Multiplying the roots in the denominators. Rationalizing Denominators Worksheet Answers and Worksheets 44 Lovely Simplifying Radical Expressions Worksheet Hd. Assume all variables have positive values. Fixing it (by making the denominator rational) is called "Rationalizing the Denominator"Note: there is nothing wrong with an irrational denominator, it still works. Algebra. Top Python Programming Interview Questions w... Exercices Corrigés Physique Chimie Seconde en PDF. Document N°4 PDF. denominator worksheet doc.rationalizing the denominator with variables.purdue Rationalizing Denominators Worksheet Answers Also Rationalizing the Denominator Worksheet New Algebra Archive July 25. Watch Queue Queue. Watch Queue Queue The multiplication of the denominator by its conjugate results in a whole number (okay, a negative, but the point is that there aren't any radicals): Python Projects for Beginners. This Fifteen Multiple Choice Rationalizing the Denominator Problems Worksheet is suitable for 10th - 12th Grade. English, science, history, and more. Multiple Choice Questions, MCQs will form a significant part of the Mathematics question paper in CBSE Class 9 Annual Exam 2020. It's easier to say even, so maybe that's another justification for rationalizing this denominator. When you encounter a fraction that contains a radical in the denominator, you can eliminate the radical by using a process called rationalizing the denominator. If the binomial occurs in the denominator we will have to use a different strategy to clear the radical. To be in "simplest form" the denominator should not be irrational!. Biological and Biomedical Earn Transferable Credit & Get your Degree, Create your account to access this entire worksheet, A Premium account gives you access to all lesson, practice exams, quizzes & worksheets. Some of the worksheets for this concept are Radicals, Rationalize the denominator and multiply with radicals, Multiply and divide radicals 1 simplify by rationalizing. software.rationalizing imaginary denominators worksheet answer key.rationalize worksheet.quiz exam test MCQ multiple choice questions and answers.simplifying A collection of linear orderings is said to provide a rationalization by multiple rationalesfor a choice function if the choice from any choice set can be rationalized by one of the orderings. Some of the worksheets displayed are Radicals, Rationalize the denominator and multiply with radicals, Multiply and divide radicals 1 simplify by rationalizing. The base of a fraction is known as the denominator. 30 Rationalizing the Denominator Worksheet Kuta Worksheet Rationalizing Denominators Worksheet Answers Also Rationalizing the Denominator solution4 . The online math tests and quizzes for rationalizing denominator with with one or two radical terms. Rationalize the denominator and multiply with Radicals Rationalizing is done to remove the radical from the denominator of a fraction. To rationalize the denominator means to eliminate any radical expressions in the denominator such as square roots and cube roots. Free worksheet(pdf) and answer key on rationalizing the denominator. Multiplication: Coefficients with Coefficients I RADicals with RADicals 2. So the first thing I'd want to do is just simplify this radical right here. Which makes me think I don't understant rationalizing the denominator. These are multiple choice so it makes it that much easier. 1/y-5 b. 5/a^2+5a+4, 4a/a^2+3a+2 . Let's do a couple more examples. $\displaystyle\frac{\sqrt{63}}{\sqrt{72}}$ When rationalizing the denominator, there are two ways to do it. Free rationalize denominator calculator - rationalize denominator of radical and complex fractions step-by-step This website uses cookies to ensure you get the best experience. 1.) {{courseNav.course.mDynamicIntFields.lessonCount}} lessons ©l s2 n0E1Q1J 9K eu ZtEa T 3Siojf Xtpw ZaYrJe Z cLTLzC k.U K yAVljl l lr1i vg thCt ysD Drqe 4s qe rMvRe5dW.b F dM sa 1d 1eL wBi4t9h 2 wI9nif niknLi lt peS hAWlag9e berBab K1 f.4-3-Worksheet by Kuta Software LLC Answers to Rationalizing the Denominator Simplify by rationalizing the denominator. Certain answers such as fractions will have multiple fields. Videos, worksheets, 5-a-day and much more Rationalize The Denominatormultiple Choice. practice with answers.rationalizing the denominator with variables worksheet.rationalizing denominator quiz.rationalisation of surds problems.foil with radicals Remember to find the conjugate all you have to do is change the sign between the two terms. Find the numerical value of 2x 7x 15 x 2 2 when x = -2. Sofsource.com includes practical resources on rationalizing trinomial denominators, denominator and square roots and other math topics. Database Downtime. Python Exercises with Solutions. Some of the worksheets for this concept are Radicals, Rationalize the denominator and multiply with radicals, Multiply and divide radicals 1 simplify by rationalizing. 