8 4 Negative And Zero Exponents
โขDownload as PPT, PDFโข
4 likesโข1,399 views
Negative exponents do not represent negative numbers. Any number to the zero power equals one, which can be proven using the division of powers property. Negative exponents can be simplified by raising the number under the exponent to a positive power equal to the negative exponent.
Report
Share
Report
Share
Recommended
Chapter4.3
The document discusses rules for operations involving exponents, including:
- Multiplying powers with the same base by adding the exponents
- Dividing powers with the same base by subtracting the exponents
- Raising a power to a power by multiplying the exponents
Examples are provided to illustrate multiplying, dividing, and raising powers to powers according to these rules. A quiz with problems applying the rules concludes the document.
Ch 1 review stations (Classwork)
The document provides instructions for completing a chapter review activity involving multiple stations. Students will work through assignments at different stations covering topics like patterns and expressions, algebraic expressions, solving equations and inequalities, and absolute value equations and inequalities. When time is called, students will move to the next station listed at the bottom of the assignment.
Position of objects and translation
Estas son las hojas de trabajo realizadas durante el desarrollo de la temรกtica de posiciรณn de objetos.
Sept. 15, 2014
This document provides information and instructions for an upcoming algebra test. It discusses Khan Academy assignments that are due by Sunday at 7:00 pm. It also details the format and content of the test, which will be tomorrow and will include topics like order of operations, integers, simplifying expressions, translating expressions and equations, and identifying parts of algebraic expressions and equations. The document provides sample problems for a warm-up review covering these test topics.
Transformation numerical
The document describes scaling a square object by 4 units in the x-direction and 3 units in the y-direction while keeping one point fixed. It provides the step-by-step process to derive the composite matrix for this scaling transformation, which involves translating the object, scaling it, and translating it back. To check if a point remains fixed after the transformation, its homogeneous coordinates are multiplied with the composite matrix. This proves the given point is fixed, while another arbitrary point is not fixed after the transformation.
Section 4.3 And 4.4 Plus Warm Ups
Section 4.3 and 4.4, plus Warm Up from Sections 4.1 and 4.2. Greatest Common Factors and Simplifying Fractions.
La potenciaciรณn
1. The document provides definitions and examples for exponents and powers. It explains the rules for addition, subtraction, multiplication and division of exponents.
2. Students are assigned problems applying the rules of exponents, including expressing powers with radicals, fractional exponents, and positive exponents.
3. The assignment is due on April 23rd and involves simplifying expressions using the rules of exponents, as well as expressing powers in different but equivalent forms.
Fun with radicals
This document provides an overview of working with radicals that include variables. It discusses simplifying radicals with variables by splitting exponents into perfect squares. It also covers multiplying and dividing radicals by separating numbers and variables, then simplifying numbers and combining like terms. Sample assessment problems are included to practice these skills.
Recommended
Chapter4.3
The document discusses rules for operations involving exponents, including:
- Multiplying powers with the same base by adding the exponents
- Dividing powers with the same base by subtracting the exponents
- Raising a power to a power by multiplying the exponents
Examples are provided to illustrate multiplying, dividing, and raising powers to powers according to these rules. A quiz with problems applying the rules concludes the document.
Ch 1 review stations (Classwork)
The document provides instructions for completing a chapter review activity involving multiple stations. Students will work through assignments at different stations covering topics like patterns and expressions, algebraic expressions, solving equations and inequalities, and absolute value equations and inequalities. When time is called, students will move to the next station listed at the bottom of the assignment.
Position of objects and translation
Estas son las hojas de trabajo realizadas durante el desarrollo de la temรกtica de posiciรณn de objetos.
Sept. 15, 2014
This document provides information and instructions for an upcoming algebra test. It discusses Khan Academy assignments that are due by Sunday at 7:00 pm. It also details the format and content of the test, which will be tomorrow and will include topics like order of operations, integers, simplifying expressions, translating expressions and equations, and identifying parts of algebraic expressions and equations. The document provides sample problems for a warm-up review covering these test topics.
Transformation numerical
The document describes scaling a square object by 4 units in the x-direction and 3 units in the y-direction while keeping one point fixed. It provides the step-by-step process to derive the composite matrix for this scaling transformation, which involves translating the object, scaling it, and translating it back. To check if a point remains fixed after the transformation, its homogeneous coordinates are multiplied with the composite matrix. This proves the given point is fixed, while another arbitrary point is not fixed after the transformation.
Section 4.3 And 4.4 Plus Warm Ups
Section 4.3 and 4.4, plus Warm Up from Sections 4.1 and 4.2. Greatest Common Factors and Simplifying Fractions.
La potenciaciรณn
1. The document provides definitions and examples for exponents and powers. It explains the rules for addition, subtraction, multiplication and division of exponents.
2. Students are assigned problems applying the rules of exponents, including expressing powers with radicals, fractional exponents, and positive exponents.
3. The assignment is due on April 23rd and involves simplifying expressions using the rules of exponents, as well as expressing powers in different but equivalent forms.
Fun with radicals
This document provides an overview of working with radicals that include variables. It discusses simplifying radicals with variables by splitting exponents into perfect squares. It also covers multiplying and dividing radicals by separating numbers and variables, then simplifying numbers and combining like terms. Sample assessment problems are included to practice these skills.
