MAT 204: Mathematics in the Modern World for Engineers Module #1 PDF

MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________...

MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Lesson title: Introduction to Algebra Materials: Pen, Notebook, Scientific Calculator Learning Targets: At the end of the module, students will be able to: References: 1. Define basic algebraic terms such as numbers, constants, College Algebra by R. David Gustafson variables and functions by providing some examples. and James Stewart 2. Classify numbers as rational, irrational, integers and non - integers through examples. 3. Solve basic arithmetic problems through boardwork and group activities. CONNECT (5 mins) A.1. Lesson Preview/Review Algebra is a fundamental branch of mathematics that deals with symbols and the rules for using those symbols. It is a powerful tool used to solve equations and describe relationships between quantities. Algebra allows us to generalize mathematical principles and solve problems in a more abstract way. Instead of dealing with specific numbers, we will work with variables that represent unknown quantities. By understanding the relationships between these variables and applying various operations, we can unlock solutions to a wide range of mathematical problems. In this module, we will recall the basics of algebra. We will start with the basic terms that we will be encountering in this course followed by classification of numbers. We will also COACH B.1. Content Notes (50 mins) Directions: Read the following text. Make sure that your pen and paper are beside you so you can readily take down the key points and concepts presented. After reading the text, answer the questions on the Skill Building Activity to reinforce what you have just learned. Remember that understanding the fundamentals is key to mastering Mathematics. In this lesson, we will be will use the comprehension strategy Summarizing and Generalization. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ INTRODUCTION TO ALGEBRA CLASSIFICATION OF NUMBERS Before we In Mathematics, a number is used to show many or where something is in a list. We use numbers to count things, measure things and do computations. Number can s called numerals Numbers can be real or imaginary. Real numbers are the ones we use everyday, like when we measure length, weight or temperature. Imaginary numbers are not seen in real life but help solve some math problems. In this course, we will only talk about real numbers. A set is a collection of objects. For example, the set contains the numbers 1, 3, 8, 9 and 10. We call these the numbers the elements or members of the set. These elements or members of a set are listed within braces. There are two basic sets of numbers: natural numbers and whole numbers. Natural numbers are used for counting and are also called positive integers or counting numbers. The set of natural numbers is. Whole numbers include zero and all natural numbers. The set of whole numbers is. Real numbers can be split into rational and irrational numbers. Rational numbers can be written as fractions, incl which means a comparison of two values, like a fraction. Examples of rational numbers are and. Irrational numbers are numbers that can be written as decimals but not as fractions. Examples of irrational numbers are (pi) and (the square root of 2). Rational numbers can be divided into integers and non - integers. Integers are numbers without decimals or fractions. The set of integers includes natural numbers, their negatives and zero. The set of integers is. Zero is an integer but is neither positive nor negative. Non - integers are numbers that are not natural numbers, negative natural numbers, or zero. Examples of non - integers are and. The next figure shows a summary on the classification of real numbers. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ EXAMPLE 1 Classify the following numbers. Identify them if they are rational, irrational, integer, or non - integer. a) b) c) d) e) f) g) h) Solution: First, let us identify if the number is rational or irrational. To make things easier, just remember this, if you can then the number is irrational. Now, if the number is rational, we can classify them further as an integer or non - integer. To determine if the number is integer or not, just check if the number has a fractional part. If the number has no fractional part, then it is an integer. If the number has a fractional part, then it is a non - integer. Let us now classify the following numbers. a) b) This number can be converted into a fraction: This number can be converted into a fraction: Since the number can be converted into a fraction, Since the number can be converted into a fraction, this number is a rational number. This number has a this number is a rational number. At the same time, fractional part (0.75). Therefore, this number is a non this number has no fractional part. Hence, this is an - integer. integer (specifically a negative integer). Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ c) d) First, let us solve for the exact value of this number. Let us first solve for the value of this number. You We can use our calculators for this. We can see that may use a calculator if you want. This number cannot be expressed as a fraction. Therefore, this number is irrational. OR Since the number can be converted into a fraction, this number is a rational number. This number has no fractional part. Therefore, this number is an integer. e) f) The given number is in fraction form. Therefore, the number. You can get the value of e using your number is rational. This number has a fractional part calculator. (the equivalent decimal number is not a whole number). Therefore, this number is a non - integer. This number cannot be expressed as a fraction since it. Therefore, this number is irrational. g) h) Let us be very careful. You might think that all non -Let us be very careful here. You might think that terminating decimals are irrational. Take note that since we have a numerator and denominator, the number is a fraction and thus a rational number. Remember that before saying that the number is Since this number can be expressed as a fraction, rational, both numerator and denominator must be this number is a rational number. At the same time, whole numbers. For this number, the denominator this number has a fraction part. Thus, this number is (3) is a whole number. However, the numerator a non - integer. is not a whole number. Since the numerator and denominator are not both whole numbers, the given number is an irrational number. FUNCTIONS function is a way to connect a set of inputs to one output each. In simpler words, a function is a relationship where each input has one specific output. We can also say that a function is a rule that links one variable (called the independent variable) to another variable (called the dependent variable). A function is often written as , the value of the function when is written as. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ A variable e depending on the math problem. Common letters used for variables are and. There are two types of variables: dependent and independent. A dependent variable changes based on the value of another number of variable. An independent variable does not change based on other values. Think of it like this: the independent variable is the input, and the dependent variable is the output of a function. When the value of the independent variable changes, the value of the dependent variable also changes. Let us take a look at the following examples. EXAMPLE 2 The relation is an example of a function. Our dependent variable (output) here is y while the independent variable (input) is x. Let us take a look at the following computations. If , then. If , then. If , then. As we can see, the value of y changes depending on the value of x. If we change the value of x, then the value of y also changes, that is why y is our dependent variable and x is our independent variable. Now, on your higher mathematics (like on Calculus), functions are sometimes written in the form. In this case, our function can be rewritten as. This means that we have a function where the independent variable is x and the output of the function can be computed by performing. EXAMPLE 3 Given the function. Find the following: a) b) c) Solution: a) This expression means that we will change all x in f(x) into 3 and then solve the function. Therefore: b) This expression means that we will change all x in f(x) into and then solve the function. Therefore: c) To solve this, we will first need to get the answer of and. After that, we will subtract the two answers. The expression means that we will change all x in f(x) into and then solve the function. Therefore: The expression means that we will change all x in f(x) into and then solve the function. Therefore: Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Now that we have solved for and , we can now solve for the given problem: We are now done with our text. I hope we are able to recall some of the basic terms in Algebra. Before reading the text, it was mentioned that we will be using the comprehension strategy Summarizing and Generalization. I hope that as you read the text, you made a summary of what you learned. Here is a sample of summary if the lesson. You may use this as a guide to make your own summary in the future. INTRODUCTION TO ALGEBRA A set is a collection of objects. We call the objects inside a set as elements or members. A number is a concept used to represent quantity or position and are used for counting, measuring and performing mathematical operations. They are represented as symbols called numerals. Number can be classified as real number, imaginary number, rational number, irrational number, integer or non - integer. Real numbers are numbers that we use in our everyday life. Imaginary numbers are numbers that we cannot physically see or count. Rational numbers are numbers that can be expressed as a ratio or fraction. If a number can be expressed as a fraction (where both numerator and denominator must be whole numbers) then it is a rational number. If we cannot do so, the number is irrational. Not all non - terminating decimal numbers - terminating decimal numbers. Rational numbers can be classified as integer or non - integers. Integers are number that have no fractional part. If a number has a fractional part, then it is a non - integer. Integers include all positive whole numbers, negative whole numbers and 0. Zero is an integer, however, it is neither positive nor negative. A function is a relation between a set of inputs having one output each. We can also define functions as an expression, rule, or law that defines a relationship between one variable (the independent variable) and another variable (the dependent variable). A function may be denoted as. If we are given the expression for example, this means that we will change all x into 3 and solve for the value of the function. A variable is a symbol or letter which is used to represent an unknown number. A dependent variable is a variable that depends on the value of some other number or variable. An independent variable does not depend on any values. As we change the value of the independent variable, the value of the dependent variable also changes. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ B.2. Skill Building Activity (10 mins) Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. 1. Classify the following numbers whether they are rational or irrational, integer or non - integer. Briefly explain why. a) b) c) d) e) f) 2. Given the function. Solve for the following: a) b) c) d) For supplementary materials (readings and videos) and practice problems, visit this website. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:foundation-algebra You may ask your instructor to give you the link via chat or email for easier access. In that website you may found additional learning materials regarding the topic. If you want more practice problems, you can take the quiz(es) and unit test found on the same page. CHECK C.1. Check for Understanding (15 mins) Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. 1. List 3 examples of irrational numbers, 3 examples of integers and 3 examples of non - integers. 2. Give an example of a non - terminating decimal that is an integer. 3. Classify the following numbers whether they are rational or irrational, integer or non - integer. Briefly explain why. a) b) c) d) e) f) 4. Given the function. Solve for the following: a) b) c) d) Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #1 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ CONCLUDE (10 mins) D.1. Summary / Frequently Asked Questions Where do irrational numbers come up in the real world? Irrational numbers show up actually all over the place. For example, the number (an irrational number) is very important in computations involving circles. The square root of 2, also an irrational number, is important in understanding right triangles. What are examples of nonterminating decimals that are also rational numbers? Repeating decimals are nonterminating decimals that are also rational numbers. These numbers occur which is equivalent to 101/999, a rational number. Where do we use functions in the real world? Functions are used in all sorts of real - world applications. For example, we use functions to model physical processes, like the motion of a car or the growth of a population. We can also use them to analyze data, like fi D.2. Thinking about Learning Three things you learned: 1. ________________________________________________________________________________ 2. ________________________________________________________________________________ 3. ________________________________________________________________________________ Two things that you would like to learn more about: 1. ________________________________________________________________________________ 2. ________________________________________________________________________________ One question you still have: 1. ________________________________________________________________________________ D.3. Answer Key (Answers to the tasks in the SAS so you can check your understanding) (Answers will be given by your professors) Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Lesson title: Solving System of Linear Equations Materials: Pen, Notebook, Scientific Calculator Learning Targets: At the end of the module, students will be able to: References: 1. Recognize the different methods for solving systems of linear College Algebra by R. David Gustafson equation. and James Stewart 2. Solve a system of linear equations using the different Algebra and Trigonometry with methods such as elimination or substitution method WileyPLUS Set by Cynthia Y. Young CONNECT (5 mins) A.1. Lesson Preview/Review Welcome to Module #2!. In our last module, we learned about the basic terms and concepts on Algebra. Today, we will learn about the different methods on solving linear equations. Read the learning targets listed above so you have To help you learn better, remember: Work with the SAS in sequence. Our SAS activities were designed so that each task will help you learn more effectively. Do all the tasks. Working on all the tasks will help you learn more. If there are changes to the tasks, your teacher will tell you. Read instructions carefully. Ask you teacher or your classmates if you have questions about the tasks. In our last lesson, we learned about functions and how they show relationships between inputs and Linear equations are everywhere in real life. They help us understand and solve problems involving things week to buy a new shirt, you can use a linear equation. When we have two or more linear equations together, we call it a system of linear equations. Solving these systems can help us make decisions when we have more than one thing to consider. For instance, if you want to figure out how many different books you can buy with a certain amount of money, a system of linear equations can help. Studying linear equations and systems of linear equations is important because it gives us tools to solve real - world problems, make better decision, and understand how many different things are connected. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ COACH B.1. Content Notes (50 mins) Directions: Read the following text. Make sure that your pen and paper are beside you so you can readily take down the key points and concepts presented. After reading the text, answer the questions on the Skill Building Activity to reinforce what you have just learned. Remember that understanding the fundamentals is key to mastering Mathematics. In this lesson, we will be Solving Systems of Linear Equations we read, we will use the comprehension strategy Summarizing and Generalization. SOLVING SYSTEMS OF LINEAR EQUATIONS EQUATIONS, EXPRESSIONS AND INEQUALITIES An expression is a combination of numbers, variables and math operations (like addition, subtraction, multiplication, and division). They do not contain an equals sign. Examples of expressions are. and An equation is a statement indicating two expressions are equal. Some examples of equations are and Also, take note that in expressing equations, we can interchange the left side and right side of the equation and the equation remains the same. This means that the equation is the same with the equation. An inequality is a mathematical statement that shows one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. It uses symbols like and. Examples of inequalities are: and Take note that unlike equations, we cannot simply interchange the left side and right side on inequality. If you want to interchange the left side and right side of an inequality, then you also need to flip the inequality symbol. For example, the inequality is not the same with. Instead, the inequality is equivalent to. Expressions help us present values. Equations help us solve problems and find exact values. Inequalities help us understand ranges and limits of values. In this module, we will be taking a look on how to solve equations and system of linear equations. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ LINEAR EQUATIONS WITH ONE VARIABLE A linear equation is a type of equation where the highest power of the variable (like x) is 1. When drawn or plotted, it looks like a straight line. A linear equation is an equation of the form , where a, b and c are constants.. An example of a linear equation is. In this equation, x is our variable and the numbers 3, 10 and 42 are constants. Notice that in the given equation, the variable x is only raised to the power of 1. Let us now recall how to solve linear equations. Let us start with solving linear equations in one variable. Example 1 In the given equation solve for x. Solution: In this problem, is the left side of the equation and 23 is the right side. We need to find the value of x that makes both sides equal. To do this, we will use some mathematical operations to get x by itself on one side of the equation. Step 1. Identify the given. The given is. Our goal is to get x alone on the left side. Step 2. Remove on the left side of the equation. Since 15 is subtracted from 12x, we need to add 15 to both sides to get rid of it on the left side. Remember, whatever we do to one side, we must do to the other side too. (Notice that will be equal to 0) Step 3. Remove on the left side of the equation. Now, we have 12x on the left side. To get x by itself, we need to divide both sides by 12. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ So, the solution to the equation is x = 19/6. Let us have another example. Example 2 Solve for y in the given equation: Step 1. Identify the given. We start with the equation. We can also write is as. Our goal is to get y alone on the left side. Step 2. Remove on the left side of the equation. Since 33 is added to , we need to subtract 33 from both sides to get rid of it on the left side. (Notice that will be equal to 0) Step 3. Remove on the left side of the equation. Now, is multiplied by y. To get y by itself, we divide both sides by. Since we have rewritten the equation so that only y is on the left side, we are done. The solution to the equation is y = 13/3. Example 3 Solve for z in the given equation: Solution: We need to get all the z terms on the one side of the we do it step-by-step: Step 1. Identify the given. We start with the equation. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Step 2. Move 7 to the right side of the equation. Since 7 is added to , we subtract 7 from both sides. Step 3. Move to the left side of the equation. Since is subtracted on the right, we add 4z to both sides. Step 4. Solve for z. Since 10 is multiplied by z, we divide both sides by 10 to find z. Since we have rewritten the equation so that only z is on the left side, we are done. The solution to the equation is. solve systems of linear equations. SYSTEMS OF LINEAR EQUATIONS A system of linear equations is when we have two or more equations with the same variables. Our goal is to find the values of these variables that make all the equations true at the same time. There are different methods to solve these systems, and we will go over the two of the most common ones: the substitution method and the elimination method. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Example 4: Using the Substitution Method Solve the system of equations: Step 1. Solve one of the equations for one variable. Step 2. Substitute this expression in the other equation. Replace y in the second equation with. Step 3. Solve for x. Use the steps we learned in solving linear equation in one variable. Step 4. Substitute x back in the equation to find y. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Therefore, the solution to the system is Before we proceed, let us check if our answer is correct. Let us substitute the values that we obtained to both equations. Both equations must be satisfied. For the first equation: The first equation is satisfied. For the second equation: The second equation is satisfied. Since both equations are satisfied, we can confirm that indeed, our solution is correct. Example 5: Using the Elimination Method Solve the system of equations: Step 1. Eliminate one of the variables by adding or subtracting the two equations. First, decide if you want to eliminate the x terms or the y terms. Based on the given, notice that the y terms just differ in sign. Thus, it is easier to eliminate the y terms. All we have to do is to add the two equations. Take note that adding two terms which only differ in sign will result into a sum of 0, essentially eliminating them. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Thus, we will get the equation. Step 2. Solve for x. Use the steps we learned in solving linear equations in one variable. Step 3. Substitute x back into one of the original equations to solve for y. Let us use the second equation Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Therefore, the solution to the system is Let us check if our answer is correct. Let us substitute the values that we obtained to both equations. Both equations must be satisfied. For the first equation: The first equation is satisfied. For the second equation: The second equation is satisfied. Since both equations are satisfied, we can confirm that indeed, our solution is correct. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Example 6: Using the Elimination Method Solve the system of equations: Step 1. Eliminate one of the variables by adding or subtracting the two equations. Make the coefficients of. The x term of the first equation is 2x while the x term of the second equation is 4x. In order for the coefficients of the x terms to be equal, let us multiply the first equation by 2. Step 2. Subtract the second equation from the modified first equation to eliminate x Thus, we will get the equation. Step 3. Substitute y = 3 back into one of the original equations to solve for x. Let us use the first equation. Therefore, the solution to the system is Let us check if our answer is correct. Let us substitute the values that we obtained to both equations. Both equations must be satisfied. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ For the first equation: The first equation is satisfied. For the second equation: The second equation is satisfied. Since both equations are satisfied, we can confirm that indeed, our solution is correct. Choosing the right method can simplify your work and make solving systems of equations more efficient. Use Substitution when it is easy to solve one equation for a single variable. Use Elimination when you can quickly cancel out a variable by adding or subtracting the equations. We are now done with our text. I hope we are able to learn some techniques in solving linear equations. Before reading the text, it was mentioned that we will be using the comprehension strategy Summarizing and Generalization. I hope that as you read the text, you made a summary of what you learned. Here is a sample of summary if the lesson. You may use this as a guide to make your own summary in the future. SOLVING SYSTEMS OF LINEAR EQUATIONS An expression is a combination of numbers, variables and math operations. An example is. An equation is a statement indicating two expressions are equal. An example is. An inequality is a mathematical statement that shows one expression is greater than, less than, greater than or equal to, or less than or equal to another expression. An example is. A linear equation is a type of equation where the highest power of the variable is 1. It is an equation of the form , where a, b and c are constants. To solve linear equation in one variable, we will be first using some mathematical operations to get the variable by itself on one side of the equation. Once all the variable terms are on one side of the equation and the constants on the other side of the equation, we can now solve for unknown variable. A system of linear equations is when we have two or more equations with the same variables. The two common methods in solving systems of linear equations are the Substitution Method and the Elimination Method. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Substitution is usually used when it is easy to solve one equation for a single variable. Elimination is usually used when you can quickly cancel out a variable by adding or subtracting the equations. To check if the solution to a system of linear equation is correct, we need to substitute the values of the variables you found back into the original equations and see if they satisfy all of them simultaneously. B.2. Skill Building Activity (15 mins) Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. Obtain the solution to the system of equations. Hint: Before applying the methods, rewrite the equations first to the standard form ax + by = c. 1. 2. 3. CHECK (15 mins) C.1. Check for Understanding Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. Obtain the solution to the system of equations. 1. 2. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #2 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ CONCLUDE (10 mins) D.1. Summary / Frequently Asked Questions What are some real - world applications of systems of equations? figure out how many adult and child tickets were sold at a movie theater, we might set up a system of equations with one equation for the total number of tickets and another equation for the total amount of money collected. Can a system of equations have more than one solution? Yes. A system of linear equations can have no solution, one solution or infinitely many solutions. Sometimes we can tell from looking at the system, and other times we may need to use substitution, elimination, or graphing to figure it out. In this course, we will only deal with systems of linear equations with one solution. D.2. Thinking about Learning Three things you learned: 1. ________________________________________________________________________________ 2. ________________________________________________________________________________ 3. ________________________________________________________________________________ Two things that you would like to learn more about: 1. ________________________________________________________________________________ 2. ________________________________________________________________________________ One question you still have: 1. ________________________________________________________________________________ D.3. Answer Key (Answers to the tasks in the SAS so you can check your understanding) (Answers will be given by your professors) Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Lesson title: Solving Worded Problems in Algebra Materials: Pen, Notebook, Scientific Calculator Learning Targets: At the end of the module, students will be able to: References: 1. Translate word problems into algebraic expressions College Algebra by R. David Gustafson 2. Apply algebra to solve worded problems and James Stewart Algebra and Trigonometry with WileyPLUS Set by Cynthia Y. Young CONNECT (5 mins) A.1. Lesson Preview/Review Welcome to Module #3! In our last module, we learned how to solve systems of linear equations. Today, we will recall how to solve worded problems in Algebra. Read the learning targets listed above so you have a To help you learn better, remember: Work with the SAS in sequence. Our SAS activities were designed so that each task will help you learn more effectively. Do all the tasks. Working on all the tasks will help you learn more. If there are changes to the tasks, your teacher will tell you. Read instructions carefully. Ask you teacher or your classmates if you have questions about the tasks. In our last lesson, we learned how to solve systems of linear equations. Now, we're going to build on that knowledge by learning how to change word problems into equations. This skill is very useful because it helps us take a problem written in words and turn it into a math problem we can solve. This makes it easier to understand and find the answer. We use this skill not only in school but also in everyday life. For example, if we want to know how much money we need to save to buy something or how long we need to wait before arriving at our destination. Learning to turn word problems into equations helps us think better and solve problems more easily, making our lives simpler and more organized. COACH B.1. Content Notes (50 mins) Directions: Read the following text. Make sure that your pen and paper are beside you so you can readily take down the key points and concepts presented. After reading the text, answer the questions on the Skill Building Activity to reinforce what you have just learned. Remember that understanding the fundamentals is key to mastering Mathematics. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ In this lesson, we will be Solving Worded Problems in Algebra we read, we will use the comprehension strategy Summarizing and Generalization. SOLVING WORDED PROBLEMS IN ALGEBRA Solving word problems in algebra can seem challenging at first, but with a clear method, it becomes much easier. The key is to translate the words into mathematical equations that you can solve. This skill is important because it helps us understand and solve real - world problems using algebra. Let us go through the steps 1. Read the Problem Carefully. Make sure you understand the problem. Read it several times if needed. 2. Identify the Unknowns. Determine what the problem is asking for. Assign variables (like x or y) to the unknowns. 3. Extract Key Information. Pick out the important numbers, relationships, and conditions given in the problem. 4. Translate Words into Equations. Convert the verbal statements into algebraic expressions or equations. Look for keywords be using addition 5. Set up the Equation. Write down the equation or equations based on the relationships and conditions identified. 6. Solve the Equation. Use algebraic methods to solve for the unknown variables. 7. Check Your Solution. Substitute the solution back into the original problem to verify that it works. 8. Write the Answer. Clearly state the answer in the context of the problem. EXAMPLE 1 The IT department has twice as many tablets as laptops. Together, there are 18 devices. How many tablets and how many laptops does the IT department have? Step 1. Read the Problem Carefully The IT department has twice as many tablets as laptops. Together, there are 18 devices. Step 2. Identify the Unknowns Let x be the number of laptops. Let y be the number of tablets. Step 3. Extract Key Information Twice as many tablets as laptops Total devices are 18 Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Step 4. Translate Words into Equations (because tables are twice the number of laptops) (there are total device) Step 5. Set Up the Equation From , substitute y in the second equation Step 6. Solve the Equation Let us solve for y by substituting x = 6. Since , Step 7. Check Your Solution Total devices: Tablets are twice the laptops: Step 8. Write the Answer The IT department has 6 laptops and 12 tablets. EXAMPLE 2 A number is three times another number. The sum of the two numbers is 48. What are the two numbers? Step 1. Read the Problem Carefully A number is three times another number. The sum of the two numbers is 48. Step 2. Identify the Unknowns Let x be the smaller number. Let y be the larger number. Step 3. Extract Key Information One number is three times the other. Their sum is 48. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Step 4. Translate Words into Equations (because the larger number is three times the smaller number) (the sum of the two numbers) Step 5. Set Up the Equation From , substitute y in the second equation. Step 6. Solve the Equation Let us solve for y by substituting x = 12. Since , Step 7. Check Your Solution Sum of the numbers: Large number is three times the smaller number: Step 8. Write the Answer The two numbers are 12 and 36. EXAMPLE 3 Student A bought some notebooks and pens for a total of PhP 60. The cost of one notebook is PhP 5 more than the cost of one pen. If the student bought 4 notebooks and 6 pens, what are the costs of a notebook and a pen? Step 1. Read the Problem Carefully Student A bought 4 notebooks and 6 pens. Total cost is PhP 60. The cost of one notebook is PhP 5 more than the cost of one pen. Step 2. Identify the Unknowns Let x be the cost of one pen. Let y be the cost of one notebook. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Step 3. Extract Key Information 4 notebooks and 6 pens cost PhP 60. The cost of one notebook is PhP 5 more than the cost of one pen. Step 4. Translate Words into Equations (because the cost of one notebook is PhP 5 more than the cost of one pen) (the total cost) Step 5. Set Up the Equation From , substitute y in the second equation. Step 6. Solve the Equation Thus, the cost of one pen is PhP 4. Let us solve for y by substituting x = 4. Since ,. Thus, the cost of one notebook is PhP 9. Step 7. Check Your Solution Total cost: The cost of one notebook is PhP 5 more than the cost of one pen: Step 8. Write the Answer The cost of one pen is and the cost of one notebook is. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ EXAMPLE 4 Student A travels from Town A to Town B, a distance of 120 miles, at an average speed of 60 miles per hour. On her way back from Town B to Town A, she travels at an average speed of 40 miles per hour. What is the total time she spends traveling to and from Town B? Step 1. Read the Problem Carefully Student A travels from Town A to Town B, a distance of 120 miles. Her average speed on the way to Town B is 60 miles per hour. Her average sped on the way back from Town B to Town A is 40 miles per hour. Step 2. Identify the Unknowns Let be the time spent travelling to town B. Let be the time spent travelling back from Town B to Town A. Step 3. Extract Key Information Distance to Town B: 120 miles Speed to Town B: 60 miles per hour Speed from Town B: 40 miles per hour. Step 4. Translate Words into Equations Recall that we can relate distance, speed and time using the formula: Hence, if we want to solve for the time, the formula becomes: (time to town B) (time from Town B) Step 5. Set Up the Equation Step 6. Solve the Equation Student A spends 2 hours traveling to Town B and 3 hours travelling back from Town B. Total time: Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Step 7. Check Your Solution Using the solved values for and , let us check if we will satisfy the given. You can either confirm if the distances are satisfied or the speeds are satisfied. For this example, let us confirm if the distances mentioned in the given will be satisfied. For the distance from Town A to Town B: For the distance from Town B to Town A: This confirms that our solution is correct since the distance from Town A to Town B and distance from Town B to Town A are the same. Step 8. Write the Answer Student A spends a total of 5 hours travelling to and from Town B. EXAMPLE 5 A teenager ran from her home to a park at an average speed of 12 km/h. She then rode an express bus from the park to her school. The bus has an average speed of 76 km/h. She traveled a total distance of 120 kilometers, and the entire trip took 2 hours. Assuming that the paths taken from home to the park and from the park to the school form a straight line, how far is her home from the park and how far is the park from her school? Step 1. Read the Problem Carefully The teenager travels for a total of 2 hours. The average speed of the teenager as she run from home to park is 12 km/h The average speed of the bus that the student rode on from park to school is 76 km/h Step 2. Identify the Unknowns Let be the distance from her home to the park Let be the distance from the park to her school Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Step 3. Extract Key Information Speed from home to the park: 12 km/h Speed from the park to the school (the speed of the bus): 76 km/h Total distance travelled: 120 km Total time taken for the trip: 2 hours Step 4. Translate Words into Equations Recall that we can relate distance, speed and time using the formula: Hence, if we want to solve for the time, the formula becomes: Time taken from home to the park Time taken from the park to school Total