1/y+5 c. 6/y+25 d. Algebra. Rationalizing the numerator and denominator. So lets divide the numerator by 2. The denominator here contains a radical, but that radical is part of a larger expression. For example, how can we rationalize the following fractions? The quiz questions will test your understanding of how to perform this operation. ton to pounds. Find the conjugate of a binomial by changing the sign that is between the 2 terms, but keep the same order of the terms. Because everything in the numerator and everything in the denominator is divisible by 2. no. In this rationalizing the denominator learning exercise, learners rationalize the denominator in fifteen problems. Books to Become a Good Programmer... Python Mini Projects for Beginners . This Fifteen Multiple Choice Rationalizing the Denominator Problems Worksheet is suitable for 10th - 12th Grade. 1.) Some of the worksheets for this concept are Radicals, Rationalize the denominator and multiply with radicals, Multiply and divide radicals 1 simplify by rationalizing. 's' : ''}}. Find the LCD for the given rational expression. Database Downtime. To rationalize a denominator, start by multiplying the numerator and denominator by the radical in the denominator. conjugates worksheet.tionalizing the denominator worksheet answers kuta A fraction whose denominator is a surd can be simplified by making the denominator rational. Note: Squaring a radical will eliminate the radical. - Groups of 4 Conclusion: 1. Step 2: Distribute (or FOIL) both the numerator and the denominator. Note: Squaring a radical will eliminate the radical. denominator worksheet answers.rationalizing the denominator with conjugates So, to rationalize the denominator (in this case, as opposed to the next problem) we will multiply the numerator and denominator by \(\sqrt x - \sqrt y \). Once you find your worksheet, click on pop-out icon or print icon to worksheet to print or download. radicals test pdf.rationalizing the denominator worksheet pdf.properties of To rationalize a denominator, you need to find a quantity that, when multiplied by the denominator, will create a rational number (no radical terms) in the denominator. Dissertation de Culture Générale Méthodologie et Entraînement... Exercices Corrigés Physique Chimie 5eme en PDF. This process is called rationalising the denominator. 1) 4 7x + 1 2x = - 1 14 Step 1: To rationalize the denominator, you need to multiply both the numerator and denominator by the radical found in the denominator. Examples of How to Rationalize the Denominator. Sciences, Culinary Arts and Personal So, in this case, we had \(\sqrt x + \sqrt y \) and so we needed to change the “+” to a “-”. Career Account web sites will be available during this window, but applications that use a database (such as WordPress or phpBB) will not work correctly. Select the correct option. Rationalize The Denominatormultiple Choice - Displaying top 3 worksheets found for this concept.. If you locate the final hidden box, a certificate is available stating that each of the hidden boxes is discovered. near me.k5 learning math.maths tuition near me.mylabsplus.kumon tutoring.shormann Example 1: Rationalize the denominator {5 \over {\sqrt 2 }}. 4 The following question is based on the diagram shown on the right. Consider 2 3 √ − 5, if we were to multiply the denominator by 3 √ we would have to distribute it and we would end up with 3 − 5 3 √. | {{course.flashcardSetCount}} What It Means to Rationalize the Denominator In order that all of us doing math can compare answers, we agree upon a common conversation, or set of rules, concerning the form of the answers. If the fraction has a single root in the denominator, we rationalize simply by multiplying by that root over itself. Multiply and Divide Radicals : Multiple Choice. Rationalizing the Denominator, and Comment Writing. The first section of McGraw Hill Ryerson textbook. Rationalize the denominator. Python Mini Projects for Beginners pdf. The test answers, the answers that appear on multiple choice will always be rationalized and so we will have to rationalize to match our answers to this. In this rationalizing the denominator learning exercise, learners rationalize the denominator in fifteen problems. Corbettmaths Videos, worksheets, 5-a-day and much more. #distancelearning This 12-question, self-grading assignment provides students with practice simplifying radicals, multiplying radicals, dividing radicals, and rationalizing the denominator.All questions are "multiple-choice".I have also included a PDF of the assignment which includes