11X1 T10 07 sum and product of roots (2010)
If ฮฑ and ฮฒ are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
Algebra 2 unit 10.4
This document provides instruction on using the Law of Sines to solve triangles. It begins with examples of using the Law of Sines to find missing side lengths or angle measures when two angles and a side, or two sides and an angle are known. It also covers cases where an ambiguous triangle could result from given side-side-angle information. The document demonstrates solving for the area of triangles using trigonometric functions. It concludes with practice problems applying the Law of Sines to find missing measurements and the number of possible triangles based on given side lengths and an angle measure.
Math12 lesson2
The document discusses trigonometric ratios and functions for right triangles. It defines the six trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of the sides of a right triangle with angle A. Reciprocal functions are also defined. The Pythagorean theorem relates the sides of a right triangle. Functions of complementary angles are equivalent to cofunctions of the original angle. Specific values are given for the trig functions of 45, 30, and 60 degrees. Examples are provided to calculate trig functions from right triangles.
50 solving equations by factoring
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
Algebra 2 unit 10.5
The document discusses the Law of Cosines and Heron's Formula. It begins by explaining that the Law of Cosines can be used to find side lengths and angle measures of a triangle when given side-angle-side or side-side-side information. It then provides examples of using the Law of Cosines to solve for missing values of triangles. It also introduces Heron's Formula, which can be used to find the area of a triangle based on its side lengths. Examples are given of using both the Law of Cosines and Heron's Formula to solve multi-step word problems involving triangles.
Mathematics 9 Lesson 6: Rational Exponents
This powerpoint presentation discusses or talks about the topic or lesson: Rational Exponents. It also discusses and explains the rules, concepts, steps and examples of Rational Exponents.
Solving Equations by Factoring
The document discusses using factoring to solve equations arising from physics problems involving height, rate, and time. It provides examples of solving equations derived from the physics formula h=rt-16t^2 for height, rate, or time given two of the three variables. Factoring allows setting each factor equal to zero to find the possible solutions, which are then checked in the original equation.
Simplifying expressions with negative and zero exponents
The document provides steps for simplifying expressions with negative exponents:
1) Move the base with the negative exponent to the top of the expression, changing the exponent's sign from negative to positive.
2) Remember that anything raised to the zero power is equal to 1.
3) Examples are provided to demonstrate simplifying expressions using these steps.
Quadratic Formula
The document discusses solving quadratic equations using the quadratic formula. It provides the general form of a quadratic equation as ax2 + bx + c = 0 and introduces the quadratic formula as x = (-b ยฑ โ(b2 - 4ac))/2a. Several examples of using the quadratic formula to solve quadratic equations are shown step-by-step with explanations of each step. The examples illustrate determining the a, b, and c coefficients and performing the calculations to find the roots of the equations.
AA Section 2-9
The document contains examples and explanations of combined and joint variation. It begins with four warm-up problems that demonstrate combined variation, where the dependent variable varies directly with one independent variable and inversely with another. It explains that to find the constant of variation k, one variable is held constant while solving for k. The document then provides an example of finding k and solving a combined variation word problem. It also defines and provides an example of joint variation, where the dependent variable varies directly with two or more independent variables. It explains finding k is done in the same way as in combined variation problems.
Solving quadratics with square roots
This document provides information on solving quadratic equations by finding square roots. It discusses how to solve equations by taking the square root of each side, and that the square root of a positive number has two answers: one positive and one negative. Properties of square roots are presented. Examples are provided on simplifying expressions involving square roots using the product and quotient properties. The document also covers rationalizing the denominator to remove radicals in fractions and solving quadratic equations by taking square roots and isolating the variable. Methods for solving word problems involving falling objects using the quadratic formula are demonstrated.
Joint variation
This document discusses different types of joint variations between quantities. A joint variation occurs when more than two quantities vary together. The document provides examples of direct and inverse joint variations, explaining that if z varies directly as x and y, with k as a constant, then z = kxy. Similarly, if z varies inversely as x and directly as y, with k as a constant, then z = k/xy. Joint variations describe the mathematical relationships between three or more varying quantities.
PC 6.1 Notes
1. The document discusses solving oblique triangles using the Law of Sines. It provides examples of solving triangles given: (1) two angles and a side (ASA case) and (2) two sides and a non-included angle (SSA case).
2. For the ASA case, it shows how to find the missing angle and sides using the given information. For the SSA case, it notes that SSA is not a unique case and there may be 0, 1, or 2 possible triangles depending on the side lengths.
3. It provides an example of solving a triangle with ASA given and finds the missing angle and sides. It also provides an example of an SSA case where
Quadratic inequalities
A quadratic inequality is an inequality involving a quadratic expression, such as ax^2 + bx + c < 0. To solve a quadratic inequality, we first find the solutions to the corresponding equation (set the inequality equal to 0) and then test values on either side of those solutions in the original inequality to determine the solutions to the inequality. The solutions to the inequality will be all values of the variable that satisfy the given relationship.
Mathematics 9 Lesson 4-A: Direct Variation
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
Trigonometry: Solving Triangles
The document summarizes two laws for solving triangles - the Law of Cosines and the Law of Sines. It also discusses an ambiguous case of the Law of Sines. The Law of Cosines can be used to find a missing side given two sides and the included angle, or to find a missing angle given all three sides. The Law of Sines can be used to find a missing side or angle given two angles and the side opposite one of them. The ambiguous case allows finding a missing angle given two sides and the angle opposite one of them.
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the Square
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using Completing the Square. It also discusses the steps in solving quadratic equations using the method of Completing the Square.
