completing the square allows quadratic expressions to be written in the form x + a^2 + b. This is useful for solving quadratic equations and finding the turning point of parabolas.
2) To solve the equation x^2 + 4x - 7 = 0 by completing the square:
a) Write the expression in the form x + a^2 + b as x + 2^2 - 11
b) Set this equal to 0 and solve for x, obtaining the solutions x = -2 ± √11.
3) Completing the square and writing quadratic expressions in the form x + a^2 + b allows them to be more easily solved and for properties like the
The document contains information about rational numbers including integers, fractions, and decimals. It provides examples of adding and subtracting rational numbers on a number line. Key points include:
- Rational numbers include integers, fractions, and decimals.
- Zero is a whole number but not a positive integer.
- Examples are given of comparing rational numbers and performing addition and subtraction on a number line.
- Properties of addition like commutativity and inverses are illustrated.
1) The document discusses properties of real numbers including integers, rational numbers, decimals, and fractions. It covers the four fundamental operations on integers - addition, subtraction, multiplication, and division.
2) Key properties of integer addition and subtraction are discussed, including closure, commutativity, associativity, and additive identity. Addition is commutative and associative, while subtraction is not commutative or associative.
3) Examples are provided to illustrate performing the four operations on integers and evaluating expressions involving integers. Rules for multiplying and dividing positive and negative integers are also explained.
The document discusses rules for evaluating sums and products of positive and negative integers:
1) When adding positive integers, add the magnitudes and keep the positive sign. When adding negative integers, add the magnitudes and keep the negative sign.
2) When adding a positive and negative integer, subtract the magnitudes and keep the sign of the integer with the largest magnitude.
3) When multiplying integers, the sign of the product is positive if there is an even number of negative factors and negative if there is an odd number of negative factors.
The document discusses integers and rational numbers including:
- All integers are rational numbers because they can be written as fractions.
- Negative numbers are expressed with a negative sign such as -77 feet below sea level.
- When multiplying integers, a positive result occurs with an even number of negatives and a negative result with an odd number.
- Coordinates are plotted on a coordinate plane and basic operations are performed with integers and rational numbers.
This document provides an introduction to integers through five parts:
Part I defines key integer vocabulary like positive and negative numbers. It discusses integer properties like opposites and compares/orders integers on number lines. Real world applications like temperature, sea level, and money are explored.
Part II covers integer addition rules - signs the same means keep the sign, signs different means subtract the numbers and keep the larger absolute value sign. Number lines demonstrate adding integers visually.
Part III explains that subtracting a negative number is the same as adding a positive number through changing operation and number signs. More examples solidify this rule.
Part IV proves this subtraction rule is true by using the same checking method as regular subtraction equations
This document provides an animated demonstration of the order of operations in mathematics (BIDMAS). It shows examples of calculations with different arrangements of addition, subtraction, multiplication and division and whether the results are correct according to the standard order of operations or not. The standard order is brackets, indices, division, multiplication, addition and subtraction from left to right. Several practice questions are provided for students to calculate following this order of operations.
This document provides an animated demonstration of the order of operations in mathematics (BIDMAS). It shows examples of calculations with different arrangements of addition, subtraction, multiplication and division and whether the results are correct according to the standard order of operations or not. The standard order is brackets, indices, division, multiplication, addition and subtraction from left to right. Several practice questions are provided for students to calculate following this order of operations.
The document discusses the order of operations in mathematics. It states that brackets, exponents, multiplication, division, addition and subtraction should be followed in that order when evaluating expressions. It provides examples such as 2 + 3 x 4 = 14 by first multiplying 3 x 4 before adding. The document also discusses order of operations with fractions, noting that fractions act as brackets. It provides practice problems and solutions to reinforce these concepts.
1. The document discusses rules and principles for working with negative numbers and algebraic expressions involving negative numbers.
2. Key ideas include defining negative multiplication, such as -3 × 2, as -6; establishing that the order of factors does not matter in negative multiplication, similar to positive numbers; and simplifying algebraic expressions by using properties such as x - (y - z) = x - y + z.
3. General principles are stated for adding and subtracting negative numbers, and techniques are described for simplifying algebraic expressions involving negative terms.
This document provides an overview of operations with integers including:
- Defining integers as positive and negative whole numbers including 0
- Ordering and comparing integers
- Absolute value and opposite of integers
- Adding and subtracting integers using number lines and sign rules
- Multiplying and dividing integers and the sign of the result
- Properties like distributive property for operations with integers
The document provides instructions on subtracting integers. It explains:
1) To subtract integers, transform the subtraction into addition by keeping the first number and changing the second number's sign.