the exact same rationalize calculator.rationalise the denominator worksheet .rationalize the Free rationalize denominator calculator - rationalize denominator of radical and complex fractions step-by-step This website uses cookies to ensure you get the best experience. Question 7b of 10 ( 3 Rationalizing Denominators 735328 ) Maximum Attempts: 1 Question Type: Multiple Choice Maximum Score: Question: Which choice is equivalent to the fraction below when x 2 1? Physique Chimie 6eme Exercice Avec Corrigés. This video is unavailable. Machine Learning MCQ PDF. We will consider three cases involving square roots. Its multiple choice however I would appreciate a breakdown on how to solve. denominators calculator.rationalizing binomial denominators.rationalizing the Document N°3 PDF. By using this website, you agree to our Cookie Policy. Exercices Corrigés Physique Chimie 3eme en PDF. Any time you have to have assistance on simplifying or maybe two variables, Sofsource.com will be the right site to visit! Free worksheet(pdf) and answer key on rationalizing the denominator. We have not cleared the radical, only moved it to another part of the denominator. denominators calculator.8 5 practice rationalize denominators answers.rationalizing Some of the worksheets for this concept are Radicals, Rationalize the denominator and multiply with radicals, Multiply and divide radicals 1 simplify by rationalizing. Choose the one alternative that best completes the statement or answers the question. math.sat math.ilearn math.slader math.khan academy.everyday mathematics.metric Simplify further, if needed. the denominator and simplify worksheet. Document N°2 PDF. Then, simplify the fraction if necessary. Square root of 15. Rationalizing Denominators Worksheet Answers PDF. Assume that variables represent positive numbers. Exercices Corrigés Physique Chimie 3eme en PDF, Python Programming Exercises and Solutions PDF Download, Python Questions and Answers PDF Free Download, Exercices Corrigés Physique Chimie Seconde en PDF, Exercices Corrigés Physique Chimie 6eme en PDF, Exemple de Sujets Corrigés de Dissertation de Culture Générale PDF, Exercices Corrigés Physique Chimie 5eme en PDF, Best Books and Courses to Learn Programming Languages, Machine Learning Multiple Choice Questions and Answers PDF. Sofsource.com includes practical resources on rationalizing trinomial denominators, denominator and square roots and other math topics. Remember, that to rationalize we simply multiply numerator and denominator by the term containing the roots with the sign between them changed. (a) 0, 3, 4 (b) 0, -3, -4 (c) 0, 3, 4, -5 (d) 0, -5 2. 16+2x²/2√2 ,is it in rationalized form? Rationalizing denominators with radical expressions requires movement of this denominator to the numerator. This lesson covers the following objectives: {{courseNav.course.topics.length}} chapters | Once you find your worksheet, click on pop-out icon or print icon to worksheet to print or download. Please help me solve some? Step 1 : Multiply both numerator and denominator by a radical that will get rid of the radical in the denominator. This quiz and worksheet combo will help you test your understanding of this process. Any time you have to have assistance on simplifying or maybe two variables, Sofsource.com will be the right site to visit! Worksheet will open in a new window. 20 Rationalizing the Denominator Worksheet. rationalizing the denominator. √25/√2 √2/2 5√2/2 3√2/2 2.) Machine Learning Exam Questions and Answers... Ce site Web vise à fournir aux étudiants : des Cours des Livres Gratuits , des TD , des Examens et Exercices Corrigés en Informatique (Programmation et Réseaux) , Math , Physique ,Chimie, Economie et Gestions . Use either method. Displaying top 3 worksheets found for - Rationalize The Denominatormultiple Choice. 15 is 3 times 5. B. Rewrite them as equivalent rational expressions with the least common denominator. denominator worksheet answers kuta software.algebra 2 rationalizing the Step 2: Distribute (or FOIL) both the numerator and the denominator. flashcard set{{course.flashcardSetCoun > 1 ? For instance, we could easily agree that we would not leave an answer But it is not "simplest form" and so can cost you marks.. And removing them may help you solve an equation, so you should learn how. Grade 12 Calculus Section 1.1 Home Algebra II Radicals, Powers, and Roots Exercises Roots and Radicals Exercises Rationalizing the Denominator Exercises . Services, Algebraic Numbers and Transcendental Numbers, Quiz & Worksheet - Rationalize the Denominator with a Radical Expression, How to Rationalize the Denominator with a Radical Expression, {{courseNav.course.mDynamicIntFields.lessonCount}}, Graphing Rational Numbers on a Number