Mathematics 9 Lesson 4-C: Joint and Combined Variation
This powerpoint presentation discusses or talks about the topic or lesson Joint and Combined Variations. It also discusses and explains the rules, concepts, steps and examples of Joint and Combined Variation
Law Of Cosines Presentation
The document explains the Law of Cosines, which is used to find the length of a side or measure of an angle in a triangle when certain other information is known. It provides two examples - one using side, angle, side information to find a missing side length, and another using side, side, side information to find a missing angle measure. Both examples show setting up and solving the Law of Cosines formula to find the unknown value.
7.7 Solving Radical Equations
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
8 3quotients Of Powers
The document discusses dividing monomials with the same base but different exponents by using the quotient of powers property. Specifically:
1) For monomials with the same base, subtract the exponent of the denominator from the exponent of the numerator.
2) This property allows simplifying expressions like 4/5 = 4^3, where 5-2 = 3.
3) Examples are provided to demonstrate dividing monomials with variables as well as negative exponents.
Section 4.7 And 4.8 Plus Warm Ups
This document discusses rules for multiplying and dividing expressions with exponents. It states that to multiply terms with the same base, add the exponents. To divide terms with the same nonzero base, subtract the exponents. If subtracting exponents results in zero, the expression equals one. Negative exponents can be written as reciprocals with positive exponents. Examples are provided to illustrate simplifying expressions using these rules.
More Related Content
Viewers also liked
11X1 T10 07 sum and product of roots (2010)
If ฮฑ and ฮฒ are the roots of the quadratic equation ax2 + bx + c = 0, then:
(1) The sum of the roots is -b/a;
(2) The product of the roots is c/a;
(3) The roots can be used to form other quadratic equations or to solve for properties of the original quadratic equation.
Algebra 2 unit 10.4
This document provides instruction on using the Law of Sines to solve triangles. It begins with examples of using the Law of Sines to find missing side lengths or angle measures when two angles and a side, or two sides and an angle are known. It also covers cases where an ambiguous triangle could result from given side-side-angle information. The document demonstrates solving for the area of triangles using trigonometric functions. It concludes with practice problems applying the Law of Sines to find missing measurements and the number of possible triangles based on given side lengths and an angle measure.
Math12 lesson2
The document discusses trigonometric ratios and functions for right triangles. It defines the six trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of the sides of a right triangle with angle A. Reciprocal functions are also defined. The Pythagorean theorem relates the sides of a right triangle. Functions of complementary angles are equivalent to cofunctions of the original angle. Specific values are given for the trig functions of 45, 30, and 60 degrees. Examples are provided to calculate trig functions from right triangles.
50 solving equations by factoring
The document discusses solving linear equations by factoring using an example of determining the number of pizzas ordered. It formulates the problem as the equation 3x + 10 = 34 where x is the number of pizzas. It solves the equation by subtracting 10 from both sides, dividing both sides by 3, and determining that x = 8 pizzas were ordered. The document then provides more details on linear equations, their structure, and their general solution method.
Algebra 2 unit 10.5
The document discusses the Law of Cosines and Heron's Formula. It begins by explaining that the Law of Cosines can be used to find side lengths and angle measures of a triangle when given side-angle-side or side-side-side information. It then provides examples of using the Law of Cosines to solve for missing values of triangles. It also introduces Heron's Formula, which can be used to find the area of a triangle based on its side lengths. Examples are given of using both the Law of Cosines and Heron's Formula to solve multi-step word problems involving triangles.
Mathematics 9 Lesson 6: Rational Exponents
This powerpoint presentation discusses or talks about the topic or lesson: Rational Exponents. It also discusses and explains the rules, concepts, steps and examples of Rational Exponents.
Solving Equations by Factoring
The document discusses using factoring to solve equations arising from physics problems involving height, rate, and time. It provides examples of solving equations derived from the physics formula h=rt-16t^2 for height, rate, or time given two of the three variables. Factoring allows setting each factor equal to zero to find the possible solutions, which are then checked in the original equation.
Simplifying expressions with negative and zero exponents
The document provides steps for simplifying expressions with negative exponents:
1) Move the base with the negative exponent to the top of the expression, changing the exponent's sign from negative to positive.
2) Remember that anything raised to the zero power is equal to 1.
3) Examples are provided to demonstrate simplifying expressions using these steps.
Quadratic Formula
The document discusses solving quadratic equations using the quadratic formula. It provides the general form of a quadratic equation as ax2 + bx + c = 0 and introduces the quadratic formula as x = (-b ยฑ โ(b2 - 4ac))/2a. Several examples of using the quadratic formula to solve quadratic equations are shown step-by-step with explanations of each step. The examples illustrate determining the a, b, and c coefficients and performing the calculations to find the roots of the equations.
AA Section 2-9
The document contains examples and explanations of combined and joint variation. It begins with four warm-up problems that demonstrate combined variation, where the dependent variable varies directly with one independent variable and inversely with another. It explains that to find the constant of variation k, one variable is held constant while solving for k. The document then provides an example of finding k and solving a combined variation word problem. It also defines and provides an example of joint variation, where the dependent variable varies directly with two or more independent variables. It explains finding k is done in the same way as in combined variation problems.
