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1. This document provides examples and explanations for writing expressions using exponents. It defines key terms like base, exponent, and power. 2. Step-by-step examples are provided for writing expressions with exponents, evaluating expressions, and applying properties of exponents. Examples include writing expressions for multiplication involving repeated factors, evaluating expressions given values for variables, and using the distributive property. 3. Exponents provide a concise way to write repeated multiplication and to evaluate large expressions. Working through examples helps understand how to use exponents to write expressions compactly and evaluate them efficiently.
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1. This document provides examples and explanations for simplifying expressions using exponents. It discusses the laws for taking the power of a power and the power of a product.
2. For the power of a power, you multiply the exponents. For the power of a product, you find the power of each factor and multiply them.
3. Several multi-step examples are worked through applying these laws to simplify expressions with variables and integers. The document also provides explanations of how rewriting numbers with exponents can make calculations easier.
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1) The document provides examples of writing expressions using exponents and evaluating those expressions. It gives the expressions 2c8, 85, and m5n3p3 as examples and evaluates the expressions 512, 216, and m5n3p3.
2) It then discusses rules for multiplying and dividing powers with the same base: to multiply powers with the same base, add the exponents, and to divide powers with the same base, subtract the exponents.
3) Several step-by-step examples demonstrate simplifying expressions using these rules to multiply and divide exponents.
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This document provides examples for writing numbers using negative exponents. It begins with examples of writing expressions like 6-3 and 4-4 using positive exponents. It then provides step-by-step examples of writing fractions with negative exponents other than -1, like writing 5-2. Further examples show simplifying expressions using the properties of exponents, like 53 • 5-5 = 53 + (-5). In the examples, writing numbers with negative exponents allows writing very small or large numbers without fractions and makes multiplying or dividing numbers with the same base easier.
Lesson 1.6 (8)
Here are the steps to write a number between 0 and 1 in scientific notation with an example:
1. Write the number in decimal form. For example, 0.000375.
2. Count the number of decimal places in the number. In this example there are 4 decimal places.
3. Move the decimal point to the left by that number of places. This gives 3.75.
4. Write the number as a coefficient and exponent of 10. Since we moved the decimal point 4 places to the left, the exponent is -4.
So, 0.000375 written in scientific notation is 3.75 × 10-4.
(8) Lesson 1.2 - Powers and Exponents
1. This document provides examples and explanations for writing expressions using exponents. It defines key terms like base, exponent, and power.
2. Step-by-step examples are provided for writing expressions with exponents, evaluating expressions, and applying properties of exponents. Examples include writing expressions for multiplication involving repeated factors, evaluating expressions given values for variables, and using the distributive property.
3. Exponents provide a compact way to write repeated multiplication and to work with large numbers that come up in areas like the metric system. Being able to rewrite numbers in exponential form and evaluate such expressions is important for simplifying calculations.
Lesson 1.7 grade 8
1) The document provides examples of writing numbers in standard form and scientific notation. It also gives examples of performing arithmetic operations such as addition, subtraction, multiplication, and division on numbers expressed in scientific notation.
2) The examples illustrate how writing very large or very small numbers in scientific notation makes it easier to perform operations with them. It also allows for estimating quantities that would otherwise be difficult to comprehend in standard form.
3) Understanding scientific notation helps to compare quantities that differ vastly in magnitude, such as populations and diameters of astronomical objects. It enables estimating ratios and relative sizes in a straightforward manner.
Lesson 1.10 grade 8
This document provides examples and explanations of concepts related to classifying and ordering different types of numbers:
1) It defines rational and irrational numbers, and provides examples of each. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
2) Exercises are included for classifying numbers as rational or irrational, comparing irrational numbers on a number line, and ordering sets of numbers from least to greatest.
3) Answering "Why is it helpful to write numbers in different ways?" the document explains that not all numbers can be written the same way, like irrational numbers that cannot be expressed as integers. Being able to classify numbers facilitates understanding and operations involving different types of numbers.
Lesson 1.8 grade 8
1. The document provides examples for evaluating expressions with scientific notation, square roots, and cube roots. It includes step-by-step workings for finding square roots of perfect squares and cube roots of perfect cubes.