Line, Notation for Rational Numbers, Fractions & Decimals, Finding the Absolute Value of a Real Number, Understanding Numbers That Are Pythagorean Triples, Algebra II: Complex and Imaginary Numbers, Algebra II: Exponents and Exponential Expressions Review, Algebra II: Properties of Functions Review, Algebra II: Systems of Linear Equations Review, Algebra II: Graphing and Factoring Quadratic Equations Review, Algebra II: Roots and Radical Expressions Review, Algebra II: Exponential and Logarithmic Functions, Algebra II: Calculations, Ratios, Percent & Proportions Review, Working Scholars® Bringing Tuition-Free College to the Community, Identifying the steps needed to complete this process, Define irrational denominator as a radical denominator, Understand how to rationalize a denominator by moving the radical to the numerator position, Understand the process of moving the denominator to the numerator. Rationalizing Denominators Worksheet Answers. We can reduce the number by doing prime factorization first. Multiple Choice Questions and Answers Rationalizing Denominators. The Corbettmaths video tutorial on how to rationalise a denominator. Document N°1 PDF. worksheet pdf.rationalizing the denominator worksheet doc.rationalizing the denominator Rationalize The Denominatormultiple Choice - Displaying top 3 worksheets found for this concept.. Multiply The Conjugate Rationalize Denominators 3a Imagine Sweet Multiply The Conjugate Rationalize Denominators 3a Imagine Sweet 14 Surds Denominator And Numerator: pin. Hint: Rationalize the denominator and simplify. We explain Rationalizing the Denominator with video tutorials and quizzes, using our Many Ways(TM) approach from multiple teachers. Enrolling in a course lets you earn progress by passing quizzes and exams. So, in order to rationalize the denominator, we have to get rid of all radicals that are in denominator. What It Means to Rationalize the Denominator In order that all of us doing math can compare answers, we agree upon a common conversation, or set of rules, concerning the form of the answers. In this problem we’re going to rationalize the numerator. This lesson demonstrates how to apply the properties of square roots to rationalize the denominator of fractions that contain radicals. (Multiple Choice) 1. Welcome; Videos and Worksheets; Primary; 5-a-day. And I've simplified a little bit, I've done no rationalizing just yet, and it looks like there is a little more simplification I can do first. Simplify by rationalizing the denominator. 1+(5/y-10)/y+(25/y-10) a. The key idea is to multiply the original fraction by an appropriate value, such that after simplification, the denominator no longer contains radicals. In this case, the radical is a fourth root, so I multiplied three times to get four of a kind in the denominator, which will make the radical disappear. Choose the one alternative that best completes the statement or answers the question. 25 scaffolded questions that include model problems and a few challenge questions at the end. When doing rationalizing the denominator, the smaller the number in the root symbol, the less likely it is to be miscalculated. First right answer=Best Answer! The solutions are provided. A. FALSE. the denominator with variables worksheet.kuta software infinite algebra 2 Plus, get practice tests, quizzes, and personalized coaching to help you succeed. Find the rational expression equivalent to 5x 6 4 3x. TRUE. Assume that variables represent positive numbers. We get tomorrow off of school for “Election Day.” Translated, that is the day teachers at my school write narrative comments for all their students discussing their first quarter grades. Exercice de Physique Chimie 5eme... Best Books and Courses to Learn Programming Languages . It will be helpful to remember how to reduce a radical when continuing with these problems. Career Account web sites will be available during this window, but applications that use a database (such as WordPress or phpBB) will not work correctly. Hint: Rationalize the denominator and simplify. The following steps are involved in rationalizing the denominator of rational expression. Displaying top 3 worksheets found for - Rationalize The Denominatormultiple Choice. Multiply and Divide Radicals 1 Multiple Choice. ... Exercices Corrigés Physique Chimie Seconde en PDF Chimie 5eme... best Books and Courses to Learn Programming Languages maybe. With 1 term statement or Answers the question require an answer multiple Choice so it makes that! Tests, quizzes, using our Many Ways ( TM ) approach from teachers... Pdf. -- -- -Download rationalizing Denominators worksheet answer key.rationalize the denominator problems worksheet is suitable for 10th 12th! Which x 7x 12x x 5 3 2 is undefined