Solving quadratics with square roots
This document provides information on solving quadratic equations by finding square roots. It discusses how to solve equations by taking the square root of each side, and that the square root of a positive number has two answers: one positive and one negative. Properties of square roots are presented. Examples are provided on simplifying expressions involving square roots using the product and quotient properties. The document also covers rationalizing the denominator to remove radicals in fractions and solving quadratic equations by taking square roots and isolating the variable. Methods for solving word problems involving falling objects using the quadratic formula are demonstrated.
Joint variation
This document discusses different types of joint variations between quantities. A joint variation occurs when more than two quantities vary together. The document provides examples of direct and inverse joint variations, explaining that if z varies directly as x and y, with k as a constant, then z = kxy. Similarly, if z varies inversely as x and directly as y, with k as a constant, then z = k/xy. Joint variations describe the mathematical relationships between three or more varying quantities.
PC 6.1 Notes
1. The document discusses solving oblique triangles using the Law of Sines. It provides examples of solving triangles given: (1) two angles and a side (ASA case) and (2) two sides and a non-included angle (SSA case).
2. For the ASA case, it shows how to find the missing angle and sides using the given information. For the SSA case, it notes that SSA is not a unique case and there may be 0, 1, or 2 possible triangles depending on the side lengths.
3. It provides an example of solving a triangle with ASA given and finds the missing angle and sides. It also provides an example of an SSA case where
Quadratic inequalities
A quadratic inequality is an inequality involving a quadratic expression, such as ax^2 + bx + c < 0. To solve a quadratic inequality, we first find the solutions to the corresponding equation (set the inequality equal to 0) and then test values on either side of those solutions in the original inequality to determine the solutions to the inequality. The solutions to the inequality will be all values of the variable that satisfy the given relationship.
Mathematics 9 Lesson 4-A: Direct Variation
This powerpoint presentation discusses or talks about the topic or lesson Direct Variations. It also discusses and explains the rules, concepts, steps and examples of Direct Variations.
Trigonometry: Solving Triangles
The document summarizes two laws for solving triangles - the Law of Cosines and the Law of Sines. It also discusses an ambiguous case of the Law of Sines. The Law of Cosines can be used to find a missing side given two sides and the included angle, or to find a missing angle given all three sides. The Law of Sines can be used to find a missing side or angle given two angles and the side opposite one of them. The ambiguous case allows finding a missing angle given two sides and the angle opposite one of them.
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the Square
This powerpoint presentation discusses or talks about the topic or lesson Solving Quadratic Equations using Completing the Square. It also discusses the steps in solving quadratic equations using the method of Completing the Square.
Mathematics 9 Lesson 4-C: Joint and Combined Variation
This powerpoint presentation discusses or talks about the topic or lesson Joint and Combined Variations. It also discusses and explains the rules, concepts, steps and examples of Joint and Combined Variation
Law Of Cosines Presentation
The document explains the Law of Cosines, which is used to find the length of a side or measure of an angle in a triangle when certain other information is known. It provides two examples - one using side, angle, side information to find a missing side length, and another using side, side, side information to find a missing angle measure. Both examples show setting up and solving the Law of Cosines formula to find the unknown value.
7.7 Solving Radical Equations
This document provides steps for solving radical equations:
1) Isolate the radical on one side of the equation by performing inverse operations
2) Raise both sides of the equation to a power equal to the index of the radical to eliminate the radical
3) Solve the remaining polynomial equation
It includes examples of solving simpler radical equations as well as more complex equations involving fractions and multiple radicals. Checking solutions is emphasized as extraneous solutions may occasionally occur. Graphing calculators can also help visualize and find solutions to radical equations.
Viewers also liked (20)
Simplifying expressions with negative and zero exponents
Simplifying expressions with negative and zero exponents
ย
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the Square
Mathematics 9 Lesson 1-A: Solving Quadratic Equations by Completing the Square
ย
Mathematics 9 Lesson 4-C: Joint and Combined Variation
Mathematics 9 Lesson 4-C: Joint and Combined Variation
ย
Similar to 8 4 Negative And Zero Exponents
8 3quotients Of Powers
The document discusses dividing monomials with the same base but different exponents by using the quotient of powers property. Specifically:
1) For monomials with the same base, subtract the exponent of the denominator from the exponent of the numerator.
2) This property allows simplifying expressions like 4/5 = 4^3, where 5-2 = 3.
3) Examples are provided to demonstrate dividing monomials with variables as well as negative exponents.
Section 4.7 And 4.8 Plus Warm Ups
This document discusses rules for multiplying and dividing expressions with exponents. It states that to multiply terms with the same base, add the exponents. To divide terms with the same nonzero base, subtract the exponents. If subtracting exponents results in zero, the expression equals one. Negative exponents can be written as reciprocals with positive exponents. Examples are provided to illustrate simplifying expressions using these rules.
0.6 Rational Exponents
This document discusses rational exponents. It reviews negative exponents and introduces rational exponents. Rational exponents are defined as the nth root of the base raised to the power of the numerator over the denominator. Examples are provided to evaluate expressions with rational exponents. Rules for simplifying, adding, and factoring terms with rational exponents are described.
Chapter 07
This document provides a summary of key concepts and examples from a lesson on exponents and polynomials:
- It introduces concepts like multiplying and dividing monomials, zero and negative exponents, and the degree of polynomials.
- Examples are provided to illustrate these concepts and their application in simplifying expressions and determining the degree of polynomials.
- Students are prompted to practice these skills through examples like arranging polynomial terms in ascending or descending order based on the exponents.