2. Key concepts explained are square roots, perfect squares, radical signs, cube roots, and perfect cubes. Examples are provided to find square roots of numbers less than 100 and cube roots of small numbers.
3. Knowing how to find the exact square roots of perfect squares helps estimate the square roots of nearby non-perfect squares. Being able to write the perfect square root without the radical provides understanding and allows for computations.
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1. This document provides examples and explanations for simplifying expressions using exponents. It discusses the laws for taking the power of a power and the power of a product.
2. For the power of a power, you multiply the exponents. For the power of a product, you find the power of each factor and multiply them.
3. Several multi-step examples are worked through applying these laws to simplify expressions with variables and integers. The document also provides explanations of how rewriting numbers with exponents can make calculations easier.
Lesson 1.3 (8)
1) The document provides examples of writing expressions using exponents and evaluating those expressions. It gives the expressions 2c8, 85, and m5n3p3 as examples and evaluates the expressions 512, 216, and m5n3p3.
2) It then discusses rules for multiplying and dividing powers with the same base: to multiply powers with the same base, add the exponents, and to divide powers with the same base, subtract the exponents.
3) Several step-by-step examples demonstrate simplifying expressions using these rules to multiply and divide exponents.
Lesson 1.5 (8)
This document provides examples for writing numbers using negative exponents. It begins with examples of writing expressions like 6-3 and 4-4 using positive exponents. It then provides step-by-step examples of writing fractions with negative exponents other than -1, like writing 5-2. Further examples show simplifying expressions using the properties of exponents, like 53 • 5-5 = 53 + (-5). In the examples, writing numbers with negative exponents allows writing very small or large numbers without fractions and makes multiplying or dividing numbers with the same base easier.
Lesson 1.6 (8)
Here are the steps to write a number between 0 and 1 in scientific notation with an example:
1. Write the number in decimal form. For example, 0.000375.
2. Count the number of decimal places in the number. In this example there are 4 decimal places.
3. Move the decimal point to the left by that number of places. This gives 3.75.
4. Write the number as a coefficient and exponent of 10. Since we moved the decimal point 4 places to the left, the exponent is -4.
So, 0.000375 written in scientific notation is 3.75 × 10-4.
(8) Lesson 1.2 - Powers and Exponents
1. This document provides examples and explanations for writing expressions using exponents. It defines key terms like base, exponent, and power.
2. Step-by-step examples are provided for writing expressions with exponents, evaluating expressions, and applying properties of exponents. Examples include writing expressions for multiplication involving repeated factors, evaluating expressions given values for variables, and using the distributive property.
3. Exponents provide a compact way to write repeated multiplication and to work with large numbers that come up in areas like the metric system. Being able to rewrite numbers in exponential form and evaluate such expressions is important for simplifying calculations.
Lesson 1.7 grade 8
1) The document provides examples of writing numbers in standard form and scientific notation. It also gives examples of performing arithmetic operations such as addition, subtraction, multiplication, and division on numbers expressed in scientific notation.
2) The examples illustrate how writing very large or very small numbers in scientific notation makes it easier to perform operations with them. It also allows for estimating quantities that would otherwise be difficult to comprehend in standard form.
3) Understanding scientific notation helps to compare quantities that differ vastly in magnitude, such as populations and diameters of astronomical objects. It enables estimating ratios and relative sizes in a straightforward manner.
Lesson 1.10 grade 8
This document provides examples and explanations of concepts related to classifying and ordering different types of numbers:
1) It defines rational and irrational numbers, and provides examples of each. Rational numbers can be expressed as ratios of integers, while irrational numbers cannot.
2) Exercises are included for classifying numbers as rational or irrational, comparing irrational numbers on a number line, and ordering sets of numbers from least to greatest.
3) Answering "Why is it helpful to write numbers in different ways?" the document explains that not all numbers can be written the same way, like irrational numbers that cannot be expressed as integers. Being able to classify numbers facilitates understanding and operations involving different types of numbers.
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1. The document provides examples for evaluating expressions with scientific notation, square roots, and cube roots. It includes step-by-step workings for finding square roots of perfect squares and cube roots of perfect cubes.