2) Examples are provided of subtracting integers with different signs and the same sign.
3) A multi-step word problem is worked out as an example of subtracting integers.
The document provides an overview of basic mathematics concepts like integers, addition and subtraction rules, percentages, and operations with positive and negative numbers. It also gives examples of calculating discounts, taxes, and percentages of quantities. The document appears to be teaching materials for a mathematics class covering fundamental numerical topics.
The document contains examples of solving problems involving addition, subtraction, multiplication and division of integers. Some key examples include:
- Finding the balance in an account after a deposit and withdrawal.
- Calculating distance traveled east and west and the final position from a starting point.
- Verifying properties like commutativity, associativity and distributivity for operations on integers.
- Solving word problems involving gains, losses, temperature changes represented as positive and negative integers.
This document provides information on factoring polynomials with a common monomial factor. It defines a common monomial factor as a number, variable, or combination that appears in each term. It outlines the steps to factor polynomials with this common factor: find the greatest common factor (GCF), divide the polynomial by the GCF, and express the factorization. Examples are provided to demonstrate this process. Students are then assigned practice problems to factor polynomials using this method and given a deadline to submit their work.
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.
- To simplify a fraction, factorize it and cancel any common factors between the numerator and denominator.
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.
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
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.
June 3, 2024 Anti-Semitism Letter Sent to MIT President Kornbluth and MIT Cor...Levi Shapiro
Letter from the Congress of the United States regarding Anti-Semitism sent June 3rd to MIT President Sally Kornbluth, MIT Corp Chair, Mark Gorenberg
Dear Dr. Kornbluth and Mr. Gorenberg,
The US House of Representatives is deeply concerned by ongoing and pervasive acts of antisemitic
harassment and intimidation at the Massachusetts Institute of Technology (MIT). Failing to act decisively to ensure a safe learning environment for all students would be a grave dereliction of your responsibilities as President of MIT and Chair of the MIT Corporation.
This Congress will not stand idly by and allow an environment hostile to Jewish students to persist. The House believes that your institution is in violation of Title VI of the Civil Rights Act, and the inability or
unwillingness to rectify this violation through action requires accountability.
Postsecondary education is a unique opportunity for students to learn and have their ideas and beliefs challenged. However, universities receiving hundreds of millions of federal funds annually have denied
students that opportunity and have been hijacked to become venues for the promotion of terrorism, antisemitic harassment and intimidation, unlawful encampments, and in some cases, assaults and riots.
The House of Representatives will not countenance the use of federal funds to indoctrinate students into hateful, antisemitic, anti-American supporters of terrorism. Investigations into campus antisemitism by the Committee on Education and the Workforce and the Committee on Ways and Means have been expanded into a Congress-wide probe across all relevant jurisdictions to address this national crisis. The undersigned Committees will conduct oversight into the use of federal funds at MIT and its learning environment under authorities granted to each Committee.
• The Committee on Education and the Workforce has been investigating your institution since December 7, 2023. The Committee has broad jurisdiction over postsecondary education, including its compliance with Title VI of the Civil Rights Act, campus safety concerns over disruptions to the learning environment, and the awarding of federal student aid under the Higher Education Act.
• The Committee on Oversight and Accountability is investigating the sources of funding and other support flowing to groups espousing pro-Hamas propaganda and engaged in antisemitic harassment and intimidation of students. The Committee on Oversight and Accountability is the principal oversight committee of the US House of Representatives and has broad authority to investigate “any matter” at “any time” under House Rule X.
• The Committee on Ways and Means has been investigating several universities since November 15, 2023, when the Committee held a hearing entitled From Ivory Towers to Dark Corners: Investigating the Nexus Between Antisemitism, Tax-Exempt Universities, and Terror Financing. The Committee followed the hearing with letters to those institutions on January 10, 202
This presentation includes basic of PCOS their pathology and treatment and also Ayurveda correlation of PCOS and Ayurvedic line of treatment mentioned in classics.
Exploiting Artificial Intelligence for Empowering Researchers and Faculty, In...Dr. Vinod Kumar Kanvaria
Exploiting Artificial Intelligence for Empowering Researchers and Faculty,
International FDP on Fundamentals of Research in Social Sciences
at Integral University, Lucknow, 06.06.2024
By Dr. Vinod Kumar Kanvaria
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!