Chapter4.4
The document provides examples for multiplying, dividing, and raising monomials to a power. It includes step-by-step work for multiplying terms like (3a2)(4a5), dividing terms like 15m5/3m2, and raising terms to a power like (2a2b6)4. The examples are followed by a short quiz assessing these skills.
9 2power Of Power
The document discusses exponents and properties related to exponents. It introduces the power of powers property, which states that when a number is raised to a power that is then raised to another power, you multiply the exponents. For example, (3^3)^4 = 3^(3ร4) = 3^12. It also discusses how to simplify expressions with negative exponents by writing them as fractions with positive exponents.
Lesson 1.9 the set of rational numbers
The document defines rational numbers and discusses their properties. Rational numbers can be expressed as fractions with integer numerators and non-zero integer denominators. Their decimal representations either terminate or repeat. Examples of rational number expressions and their equivalent decimal and fractional forms are provided.
Chapter 1 Review
- To write fractions as decimals and decimals as fractions, it is helpful to write numbers in different ways. This allows for easier comparisons and calculations.
- Rational numbers can be written as ratios of integers, repeating decimals can be written as fractions, and terminating decimals can be written as fractions.
- Writing numbers in different forms, such as fractions and decimals, allows for easier evaluation of expressions and calculations.
5.4 dividing fractions updated
This document contains instructions and examples for dividing fractions. It explains that to divide fractions, you change the fractions to improper fractions, multiply by the reciprocal, divide out common factors, and then multiply the numerators and denominators. It provides step-by-step worked examples of dividing fractions by fractions and fractions by whole numbers. The document concludes with homework problems for students to practice dividing fractions.
1.3 Radicals and Rational Exponents
* Evaluate square roots.
* Use the product rule to simplify square roots.
* Use the quotient rule to simplify square roots.
* Add and subtract square roots.
* Rationalize denominators.
* Use rational roots.
0011 chapter iv
This document provides instruction on solving various types of equations, including quadratic equations, equations containing radicals, and equations that can be reduced to quadratic form. It includes examples of solving equations that are quadratic in form by rewriting them in standard form and substituting a variable, then solving the resulting quadratic equation. It also provides examples of solving equations containing radicals by isolating radicals and raising both sides of the equation to appropriate powers. Students are expected to learn to identify different equation types, select appropriate solution methods, and solve various quadratic and radical equations.
integers
This document is a lesson on integers from an online math course. It includes examples and explanations of key concepts:
- Integers include whole numbers and their opposites, and can represent elevations above and below sea level.
- The opposite of a number has the same distance from 0 on the number line but on the other side.
- Integers can be compared and ordered using a number line by their distance from 0.
- Absolute value is the distance of a number from 0 on the number line, and is always positive or zero. Examples are used to illustrate finding opposites, comparing, ordering, and absolute values of integers.
Chapter3.6
The document provides examples and explanations for solving linear inequalities by adding or subtracting the same number from both sides of the inequality. It demonstrates how to isolate the variable and graph the solution set. Examples include solving inequalities involving variables like x, m, r, n, and p. Practice problems ask students to solve and graph inequalities, and one word problem asks how many more letters must be folded given that at least 300 letters exist and 125 have already been folded.
Merge sort
Merge sort is a sorting algorithm that uses a divide and conquer approach. It divides an array into halves, recursively sorts the halves, and then merges the sorted halves into a single sorted array. The key steps are dividing, conquering by recursively sorting subarrays, and combining the sorted subarrays through a merging process. It has advantages of being a stable sort and having no worst-case scenario, though it requires more memory than other sorts.
Computational skills
The document discusses different types of numbers and operations involving positive and negative numbers. It explains rules for addition, subtraction, multiplication, and division of positive and negative numbers. It also covers order of operations using PEMDAS and provides examples of solving expressions using proper order. Finally, it discusses properties and rules for exponents, including adding, subtracting, multiplying, and dividing terms with the same base and combining exponents.
Msm1 fl ch11_01
The document is a math lesson on integers that includes examples and explanations of positive and negative numbers, opposites, integers, and absolute value. It provides practice problems identifying positive and negative numbers in real world contexts, graphing integers on a number line, and finding the absolute value of integers using a number line. The lesson concludes with a quiz reviewing these concepts.
just reference
This document provides an introduction to basic arithmetic concepts including the four fundamental operations of addition, subtraction, multiplication, and division. It discusses topics such as whole numbers, fractions, mixed numbers, and changing between numerical representations. Examples and exercises are provided to demonstrate key concepts like performing the four operations, reducing fractions, and converting between whole numbers and fractions. The goal is to lay the foundation for understanding modern mathematics.
Basic math
This document provides an introduction to basic arithmetic concepts including the four fundamental operations of addition, subtraction, multiplication, and division. It explains key terminology like digits, whole numbers, fractions, and decimal numbers. Examples are provided for how to perform each operation by hand including adding and subtracting with carrying and borrowing, multi-digit multiplication, and long division. The document also covers converting between whole numbers and fractions, reducing fractions, and changing between improper and mixed numbers. Practice problems with answers are included throughout to help reinforce the concepts.
basics
This document provides an introduction to basic arithmetic concepts including the four fundamental operations of addition, subtraction, multiplication, and division. It discusses topics such as whole numbers, fractions, mixed numbers, and changing between numerical representations. Examples and practice problems are provided to demonstrate key concepts like reducing fractions to lower or lowest terms and changing between whole numbers, fractions, and mixed numbers. The goal is to lay the foundation for modern mathematics by explaining the simplest and most basic forms.
upload
This document provides an introduction to basic arithmetic concepts including the four fundamental operations of addition, subtraction, multiplication, and division. It explains key terminology like digits, whole numbers, fractions, and decimal numbers. Examples are provided for how to perform each operation by hand in columns as well as how to check answers. The document also covers converting between whole numbers and fractions, reducing fractions, and changing mixed numbers to improper fractions and vice versa. Practice problems with answers are included throughout to illustrate each concept.