2. Key concepts explained are square roots, perfect squares, radical signs, cube roots, and perfect cubes. Examples are provided to find square roots of numbers less than 100 and cube roots of small numbers.
3. Knowing how to find the exact square roots of perfect squares helps estimate the square roots of nearby non-perfect squares. Being able to write the perfect square root without the radical provides understanding and allows for computations.
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- Raising a power to a power by multiplying the exponents
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FINAL EXPONENT.ppt
This document provides information about exponents and expressions containing exponents. It defines key terms like base, exponent, power, and factor form vs standard form. It includes examples of evaluating expressions using exponents by finding the product of the base repeated by the exponent. The examples demonstrate simplifying expressions by following the order of operations and evaluating exponents before other operations.
Chapter2.3
The document provides examples and explanations for performing arithmetic operations with rational numbers including integers, fractions, and decimals. It demonstrates how to add, subtract, multiply, and divide rational numbers, as well as how to evaluate expressions involving rational numbers. The examples are aligned with California state math standards for performing operations with rational numbers.
Math Gr4 Ch5
This chapter discusses multiplication and division facts. It includes 10 lessons: relating multiplication and division; algebra properties; facts through 5; problem solving skills; facts through 10; multiplying with 11 and 12; problem solving investigations; multiplying three numbers; factors and multiples; and prime and composite numbers. The lessons provide examples and practice with multiplication and division concepts and skills.
Teacher Lecture
The document discusses exponents and order of operations. It defines exponents as indicating how many times the base is used as a factor. It provides examples of evaluating exponential expressions by writing repeated factors with exponents. Rules for exponents include: any number to the power of 0 equals 1; any number to the power of 1 equals the number; and multiplying exponents when the bases are the same. The order of operations is explained as: exponents, multiplication/division from left to right, and addition/subtraction from left to right. Grouping symbols like parentheses and fraction bars dictate that operations within are completed first. Several examples demonstrate applying these rules to simplify expressions.
G6 m4-a-lesson 1-t
This lesson teaches students about the relationship between addition and subtraction through the use of tape diagrams and algebraic expressions. Students explore identities such as w - x + x = w by building tape diagrams that represent adding and then subtracting the same amount. They recognize that regardless of the numbers used, adding a number and then subtracting the same number will result in the original amount. Students practice writing number sentences to represent identities and evaluating expressions that involve addition and subtraction of the same terms. The lesson aims to help students understand that identities will always be true no matter what numbers are substituted in.
(7) Lesson 5.5 - Simplify Algebraic Expressions
To simplify the expression 5n + n:
1. Identify the terms: 5n and n.
2. Note that 5n and n are like terms since they have the same variable, n.
3. Use the distributive property to combine the like terms:
5n + n = (5 + 1)n
4. Simplify:
(5 + 1)n = 6n
So, the simplified expression is 6n.
(7) Lesson 5.6 - Simplify Algebraic Expressions
To add two linear expressions:
1. Arrange the terms of each expression in columns with like terms aligned.
2. Combine like terms by adding their coefficients.
3. Simplify the resulting expression by removing zero pairs and writing the expression in simplest form.
3.2 powerpoint
This document provides instruction on exponents and properties of exponents, including:
1) Multiplying powers with the same base by keeping the base and adding the exponents.
2) Dividing powers with the same base by keeping the base and subtracting the exponents.
3) Raising a power to a power by multiplying the exponents.
Examples are provided and students complete practice problems applying these properties of exponents. A brief quiz at the end reviews key concepts from the lesson. The assignment is to complete problems 1-36 applying exponent properties.
Unit 2 Fifth Grade cs 2012 2013
This document outlines the fifth grade mathematics curriculum for Unit 2 on prisms and pyramids. The unit focuses on determining the volume of 3D objects and relating volume to multiplication and addition. Students will learn about rectangular prisms, pyramids, cylinders and cones. They will measure volume using cubic units and apply formulas to find the volume of right rectangular prisms. Students will also explore how volume is additive by finding the total volume of objects made of two combined prisms. The unit aims to develop students' geometric reasoning and quantitative skills through 13 sessions of instruction and activities.