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.
MATATAG CURRICULUM: ASSESSING THE READINESS OF ELEM. PUBLIC SCHOOL TEACHERS I...NelTorrente
In this research, it concludes that while the readiness of teachers in Caloocan City to implement the MATATAG Curriculum is generally positive, targeted efforts in professional development, resource distribution, support networks, and comprehensive preparation can address the existing gaps and ensure successful curriculum implementation.
2. If we have two $5-bills and two $10-bills,
Order of Operations
3. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars.
Order of Operations
4. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
Order of Operations
5. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
Order of Operations
6. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills,
Order of Operations
7. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
Order of Operations
8. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first,
Order of Operations
9. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
10. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
This motivates us to set the rules for the order of operations.
11. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
Order of Operations (excluding raising power)
This motivates us to set the rules for the order of operations.
12. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
Order of Operations (excluding raising power)
Given an arithmetic expression, we perform the operations in
the following order .
This motivates us to set the rules for the order of operations.
13. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
Order of Operations (excluding raising power)
Given an arithmetic expression, we perform the operations in
the following order .
1st . Do the operations within grouping symbols, starting with
the innermost grouping symbol.
This motivates us to set the rules for the order of operations.
14. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
Order of Operations (excluding raising power)
Given an arithmetic expression, we perform the operations in
the following order .
1st . Do the operations within grouping symbols, starting with
the innermost grouping symbol.
2nd. Do multiplications and divisions (from left to right).
This motivates us to set the rules for the order of operations.
15. If we have two $5-bills and two $10-bills, we have the total of
2(5) + 2(10) = 30 dollars. To get the correct answer 30,
we multiply the 2 and the 5 and multiply the 2 and the10 first,
then we add the products 10 and 20.
If I have three $10-bills and you have four $10-bills, we have
3 + 4 = 7 $10-bills, and we have a total of (3 + 4)10 = 70 $.
In this case, we group the 3 + 4 in the “( )” to indicate that we
are to add them first, then multiply the sum to 10.
Order of Operations
Order of Operations (excluding raising power)
Given an arithmetic expression, we perform the operations in
the following order .
1st . Do the operations within grouping symbols, starting with
the innermost grouping symbol.
2nd. Do multiplications and divisions (from left to right).
3rd. Do additions and subtractions (from left to right).
This motivates us to set the rules for the order of operations.
20. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
Order of Operations
b. 4 + 3(5 + 2)
21. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
Order of Operations
b. 4 + 3(5 + 2)
22. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
23. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
24. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
25. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
26. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
27. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
28. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
29. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
= 9 – 2[7 – 21]
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
30. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
= 9 – 2[7 – 21]
= 9 – 2[ –14 ]
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
31. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
= 9 – 2[7 – 21]
= 9 – 2[ –14 ]
= 9 + 28
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
32. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
= 9 – 2[7 – 21]
= 9 – 2[ –14 ]
= 9 + 28
= 37
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
33. Example A.
a. 4(–8) + 3(5)
= –32 + 15
= –17
c. 9 – 2[7 – 3(6 + 1)]
= 9 – 2[7 – 3(7)]
= 9 – 2[7 – 21]
= 9 – 2[ –14 ]
= 9 + 28
= 37
(Don’t perform “4 + 3” or “9 – 2” in the above problems!!)
Order of Operations
b. 4 + 3(5 + 2)
= 4 + 3(7)
= 4 + 21
= 25
34. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
35. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
0 + x + x + x .. + x = Nx
N copies added
36. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
So 0 = 0x
0 + x + x + x .. + x = Nx
N copies added
0 + x = 1x
0 + x + x = 2x
0 + x + x + x = 3x
0 + x + x + x + x = 4x
.
37. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
So 0 = 0x
0 + x + x + x .. + x = Nx
N copies added
0 + x = 1x
0 + x + x = 2x
0 + x + x + x = 3x
The number N of added copies is called the coefficient.
0 + x + x + x + x = 4x
.
38. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
So 0 = 0x
0 + x + x + x .. + x = Nx
N copies added
0 + x = 1x
0 + x + x = 2x
0 + x + x + x = 3x
The number N of added copies is called the coefficient.
So the coefficient of 3x is 3.
0 + x + x + x + x = 4x
.
39. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
So 0 = 0x
0 + x + x + x .. + x = Nx
N copies added
0 + x = 1x
0 + x + x = 2x
0 + x + x + x = 3x
The number N of added copies is called the coefficient.
So the coefficient of 3x is 3.
Similarly ab + ab + ab + ab is 4ab, with coefficient 4,
0 + x + x + x + x = 4x
.
40. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
So 0 = 0x
0 + x + x + x .. + x = Nx
N copies added
0 + x = 1x
0 + x + x = 2x
0 + x + x + x = 3x
The number N of added copies is called the coefficient.
So the coefficient of 3x is 3.
Similarly ab + ab + ab + ab is 4ab, with coefficient 4,
and that 3(x + y) is (x + y) + (x + y) + (x + y).
0 + x + x + x + x = 4x
.
41. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Coefficients
Starting with 0, adding N copies
of x’s to 0 is written as Nx:
So 0 = 0x
0 + x + x + x .. + x = Nx
N copies added
0 + x = 1x
0 + x + x = 2x
0 + x + x + x = 3x
The number N of added copies is called the coefficient.
So the coefficient of 3x is 3.
Similarly ab + ab + ab + ab is 4ab, with coefficient 4,
and that 3(x + y) is (x + y) + (x + y) + (x + y). Note that 0x = 0.
0 + x + x + x + x = 4x
.
42. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
43. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
1* x * x * x…* x as xN
N copies of x’s
44. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
1 * x * x * x * x = x4
.
So
45. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
1 * x * x * x * x = x4
.
So 1 = x0 (x ≠ 0)
46. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
The number of multiplied copies N of xN is called the exponent.
1 * x * x * x * x = x4
.
So 1 = x0 (x ≠ 0)
47. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
The number of multiplied copies N of xN is called the exponent.
So the exponent of x3 is 3.
1 * x * x * x * x = x4
.
So 1 = x0 (x ≠ 0)
48. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
The number of multiplied copies N of xN is called the exponent.
So the exponent of x3 is 3.
An exponent applies only to the quantity directly under it.
1 * x * x * x * x = x4
.
So 1 = x0 (x ≠ 0)
49. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
So 1 = x0 (x ≠ 0)
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
The number of multiplied copies N of xN is called the exponent.
So the exponent of x3 is 3.
An exponent applies only to the quantity directly under it.
So ab3 = a*b*b*b and that (ab)3 = ab*ab*ab.
1 * x * x * x * x = x4
.
50. Exercise: Don’t do the part that you shouldn’t do!
1. 6 + 3(3 + 1) 2. 10 – 4(2 – 4)
3. 5 + 2[3 + 2(1 + 2)] 4. 5 – 2[3 + 2(5 – 9)]
Order of Operations
Ans: a. 18 b. 18 c. 23 4. 15
Exponents
Starting with 1, multiplying N copies
of x’s to 1 is written as xN.
So 1 = x0 (x ≠ 0)
1* x * x * x…* x as xN
N copies of x’s
1 * x = x1
1 * x * x = x2
1 * x * x * x = x3
The number of multiplied copies N of xN is called the exponent.
So the exponent of x3 is 3.
An exponent applies only to the quantity directly under it.
So ab3 = a*b*b*b and that (ab)3 = ab*ab*ab. Note that x0 =1.
1 * x * x * x * x = x4
.
52. Order of Operations
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
53. Order of Operations
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3)
54. Order of Operations
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
55. Order of Operations
b. Expand – 32
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
56. Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
57. Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3)
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
58. Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
59. c. Expand (3*2)2 and simplify the answer.
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
60. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
61. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Hence(3*2)2 is (3*2)(3*2)
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
62. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Hence(3*2)2 is (3*2)(3*2) = (6)(6) = 36
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
63. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Hence(3*2)2 is (3*2)(3*2) = (6)(6) = 36
d. Expand 3*22 and simplify the answer.
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
64. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Hence(3*2)2 is (3*2)(3*2) = (6)(6) = 36
d. Expand 3*22 and simplify the answer.
The base for the 2nd power is 2.
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
65. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Hence(3*2)2 is (3*2)(3*2) = (6)(6) = 36
d. Expand 3*22 and simplify the answer.
The base for the 2nd power is 2.
Hence 3*22 means 3*2*2
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
66. c. Expand (3*2)2 and simplify the answer.
The base for the 2nd power is (3*2).
Hence(3*2)2 is (3*2)(3*2) = (6)(6) = 36
d. Expand 3*22 and simplify the answer.