Similar to 8 4 Negative And Zero Exponents (20)
More from taco40
Similar Triangles
The document discusses similar triangles and how to determine if two triangles are similar. It explains that two triangles are similar if corresponding angles are congruent. It provides examples of using the Angle-Angle similarity criterion to show triangles are similar and using ratios to find a missing side of a similar triangle. The lesson covered properties of congruent and similar triangles, various similarity criteria like AA and SAS, and how to prove triangles are similar.
Lesson 8 8 A Just Started
This document discusses properties of two special right triangles: 45/45/90 triangles and 30/60/90 triangles. For 45/45/90 triangles, the hypotenuse is โ2 times the length of either leg. For 30/60/90 triangles, the longer leg is twice the length of the shorter leg. Examples are given to illustrate these properties and how to find missing side lengths.
8 8b Trig Intro
SOHCAHTOA was a wise Trigonometric Tribe chief who helped people solve problems using right triangles. He discovered the relationships between opposite, adjacent, and hypotenuse sides and angles, which are remembered through the mnemonic "SOHCAHTOA". The document then explains the definitions of opposite, adjacent, and hypotenuse sides relative to a given angle in a right triangle, and introduces the sine, cosine, and tangent ratios used in trigonometry.
8 8b Trig Intro2
This document provides examples of using trigonometric ratios (sine, cosine, tangent) to solve for unknown sides of right triangles when given angle and side measurements. Several examples are worked through, identifying opposite, adjacent, and hypotenuse sides based on a given angle and using trig functions (sine, cosine, tangent) to calculate values or unknown sides. The document demonstrates setting up and solving trigonometric ratios to find missing values in right triangles.
8 3 Similar Triangles
The document discusses similar triangles and how to determine if two triangles are similar. It explains that two triangles are similar if corresponding angles are congruent. It provides examples of using the Angle-Angle similarity criterion to show triangles are similar and using proportions to find missing sides of similar triangles. The lesson covered congruent triangles, similar triangles, determining similarity using corresponding angles, and applying similarity to find unknown lengths.
8 3similar Triangles
The document discusses similar triangles and how to determine if triangles are similar. It explains that similar triangles have congruent angles, while congruent triangles have both congruent angles and sides. It provides examples of using the Angle-Angle similarity criterion to determine if triangles are similar and setting up ratio proportions to calculate missing sides of similar triangles. The lesson objectives are to apply properties of similar triangles and prove that triangles are similar.
8 2 Triangle Sum Theorem
The document discusses several geometry theorems including:
1. The Triangle Sum Theorem which states that the sum of the interior angles of any triangle is 180 degrees.
2. The theorem that the sum of the interior angles of any quadrilateral is 360 degrees.
3. The Exterior Angle Theorem which states that an exterior angle of a triangle is equal to the sum of the two remote interior angles.
The document provides examples of using these theorems to prove other statements and find missing angle measures in geometric figures. It also assigns practice problems for students.
4 6 Probablitiy
This document reviews probability and statistics concepts from Algebra I Chapter 4.3. It discusses experimental versus theoretical probability and provides examples of calculating probability using fractions, decimals, and percentages. Sample spaces and probability trees are presented for coin flips, rolling dice, and drawing colored buttons from a bag. Calculating theoretical probability is defined as the number of favorable outcomes divided by the total number of possible outcomes.
4 5b Histograms
The document discusses histograms and how they are used to organize and summarize data. Histograms divide a continuous range of data into bins of equal size to display the frequency distribution. They show the frequency of data values within each discrete interval. The document provides an example of students measuring the density of an unknown liquid and creating a histogram to analyze the results. It outlines the steps to organize the density data values, determine the appropriate bin ranges, count the data into groups, and interpret the frequency table and histogram created.
4 1 Proportions
The document provides examples of using proportions to solve various math problems involving ratios, fractions, rates, and geometric similarities. It includes word problems about book sales, fertilizer amounts, heights and shadows, and finding missing values in proportions. The examples demonstrate setting up and solving proportions to find unknown quantities.
4[1].4central Tendencies
The document discusses employee absences and how statistics can be used to analyze absence data. It notes that 40% of illness-related absences occur on Mondays and Fridays, potentially reducing workplace effectiveness. It asks how a boss would feel about this statistic and notes that only 2/5 of absences occur at the start and end of the work week. The document also provides an overview of measures of central tendency, including mean, median, mode, and range.
4[.5a Box Whiskers
The document provides instructions for building a box and whiskers plot. It explains that box and whisker plots use the median, quartiles, minimum and maximum values of a dataset. The instructions say to line up the numbers, find the median of the top and bottom halves, and note the minimum and maximum. A number line is drawn and boxes are placed around the middle values with whiskers extending to the minimum and maximum to complete the plot. An example is provided of calculating the values for a box and whiskers plot from a dataset.
4.5a Box Whiskers
This document provides instructions for building a box and whiskers plot. It explains that you should line up and find the median of the numbers, note the minimum and maximum values, and build the plot with a box covering the middle three quartiles and whiskers extending from the box. An example number line is given that could be used as a pattern for the box and whiskers plot, along with instructions to find the minimum, maximum, median, and first and third quartiles.