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3rd Semester Computer Science and Engineering (ACU) Question papers
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Chapter 3
- Mauldin Middle School wants to make $4,740 from yearbook sales. Print yearbooks cost $60 each and digital yearbooks cost $15 each.
- This relationship can be represented by the equation 60x + 15y = 4,740.
- Setting y = 0 in the equation yields an x-intercept of 79, meaning 79 print yearbooks would earn the school $4,740.
- Setting x = 0 yields a y-intercept of 316, meaning 316 digital yearbooks would earn the school $4,740.
Chapter 2 inquiry_lab_presentation_course_3 (1)
A bar diagram helps solve a two-step equation problem by visually representing the quantities and their relationships. The bar diagram shows the total number of postcards Miranda bought and the total cost. It can be used to write an equation relating the unknown cost of a large postcard to the total quantity and cost. Solving this equation reveals that the cost of one large postcard is $2.50.
Chapter 2 lesson_4_presentation_course_3
The document provides examples for solving linear equations using the work backward strategy. It includes 4 example equations with the step-by-step work shown. The answers are then provided. Additional material is included on equivalence, solving equations with variables on both sides, and using properties of equality.
Chapter 5
This document provides examples and explanations for using algebraic concepts in geometry, specifically regarding angles formed when lines are cut by a transversal. It defines key terms like parallel lines, transversal, interior angles, exterior angles, alternate interior angles, alternate exterior angles, and corresponding angles. Examples are given to classify pairs of angles and find missing angle measures. Step-by-step examples walk through using properties of parallel lines cut by a transversal to find missing angles. The document also explains how to prove conjectures using paragraph and two-column proofs.
Lesson 5.1
This document discusses classifying angles formed when parallel lines are cut by a transversal. It defines key terms like alternate interior angles, alternate exterior angles, and corresponding angles. It then provides examples of classifying angle pairs based on their relationship. Sample problems are worked through, applying the concepts to find missing angle measures by justifying answers using angle properties of parallel lines cut by a transversal.
Chapter 7
The document provides examples and step-by-step explanations for solving geometry problems involving similar and congruent shapes using transformations and properties of similarity and congruence. In one example, it is determined that two triangles are similar because corresponding angles are congruent. In another example, transformations of a rotation and translation are used to show that two figures are congruent.
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This document provides instruction on determining if two figures are similar using transformations. It includes examples of determining if figures are similar by checking if corresponding angles are congruent and if corresponding sides are proportional. It also covers finding missing measures in similar figures using scale factors and proportions. Determining similarity involves rotations, reflections, translations, and dilations, while congruence only involves rigid motions.
Lesson 7.3 (8)
This document provides instruction and examples for determining congruence and similarity using transformations. It explains that two figures are similar if one can be obtained from the other by a sequence of rotations, reflections, translations, and dilations. Examples show using transformations and writing side ratios to determine if figures are similar. The last example finds the dimensions of an image that is enlarged twice by different scale factors, showing the results are similar.
Lesson 7.2
This document contains a lesson on determining congruence between figures using transformations. It provides examples of identifying congruent triangles and determining the transformations needed to map one onto the other. It also gives examples of using congruence to find missing angle measures. The lesson explains that two figures are congruent if their corresponding sides are congruent and corresponding angles are congruent.
Pat05 ppt 0201
This document provides an overview of functions and concepts covered in Chapter 2, including:
- Graphing functions to identify intervals where they are increasing, decreasing, or constant.
- Modeling real-world applications with appropriate functions.
- Graphing functions defined piecewise using different formulas for different parts of the domain.
- Identifying relative maximum and minimum values of functions.
- Defining functions as increasing, decreasing, or constant on intervals.
- Describing the greatest integer function.
Chapter 6
This document provides examples and explanations for describing geometric transformations on a coordinate plane, including translations, reflections, rotations, and dilations. It defines key terms like preimage, image, and scale factor. Examples show how to perform and describe each type of transformation by:
1) Graphing a figure and identifying its vertices with coordinates
2) Performing the transformation by applying the rules for that specific transformation
3) Graphing the transformed image and identifying the new coordinates of its vertices
The document ensures understanding through multiple examples of each transformation type and encourages practicing the skills on additional example problems. It emphasizes that transformations preserve properties like distance and angle measures.