The base for the 2nd power is 2.
Hence 3*22 means 3*2*2 = 12
Order of Operations
b. Expand – 32
The base of the 2nd power is 3.
Hence – 32 means – (3*3) = – 9
Example B. (Exponential Notation)
a. Expand (–3)2 and simplify the answer.
The base is (–3).
Hence (–3)2 is (–3)(–3) = 9
69. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
70. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
71. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
72. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
From part b above, we see that the power is to be carried out
before multiplication.
73. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
From part b above, we see that the power is to be carried out
before multiplication. Below is the complete rules of order of
operations.
74. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
Order of Operations (PEMDAS)
From part b above, we see that the power is to be carried out
before multiplication. Below is the complete rules of order of
operations.
75. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
From part b above, we see that the power is to be carried out
before multiplication. Below is the complete rules of order of
operations.
76. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation
From part b above, we see that the power is to be carried out
before multiplication. Below is the complete rules of order of
operations.
77. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation
3rd. (Multiplication and Division) Do multiplications and
divisions in order from left to right.
From part b above, we see that the power is to be carried out
before multiplication. Below is the complete rules of order of
operations.
78. Order of Operations
e. Expand (–3y)3 and simplify the answer.
(–3y)3
= (–3y)(–3y)(–3y) (the product of three negatives number is negative)
= –(3)(3)(3)(y)(y)(y)
= –27y3
Order of Operations (PEMDAS)
1st. (Parenthesis) Do the operations within grouping symbols,
starting with the innermost one.
2nd. (Exponents) Do the exponentiation
3rd. (Multiplication and Division) Do multiplications and
divisions in order from left to right.
4th. (Addition and Subtraction) Do additions and
subtractions in order from left to right.
From part b above, we see that the power is to be carried out
before multiplication. Below is the complete rules of order of
operations.
80. Example C. Order of Operations
a. 52 – 32
= 25 – 9
Order of Operations
81. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
Order of Operations
82. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
Order of Operations
83. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
Order of Operations
84. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
Order of Operations
85. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
Order of Operations
86. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
Order of Operations
87. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
Order of Operations
88. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
= 18
Order of Operations
89. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
= 18
d. –32 – 5(3 – 6)2
Order of Operations
90. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
= 18
d. –32 – 5(3 – 6)2
= –9 – 5(–3)2
Order of Operations
91. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
= 18
d. –32 – 5(3 – 6)2
= –9 – 5(–3)2
= –9 – 5(9)
Order of Operations
92. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
= 18
d. –32 – 5(3 – 6)2
= –9 – 5(–3)2
= –9 – 5(9)
= –9 – 45
Order of Operations
93. Example C. Order of Operations
a. 52 – 32
= 25 – 9
= 16
b. – (5 – 3)2
= – (2)2
= – 4
c. –2*32 + (2*3)2
= –2*9 + (6)2
= –18 + 36
= 18
d. –32 – 5(3 – 6)2
= –9 – 5(–3)2
= –9 – 5(9)
= –9 – 45 = –54
Order of Operations
94. Make sure that you interpret the operations correctly.
Exercise A. Calculate the following expressions.
Order of Operations
7. 1 + 2(3) 8. 4 – 5(6) 9. 7 – 8(–9)
1. 3(–3) 2. (3) – 3 3. 3 – 3(3) 4. 3(–3) + 3
5. +3(–3)(+3) 6. 3 + (–3)(+3)
B.Make sure that you don’t do the ± too early.
10. 1 + 2(3 – 4) 11. 5 – 6(7 – 8) 12. (4 – 3)2 + 1
13. [1 – 2(3 – 4)] – 2 14. 6 + [5 + 6(7 – 8)](+5)
15. 1 + 2[1 – 2(3 + 4)] 16. 5 – 6[5 – 6(7 – 8)]
17. 1 – 2[1 – 2(3 – 4)] 18. 5 + 6[5 + 6(7 – 8)]
19. (1 + 2)[1 – 2(3 + 4)] 20. (5 – 6)[5 – 6(7 – 8)]
C.Make sure that you apply the powers to the correct bases.
23. (–2)2 and –22 24 (–2)3 and –23 25. (–2)4 and –24
26. (–2)5 and –25 27. 2*32 28. (2*3)2
21. 1 – 2(–3)(–4) 22. (–5)(–6) – (–7)(–8)