4.4central Tendencies
The document discusses employee absences and productivity. It notes that 40% of illness-related absences occur on Mondays and Fridays, reducing workplace effectiveness. It asks how a boss would feel about this statistic and notes that only 2/5 of absences occur at the start and end of the work week. The document also provides an overview of measures of central tendency, including mean, median, mode, and range.
3 3two Step Equations
This document provides instructions and examples for solving two-step equations. It explains that the two steps are: 1) add or subtract on both sides, and 2) multiply or divide on both sides. It then provides examples of solving equations using these two steps, showing the process of isolating the variable and using the inverse operation. Finally, it gives some practice problems for the reader to solve two-step equations.
2 5math Laws
The document discusses various mathematical properties including:
- Commutative and associative properties of addition and multiplication which allow changing the order or grouping of terms.
- Identity properties which show that adding or multiplying a number by its identity (0 for addition, 1 for multiplication) does not change the number.
- The opposite property which involves changing all addition signs to subtraction or multiplying the entire expression by -1.
- The distributive property which allows multiplying a number by the terms inside parentheses.
- Properties of equality including reflexive, symmetric, transitive, and substitution properties which define when expressions can be said to be equal.
2 3 Subtract Integers
1) Subtracting real numbers is the same as adding the opposite (negative) number.
2) To subtract, change the operation to addition and change the sign of the number being subtracted.
3) For example, 4 - 5 would become 4 + (-5), where -5 is the opposite (negative) of 5.
2 4mult Pos By Neg
The document discusses multiplication and division of integers. It explains that:
- When multiplying integers, the sign of the product depends on the number of negative factors. An even number of negatives gives a positive product, and an odd number gives a negative product.
- Division of integers can be performed by rewriting it as multiplication using the reciprocal of the second factor.
- Fractions and their reciprocals are also discussed as ways to rewrite division of fractions as multiplication.
2 2bpos Neg Fractionss
This document discusses adding and subtracting positive and negative integers without calculators. It provides warm up problems asking students to add opposites and lists problems from the textbook for students to work on, including problems subtracting odd integers from pages 47 and 71 of the textbook.
2 2 Pos Neg Numbers
This document discusses adding positive and negative numbers. It begins with warm up questions asking students to find sums and differences without calculators. It then reviews homework answers from the previous day. The document explains how to add positive and negative numbers using colored buttons as an example. It shows -7 + 4 equals 3 negatives. In 3 sentences or less.
More from taco40 (20)
Recently uploaded
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
(๐๐๐ ๐๐๐) (๐๐๐ฌ๐ฌ๐จ๐ง ๐)-๐๐ซ๐๐ฅ๐ข๐ฆ๐ฌ
๐๐ข๐ฌ๐๐ฎ๐ฌ๐ฌ ๐ญ๐ก๐ ๐๐๐ ๐๐ฎ๐ซ๐ซ๐ข๐๐ฎ๐ฅ๐ฎ๐ฆ ๐ข๐ง ๐ญ๐ก๐ ๐๐ก๐ข๐ฅ๐ข๐ฉ๐ฉ๐ข๐ง๐๐ฌ:
- Understand the goals and objectives of the Edukasyong Pantahanan at Pangkabuhayan (EPP) curriculum, recognizing its importance in fostering practical life skills and values among students. Students will also be able to identify the key components and subjects covered, such as agriculture, home economics, industrial arts, and information and communication technology.
๐๐ฑ๐ฉ๐ฅ๐๐ข๐ง ๐ญ๐ก๐ ๐๐๐ญ๐ฎ๐ซ๐ ๐๐ง๐ ๐๐๐จ๐ฉ๐ ๐จ๐ ๐๐ง ๐๐ง๐ญ๐ซ๐๐ฉ๐ซ๐๐ง๐๐ฎ๐ซ:
-Define entrepreneurship, distinguishing it from general business activities by emphasizing its focus on innovation, risk-taking, and value creation. Students will describe the characteristics and traits of successful entrepreneurs, including their roles and responsibilities, and discuss the broader economic and social impacts of entrepreneurial activities on both local and global scales.
Wound healing PPT
This document provides an overview of wound healing, its functions, stages, mechanisms, factors affecting it, and complications.
A wound is a break in the integrity of the skin or tissues, which may be associated with disruption of the structure and function.
Healing is the bodyโs response to injury in an attempt to restore normal structure and functions.
Healing can occur in two ways: Regeneration and Repair
There are 4 phases of wound healing: hemostasis, inflammation, proliferation, and remodeling. This document also describes the mechanism of wound healing. Factors that affect healing include infection, uncontrolled diabetes, poor nutrition, age, anemia, the presence of foreign bodies, etc.
Complications of wound healing like infection, hyperpigmentation of scar, contractures, and keloid formation.
Stack Memory Organization of 8086 Microprocessor
The stack memory organization of 8086 microprocessor.
BIOLOGY NATIONAL EXAMINATION COUNCIL (NECO) 2024 PRACTICAL MANUAL.pptx
Practical manual for National Examination Council, Nigeria.
Contains guides on answering questions on the specimens provided
Temple of Asclepius in Thrace. Excavation results
The temple and the sanctuary around were dedicated to Asklepios Zmidrenus. This name has been known since 1875 when an inscription dedicated to him was discovered in Rome. The inscription is dated in 227 AD and was left by soldiers originating from the city of Philippopolis (modern Plovdiv).