Pat05 ppt 0106
This document discusses solving linear inequalities and compound inequalities. It provides examples of solving various inequalities algebraically and graphing their solution sets. It also gives an example of setting up and solving an inequality to model a real-world situation about payment plans for a house painting job.
Pat05 ppt 0105
This document discusses linear equations and functions. It provides examples of solving linear equations, including special cases where equations have no solution or infinitely many solutions. Applications involving linear models for distance, rate, and time as well as simple interest are presented. The key concepts of zeros of linear functions and finding the zero of a given linear function are also covered.
Lesson 1.7 grade 8
1. The document provides examples of writing numbers in standard form and scientific notation. It also gives examples of performing arithmetic operations such as addition, subtraction, multiplication, and division on numbers expressed in scientific notation.
2. The examples calculate population sizes, planet diameters, and perform other calculations to illustrate how scientific notation makes working with very large and small numbers easier.
3. Writing numbers in scientific notation allows for easier operations like multiplication and division of very large and small quantities, as well as comparing the relative sizes of numbers like population and distances in the solar system.
Lesson 1.1 (8)
This document contains a mathematics lesson on writing numbers in different forms, including:
1. Converting fractions to decimals and decimals to fractions
2. Examples of solving proportions and writing decimals as fractions
3. The lesson discusses why it is helpful to write numbers in different ways, such as for measurements or when dealing with money.
Chapter 9 Study Guide
This document provides examples for constructing and interpreting scatter plots, line of best fit, two-way tables, and univariate data measures. It includes step-by-step examples and answers for constructing scatter plots using bivariate data, finding the line of best fit and writing the equation, using the line of best fit to make predictions, constructing and interpreting two-way tables and relative frequencies, and describing univariate data using measures of center and the five-number summary. The document is intended to teach statistical concepts and provide practice examples.
(8) Lesson 9.3
The document provides examples and explanations for describing distributions of data sets. It discusses identifying symmetry, clusters, gaps, peaks, and outliers. It explains that the median and interquartile range should be used to describe the center and spread of non-symmetric distributions, while the mean and mean absolute deviation can describe symmetric distributions. Students are told that patterns in distributions help compare two quantities and that describing distributions involves noting their shape, center, and spread.
(8) Lesson 9.5
Here are step-by-step directions for finding the mean absolute deviation of a data set:
1. Find the mean (average) of the data set. To do this, add up all the values and divide by the number of values.
2. For each value in the original data set, subtract the mean. This will give you the difference between that value and the mean.
3. Take the absolute value of each difference. The absolute value removes negative signs, so all differences will be positive numbers.
4. Find the mean (average) of the absolute differences. To do this, add up all the absolute differences and divide by the number of values.
5. The result is the mean
(8) Lesson 9.4
Here are step-by-step directions for finding the mean, median, mode, and range of a data set:
1. List all data values from the data set.
2. To find the median, arrange the data values in numerical order from lowest to highest.
3. If there is an odd number of values, the median is the middle number. If there is an even number of values, the median is the average of the two middle numbers.
4. To find the mode, identify the data value(s) that occur most frequently. There may be more than one mode.
5. To find the mean, add all the data values and divide the sum by the total number of
(8) Lesson 9.3
This document provides examples and explanations for constructing and interpreting two-way tables and relative frequencies. It includes two examples of creating two-way tables from survey data involving two categorical variables. It also shows how to find and interpret relative frequencies from the tables. The final questions discuss how patterns can be compared using two-way tables and relative frequencies, and how a Venn diagram can be used to construct a two-way table.