time for the trip: Total distance travelled: Step 5. Set Up the Equation We have a system of two equations: Step 6. Solve the Equation Let us use elimination method. Let us eliminate. In order to do so, let us multiply the first equation by 12. Next, let us subtract the second equation from the modified first equation. Thus, we have Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Let us solve for Let us now solve for From the second equation: Step 7. Check Your Solution The total distance travelled is:. This is the same as the given distance. Step 8. Write the Answer is 114 km. We are now done with our text. I hope we are able to learn some techniques in solving worded problems in Algebra. Before reading the text, it was mentioned that we will be using the comprehension strategy Summarizing and Generalization. I hope that as you read the text, you made a summary of what you learned. Here is a sample of summary if the lesson. You may use this as a guide to make your own summary in the future. SOLVING WORDED PROBLEMS IN ALGEBRA The steps in solving worded problems in Algebra are: 1. Read the Problem Carefully 2. Identify the Unknowns 3. Extract Key Information 4. Translate Words into Equations 5. Set up the Equation Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ 6. Solve the Equation 7. Check Your Solution 8. Write the Answer The speed of an object, distance travelled by the object and time of travel can be related by the formula: From this formula, we can derive two more formulas: B.2. Skill Building Activity (15 mins) Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. Solve the following problems. 1. A company bought a total of 10 electronic devices, consisting of printers and computers. The total cost of the printers was $400, and the total cost of the computers was $1200. If each printer costs $100 less than each computer, how many printers and how many laptops did the company buy? 2. At a supermarket, the bulk price for honey is PhP 2.50 per gram, with a minimum purchase of 20 grams. If Buyer A paid PhP 80 for some honey, by how many pounds did Buyer A purchase exceed the minimum? 3. A number is four times another number. The sum of the two numbers is 60. What are the two numbers? 4. The sum of a number and three more than twice the number is 36. What is the number? 5. Separate the number 20 into two parts so that five times the smaller part plus eight is equal to the larger part. What are the two numbers? 6. An executive traveled a total of 4 hours and 875 miles by car and by plane. Driving to the airport by car, she averaged 50 miles per hour. In the air, the plane averaged 320 miles per hour. How long did it take her to drive to the airport? 7. The difference of twice a smaller integer and 7 times a larger is 4. When 5 times the larger integer is subtracted from 3 times the smaller, the result is. Find the integers. 8. Student A and Student B start from the same point and walk in opposite directions. Student A walks 2 km/hr faster than Student B. After 3 hours, they are 30 km apart. How fast did each walk? 9. Student A has more money than Student B. If Student A gave Student B PhP 20, they would have the same amount. While if Student B gave Student A PhP 22, Student A would then have twice as much as Student B. How much does Student B have? 10. There are 1000 tickets sold. The price of an adult ticket is $8.50 while the price of a child ticket is $4.50. A total of $7300 was collected. How many tickets of each kind were sold? Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ CHECK (15 mins) C.1. Check for Understanding Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. Solve the following problems. 1. Student A lived in Tokyo and Hokkaido for a total period of 14 months in order to learn Japanese. He learned an average of 130 new words per month when he lived in Tokyo and an average of 150 new words per month when he lived in Hokkaido. In total, he learned 1920 new words. How long did he live in Tokyo and how long did he live in Hokkaido? 2. A teenager ran from her home to a park at an average speed of 12 km/h. She then rode an express bus from the park to her school. The bus has an average speed of 76 km/h. She traveled a total distance of 120 kilometers, and the entire trip took 2 hours. How long did she spend running, and how long did she spend riding the bus? 3. The difference of two integers is 11. When twice the larger is subtracted from 3 times the smaller, the result is 3. Find the integers. CONCLUDE (10 mins) D.1. Summary / Frequently Asked Questions What other types of word problems, not covered in this module, should we understand before delving into Calculus? Before delving into Calculus, it's recommended that you also grasp concepts in geometry, including calculating areas and perimeters of plane figures, as well as volumes and surface areas of solids. Additionally, familiarity with concepts in Analytic Geometry, such as generating equations of lines and conic sections, is essential. All of these will be taught in the next modules. D.2. Thinking about Learning Three things you learned: 1. ________________________________________________________________________________ 2. ________________________________________________________________________________ 3. ________________________________________________________________________________ Two things that you would like to learn more about: 1. ________________________________________________________________________________ 2. ________________________________________________________________________________ Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #3 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ One question you still have: 1. ________________________________________________________________________________ D.3. Answer Key (Answers to the tasks in the SAS so you can check your understanding) (Answers will be given by your professors) Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Lesson title: Exponents and Radicals Materials: Pen, Notebook, Scientific Calculator Learning Targets: At the end of the module, students will be able to: References: 1. Explain exponents, radicals, exponential functions and College Algebra by R. David Gustafson radical functions by providing examples. and James Stewart 2. Simplify exponential and radical expressions by applying the Algebra and Trigonometry with various laws on exponents and radicals WileyPLUS Set by Cynthia Y. Young CONNECT (5 mins) A.1. Lesson Preview/Review Welcome to Module #4! In our last module, we learned how to solve worded problems in Algebra. Today, we will learn about exponents and radicals. Read the learning targets listed above so you have a good idea of To help you learn better, remember: Work with the SAS in sequence. Our SAS activities were designed so that each task will help you learn more effectively. Do all the tasks. Working on all the tasks will help you learn more. If there are changes to the tasks, your teacher will tell you. Read instructions carefully. Ask you teacher or your classmates if you have questions about the tasks. tart learning! Today, we will learn about exponents, radicals, exponential functions and radical functions. These concepts are fundamental in mathematics and have wide - ranging applications. Exponents are shorthand for repeated multiplication, while radicals are the inverse operation, representing roots of numbers. Exponential functions involve a base raised to a variable exponent are used to model growth, decay and more. Radical functions deal with expressions containing radicals, crucial in geometry and engineering. Understanding these concepts is vital for solving mathematical problems across various disciplines. Let us take a look at their properties and applications. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ COACH B.1. Content Notes (50 mins) Directions: Read the following text. Make sure that your pen and paper are beside you so you can readily take down the key points and concepts presented. After reading the text, answer the questions on the Skill Building Activity to reinforce what you have just learned. Remember that understanding the fundamentals is key to mastering Mathematics. In this lesson, we will be Exponents and Radicals will use the comprehension strategy Summarizing and Generalization. Exponents and Radicals Exponents and its Properties An exponent is a mathematical notation that tells us how many times a number, called the base, is multiplied by itself. It is a way of showing repeated multiplication in a compact form. The process of raising a number to a power is known as involution. For example, in the expression , where is the base and is the exponent, we multiply by itself times. So, if and , then means which equals 8. Exponents represent Here are additional examples. One essential skill to master is simplifying terms involving exponents. Let us explore this further by examining the following properties of exponents. An exponential function is a mathematical function where the exponent contains a variable. Examples of exponential functions are and. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Mathematics Notation (let a and b be nonzero Name of Property Description Example real numbers and m and be n integer) When multiplying exponentials with the Product Property same base, add the exponents. When dividing exponentials with the Quotient Property same base, subtract the exponents (numerator - Where denominator) When raising an Power Property exponential to a power, multiply exponents A product raised to a Product to a Power power is equal to the Property product of each factor raised to the power A quotient raised to a Quotient to a Power power is equal to the Property quotient of the factors Where Where raised to the power. A base raised to a negative - integer exponent is equivalent to Negative - Integer the reciprocal of the base Exponent Property raised to the opposite Where (positive) integer exponent. Any nonzero number Zero Exponent Property raised to the power of Where zero is 1. For any number x, one One Base Property raised to x is equal to 1. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Here are some common mistakes when simplifying exponential expressions. Let us be very careful as not to commit these errors. INCORRECT WHAT SHOULD HAVE CORRECT ERROR COMMITTED SIMPLIFICATION BEEN DONE SIMPLIFICATION The exponents were The exponents should be multiplied added The exponents were The exponents should be divided subtracted The exponent was The exponents should be raised to a power (2 multiplied was raised to 3) Both the factors (2 and x) Only x was raised to 3 should be raised to 3 In applying the product The common base was property, the common multiplied base must be retained Nothing. We cannot apply the Product Rule nor The two distinct base Quotient Rule if the base of (this term is already in are multiplied the terms are not the simplest form) same. The various properties that we have discussed are used to simplify exponential expressions. An exponential expression is simplified when: All parentheses or groupings have been eliminated A base appears only once No powers are raised to other powers All exponents are positive EXAMPLE 1 Simplify the following expressions (assume all variables are nonzero). Solution: In this example, we will learn how to simplify exponential expressions that only involves multiplication, division and involution. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Since we are just multiplying terms, we can start by rearranging the terms in such a way that the constants are together, the x terms are together and the y terms are together. Then, let us multiply the constants together, multiply the x terms together and multiply the y terms together. For the product of the x terms and y terms, we will be using the Product Property. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is. From the given, we can see that we can use the Product to a Power Property. Let us apply that property first. Next, let us apply the Power Property on and. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is. Since we are just multiplying and dividing terms, we can start by rearranging the terms in such a way that the constants are together, the x terms are together, and the y terms are together. If there are constants that can be multiplied together, let us also obtain their product. Then, let us divide the constants together, divide the x terms and divide the y terms. For the quotient of the x terms and y terms, we will be using the Quotient Property. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Since we have a term with negative exponent, let us use the Negative - Integer Exponent Property. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is. EXAMPLE 2 Simplify the following expressions (assume all variables are nonzero). Solution: Solution 1: Since we are given two x terms that are multiplied together, we can just apply the Product Property. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is. Solution 2: Since we have a term with negative exponent, we can apply the Negative - Integer Exponent Property first. We now have a quotient of two x terms. Thus, we can use the Quotient Property. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Notice that with this problem, there are two ways to solve it. However, it's clear that Solution 1 is the quicker option. This emphasizes why practicing more problems is important. Doing so helps you get better at picking the best solution for each problem you encounter. From the given, we can see that we can use the Product to a Power Property and the Quotient to a Power Property. Let us apply those properties first. Next, let us apply the Power Property on terms with powers raised to another power. Since we are now just multiplying and dividing terms, let us rearrange the terms in such a way that the q terms are together, and the p terms are together. Then, let us apply the Quotient Property. Since we have a term with negative exponent, let us use the Negative - Integer Exponent Property. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ If we refer to our checklist, what we got is already in simplest form. Thus, the answer is From the given, we can see that we can use the Product to a Power Property and the Quotient to a Power Property. Let us apply those properties first. Next, let us apply the Power Property on terms with powers raised to another power. Since we are now just multiplying and dividing terms, let us rearrange the terms in such a way that the m terms are together, and the n terms are together. Then, let us apply the Quotient Property. Since we have a term with negative exponent, let us use the Negative - Integer Exponent Property. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ If we refer to our checklist, what we got is already in simplest form. Thus, the answer is Radicals and its Properties Radicals involve roots, like square roots and cube roots. The radical symbol shows the root. The number inside the radical is called the radicand, and the small number above and to the left is the index. If For example, the expression means the fourth root of , with a radicand of and an index of 4. means the square root of 3x with a radicand of 3x and index of 2. original number. For instance, the square root of 9 is 3 because. The cube root of 8 is 2 because. Radical functions are mathematical functions that involve radicals. Examples are and. Radicals, like exponents, have rules to simplify them. Mathematics Notation (let a and b be nonzero Name of Property Description Example real numbers and m and be n integer) The nth root of the product of Product Property two numbers is equal to the product of their nth roots The nth root of the quotient Quotient Property of two numbers is equal to the quotient of their nth roots Raising an nth root to a power is equivalent to raising the radicand to the same power. Power Property Raising an nth root to the power of n removes the radical, leaving just the radicand. For any real number x and Radical - Exponential positive integer n, the nth Property root of x can be expressed as the form Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Let us use these properties to simplify some radical expressions. A radical expression is simplified when: All exponents in the radicand must be less than the index. Any exponents in the radicand can have no factors in common with the index. No fraction appears under a radical. No radicals appear in the denominator of a fraction. Let us take a look at the following examples. Example 3 Simplify the following expressions (assume all variables are positive). Solution: The exponent of the radicand (7) is larger than the index (2). This violates the first rule. Let us use the Product Property and Power Property. Since the index is 2, let us rewrite in a way that includes an exponent of 2. By the Product Property and the Power Property of exponents Next, substitute back in to the radical. After that, let us apply the Product Property of radicals. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is This radical violates the second simplification rule since the index (9) and the exponent of the radicand (6) have a common factor of 3. To simplify problems like this, let us use the Radical - Exponential Property to simplify it. First, convert the radical expression to an exponential expression. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ Next, simplify the exponent: Finally, convert the simplified exponential expression back to a radical expression: If we refer to our checklist, what we got is already in simplest form. Thus, the answer is First, let us apply the Product Property to break - down the given expression. Let us use the steps that we used in the first two examples to simplify each expression. If we refer to our checklist, what we got is already in simplest form. Thus, the answer is Example 4 Rationalize the denominator for each of the following. Assume the variables are positive. Solution: In radicals, rationalization is used to remove radical expressions from the denominator of a fraction. In this case, we are going to apply the Power Property, specifically the fact that. Since the denominator is a square root, we can multiply the same square root once in the denominator. Also, the term that you multiplied in the denominator must also be multiplied in the numerator. Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ If we refer to our checklist, what we got is already in simplest form. Thus, the answer is To rationalize the denominator when it is a sum or difference involving a radical, multiply both the numerator and the denominator by the conjugate of the denominator (change subtraction to addition or vice versa): If we refer to our checklist, what we got is already in simplest form. Thus, the answer is We are now done with our text. I hope that you are able to learn about exponents and radicals and how to simplify exponential and radical expressions. Before reading the text, it was mentioned that we will be using the comprehension strategy Summarizing and Generalization. I hope that as you read the text, you made a summary of what you learned. Here is a sample of summary if the lesson. You may use this as a guide to make your own summary in the future. EXPONENTS AND RADICALS An exponent is a mathematical notation that tells us how many times a number, called the base, is multiplied by itself. The process of raising a number to a power is known as involution. An exponential function is a mathematical function where the exponent contains a variable. Examples of exponential functions are and. Properties of Exponents Product Property: Product to a Power Property: Quotient Property: Quotient to a Power Property: Power Property: Negative - Integer Exponent Property: Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ An exponential expression is simplified when: a. All parentheses or groupings have been eliminated b. A base appears only once c. No powers are raised to other powers d. All exponents are positive Radicals are expressions that involve roots, such as square roots, cube roots, and higher - order roots. The radical symbol is the symbol used to denote roots. The radicand is the number or expression inside the radical symbol. The index is the small number written just above and to the left of the radical symbol indicating which root to take. The process of getting the root is known as evolution. Radical functions are mathematical functions that involve radicals. Examples are and. Properties of Radicals Product Property: Quotient Property: Power Property: Radical - Exponential Property: A radical expression is simplified when: a. All exponents in the radicand must be less than the index. b. Any exponents in the radicand can have no factors in common with the index. c. No fraction appears under a radical. d. No radicals appear in the denominator of a fraction. B.2. Skill Building Activity (15 mins) Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. 1. Simplify the following expressions: 2. Rationalize the denominator of the following expressions: Learning Modules by PHINMA Education is licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.  MAT 204: Mathematics in the Modern World for Engineers Module #4 Name: _________________________________________________________________ Class number: _______ Section: ____________ Schedule: ________________________________________ Date: ________________ For supplementary materials (readings and videos) and practice problems, visit this website. https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:rational-exponents-radicals CHECK C.1. Check for Understanding (15 mins) Directions: Answer the following problems. Use the concept that was taught in the text that you have just read. 1. Simplify the following expressions: 2. Rationalize the denominator of the following expressions: CONCLUDE (10 mins) D.1. Summary / Frequently Asked Questions What are the real world application of exponents? Exponents are used in many real-world scenarios across different fields like science, finance, engineering, and daily life. Examples include calculating compound interest in finance, modeling population growth, radioactive decay, and various physics formulas. In engineering, exponents help calculate power and energy. In medicine, they model the growth of bacteria. In computer science, exponents are used in algorithms, and in environmental science, they help model the spread of pollutants. What are the real world application of exponents? Radicals have numerous real-world applications across various fields like science, engineering, finance, and everyday life. In architecture and construction, the Pythagorean theorem, which involves radicals, is widely used. In physics, radicals help calculate speeds and forces. Engineering applications include determining stresses and strains in materials and calculating the root mean square (RMS) of alternating current. In medicine, radicals are used in do