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Geography as discipline
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
The History of NZ 1870-1900.
Making of a Nation.
From the NZ Wars to Liberals,
Richard Seddon, George Grey,
Social Laboratory, New Zealand,
Confiscations, Kotahitanga, Kingitanga, Parliament, Suffrage, Repudiation, Economic Change, Agriculture, Gold Mining, Timber, Flax, Sheep, Dairying,
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering
Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
3. Ketki Rangwala Dalal, Essentials of Civil Engineering, Charotar Publishing House.
4. BCP, Surveying volume 1
Mule event processing models | MuleSoft Mysore Meetup #47
Mule event processing models | MuleSoft Mysore Meetup #47
Event Link:- https://meetups.mulesoft.com/events/details/mulesoft-mysore-presents-mule-event-processing-models/
Agenda
โ What is event processing in MuleSoft?
โ Types of event processing models in Mule 4
โ Distinction between the reactive, parallel, blocking & non-blocking processing
For Upcoming Meetups Join Mysore Meetup Group - https://meetups.mulesoft.com/mysore/YouTube:- youtube.com/@mulesoftmysore
Mysore WhatsApp group:- https://chat.whatsapp.com/EhqtHtCC75vCAX7gaO842N
Speaker:-
Shivani Yasaswi - https://www.linkedin.com/in/shivaniyasaswi/
Organizers:-
Shubham Chaurasia - https://www.linkedin.com/in/shubhamchaurasia1/
Giridhar Meka - https://www.linkedin.com/in/giridharmeka
Priya Shaw - https://www.linkedin.com/in/priya-shaw
Bossa Nโ Roll Records by Ismael Vazquez.
Bossa N Roll Records presentation by Izzy Vazquez for Music Retail and Distribution class at Full Sail University
HYPERTENSION - SLIDE SHARE PRESENTATION.
IT WILL BE HELPFULL FOR THE NUSING STUDENTS
IT FOCUSED ON MEDICAL MANAGEMENT AND NURSING MANAGEMENT.
HIGHLIGHTS ON HEALTH EDUCATION.
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
The recent surge in pro-Palestine student activism has prompted significant responses from universities, ranging from negotiations and divestment commitments to increased transparency about investments in companies supporting the war on Gaza. This activism has led to the cessation of student encampments but also highlighted the substantial sacrifices made by students, including academic disruptions and personal risks. The primary drivers of these protests are poor university administration, lack of transparency, and inadequate communication between officials and students. This study examines the profound emotional, psychological, and professional impacts on students engaged in pro-Palestine protests, focusing on Generation Z's (Gen-Z) activism dynamics. This paper explores the significant sacrifices made by these students and even the professors supporting the pro-Palestine movement, with a focus on recent global movements. Through an in-depth analysis of printed and electronic media, the study examines the impacts of these sacrifices on the academic and personal lives of those involved. The paper highlights examples from various universities, demonstrating student activism's long-term and short-term effects, including disciplinary actions, social backlash, and career implications. The researchers also explore the broader implications of student sacrifices. The findings reveal that these sacrifices are driven by a profound commitment to justice and human rights, and are influenced by the increasing availability of information, peer interactions, and personal convictions. The study also discusses the broader implications of this activism, comparing it to historical precedents and assessing its potential to influence policy and public opinion. The emotional and psychological toll on student activists is significant, but their sense of purpose and community support mitigates some of these challenges. However, the researchers call for acknowledging the broader Impact of these sacrifices on the future global movement of FreePalestine.
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...PsychoTech Services
A proprietary approach developed by bringing together the best of learning theories from Psychology, design principles from the world of visualization, and pedagogical methods from over a decade of training experience, that enables you to: Learn better, faster!ย
Recently uploaded (20)
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
NEWSPAPERS - QUESTION 1 - REVISION POWERPOINT.pptx
ย
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
Philippine Edukasyong Pantahanan at Pangkabuhayan (EPP) Curriculum
ย
BIOLOGY NATIONAL EXAMINATION COUNCIL (NECO) 2024 PRACTICAL MANUAL.pptx
BIOLOGY NATIONAL EXAMINATION COUNCIL (NECO) 2024 PRACTICAL MANUAL.pptx
ย
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
Geography as a Discipline Chapter 1 __ Class 11 Geography NCERT _ Class Notes...
ย
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
Level 3 NCEA - NZ: A Nation In the Making 1872 - 1900 SML.ppt
ย
Chapter wise All Notes of First year Basic Civil Engineering.pptx
Chapter wise All Notes of First year Basic Civil Engineering.pptx
ย
Mule event processing models | MuleSoft Mysore Meetup #47
Mule event processing models | MuleSoft Mysore Meetup #47
ย
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
THE SACRIFICE HOW PRO-PALESTINE PROTESTS STUDENTS ARE SACRIFICING TO CHANGE T...
ย
Gender and Mental Health - Counselling and Family Therapy Applications and In...
Gender and Mental Health - Counselling and Family Therapy Applications and In...
ย
SWOT analysis in the project Keeping the Memory @live.pptx
SWOT analysis in the project Keeping the Memory @live.pptx
ย
8 4 Negative And Zero Exponents
- 1. Negative and Zero Exponents Algebra I Chapter 8.4
- 2. Negative Integers do NOT mean negative numbers
- 9. ย