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- 1. Course 3, Lesson 1-2 Write each fraction or mixed number as a decimal. 1. 2. 3. Write each decimal as a fraction or mixed number in simplest form. 4. 0.08 5. 1.375 6. The largest moth is the Atlas moth. The Atlas moth is 11.8 inches long. What is the length of an Atlas moth written as a mixed number? 7 8 5 12 3 2 20
- 2. Course3, Lesson 1-2 ANSWERS 1. 0.875 2. 3. 2.15 4. 5. 6. 0.416 2 25 3 1 8 4 11 in. 5
- 3. WHY is it helpful to write numbers in different ways? The Number System Course3, Lesson 1-2
- 4. Course3, Lesson 1-2 Common Core State Standards © Copyright 2010. National Governors Association Center for Best Practices and Council of Chief State School Officers. All rights reserved. The Number System • 8.EE.1 Know and apply the properties of integer exponents to generate equivalent numerical expressions. Mathematical Practices 1 Make sense of problems and persevere in solving them. 3 Construct viable arguments and critique the reasoning of others. 4 Model with mathematics. 8 Look for and express regularity in repeated reasoning.
- 5. • To write and evaluate expressions using exponents Course3, Lesson 1-2 The Number System
- 6. • power • base • exponent Course3, Lesson 1-2 The Number System
- 7. 1 Need Another Example? 2 Step-by-Step Example 1. Write the expression using exponents. (–2) • (–2) • (–2) • 3 • 3 • 3 • 3 The base –2 is a factor 3 times, and the base 3 is a factor 4 times. (–2) • (–2) • (–2) • 3 • 3 • 3 • 3 = (–2)3 • 34
- 8. Answer Need Another Example? Write (–9) • (–9) • (–9) • (–9) • (–9) • 4 • 4 • 4 as an expression using exponents. (–9)5 • 43
- 9. 1 Need Another Example? 2 3 Step-by-Step Example 2. Write the expression using exponents. a • b • b • a • b Use the properties of operations to rewrite and group like bases together. The base a is a factor 2 times, and the base b is a factor 3 times. a • b • b • a • b = a • a • b • b • b = a2 • b3
- 10. Answer Need Another Example? Write x • y • x • y • x • x as an expression using exponents. x4 • y2
- 11. 1 Need Another Example? 2 Step-by-Step Example 3. Evaluate . = Write the power as a product. Multiply
- 13. 1 Need Another Example? 2 3 4 Step-by-Step Example 4. The deck of a skateboard has an area of about 25 • 7 square inches. What is the area of the skateboard deck? 25 • 7 = 2 • 2 • 2 • 2 • 2 • 7 Write the power as a product. Associative Property= (2 • 2 • 2 • 2 • 2) • 7 = 32 • 7 or 224 The area of the skateboard deck is about 224 square inches. Multiply.
- 14. Answer Need Another Example? A racquetball court has an area of 2 • 42 • 52 square feet. What is the area of the racquetball court? 800 ft2
- 15. 1 Need Another Example? 2 3 Step-by-Step Example 5. Evaluate the expression if a = 3 and b = 5. a2 + b4 a2 + b4 = 32 + 54 Replace a with 3 and b with 5. Write the powers as products.= (3 • 3) + (5 • 5 • 5 • 5) = 9 + 625 or 634 Add.
- 16. Answer Need Another Example? Evaluate x3 + y5 if x = 4 and y = 2. 96
- 17. 1 Need Another Example? 2 3 Step-by-Step Example 6. Evaluate the expression if a = 3 and b = 5. (a – b)2 (a – b)2 = (3 – 5)2 Replace a with 3 and b with 5. Perform operations in the parentheses first. = (–2)2 = (–2) • (–2) or 4 Write the powers as products. Then simplify.
- 18. Answer Need Another Example? Evaluate (x + y)2 if x = 4 and y = 2. 36
- 19. How did what you learned today help you answer the WHY is it helpful to write numbers in different ways? Course 3, Lesson 1-2 The Number System
- 20. How did what you learned today help you answer the WHY is it helpful to write numbers in different ways? Course 3, Lesson 1-2 The Number System Sample answers: • Writing numbers using powers is a short way of writing very large numbers. • You would use exponents when you work with the metric system since the metric system is based on powers of ten.
- 21. Write about how you think powers and exponents connect with multiplying and dividing monomials. Ratios and Proportional RelationshipsThe Number System Course 3, Lesson 1-2