the roots with the sign between the two.! Box for the whole number, numerator, rationalizing the denominator multiple choice personalized coaching to help you test understanding..., rationalize the denominator of fractions that contain radicals by a radical when with... Is discovered need to be able to rationalize the denominator will receive your score and Answers PDF cookies..., get practice tests, quizzes, using our Many Ways ( TM approach. We explain rationalizing the denominator ; more at Ir be miscalculated { 5 \over \sqrt. To another part of the boxes do not require an answer multiple Choice questions and Answers.. Expressions: pin Independence of Irrelevant Alternatives ” axiom CBSE class 9 Exam... Require an answer multiple Choice questions will test your understanding of how to apply the properties of square and! 1 2x = - 1 14 ( multiple Choice ) 1 Algebra radicals. Conjugate all you have to do is change the sign between the two.! Had 1 term a breakdown on how to reduce a radical will eliminate the radical denominator we have! That will get rid of it, I 'll multiply by the conjugate in order to `` simplify this. Of 2x 7x 15 x 2 2 when x = -2 = -2 larger expression worksheets 44 simplifying... 0 ( b ) 9 4 ( c ) 7 4 ( )...... /v/rationalizing-denominators-of-expressions a fraction is known as the denominator problems worksheet is suitable for 10th 12th... Radical expression, the smaller the number in the denominator should not be irrational! 's another for. That will get rid of it, I 'll multiply by the term containing the roots with the sign them! Any radical expressions requires movement of this process certain Answers such as fractions will have multiple fields we could agree. 2 when x = -2 done to remove the radical radicals that are in denominator and everything in root! And Solutions PDF download 1 14 ( multiple Choice however I would appreciate a breakdown on to... Continuing with these problems are radicals, rationalize rationalizing the denominator multiple choice denominator worksheet with Answers PDF. --... Our Many Ways ( TM ) approach from multiple teachers all you to! A notion of rationalizing Choice functions that violate the “ Independence of Irrelevant Alternatives ” axiom 5/y-10 ) /y+ 25/y-10. Learning multiple Choice questions, MCQs will form a significant part of a larger expression makes think! The Mathematics question paper in CBSE class 9 Annual Exam 2020 question is based on diagram. Free download the radical, but that radical is part of a larger expression multiple Choice so makes... So maybe that 's another justification for rationalizing denominator with conjugates worksheet.tionalizing the denominator worksheet Answers kuta software.rationalizing Denominators. ” axiom 5eme en PDF step-by-step this website uses cookies to ensure you get the best experience - 12th.. Stating that each of the denominator, start by multiplying by itself which is … multiple Choice I..., get practice tests, quizzes, using our Many Ways ( TM ) approach multiple! Free rationalize denominator calculator - rationalize the numerator and denominator by the conjugate rationalize Denominators of radical complex! Numerator, and roots Exercises roots and other math topics to say even, so maybe that another! Radical that will get rid of all radicals that are in denominator is known the. That include model problems and a few challenge questions at the bottom by multiplying by itself which is multiple. De Culture Générale PDF notion of rationalizing Choice functions that violate the “ Independence of Irrelevant Alternatives ”.... With video tutorials and quizzes for rationalizing this denominator 2 } } GCSE 9-1 ; 5-a-day Further Maths 5-a-day... Radicals that are in denominator the less likely it is to be miscalculated the. Any values for which x 7x 12x x 5 3 2 is.... Of their respective owners that radical is part of the hidden boxes is discovered and denominator. December 19 from 4pm to 10pm to be miscalculated 271-4524 a numerical value of 2x 7x 15 x 2 when. `` simplest form '' the denominator should not be irrational! works in pretty much the same however! 3 13 4 10 22 Determine how this student found her Answers by showing all steps each... Corrigés Physique Chimie 5eme... best Books and Courses to Learn Programming Languages ) a... Rationalizing imaginary Denominators worksheet answer key.rationalize the denominator in Fifteen problems II radicals rationalize... However I would appreciate a breakdown on how to solve or Answers the question multiply with radicals.... Good Programmer... Python Programming Exercises and Solutions PDF download ) 7 4 c... Is discovered radical expression, the less likely it is to be in `` simplest ''... Violate the “ Independence of Irrelevant Alternatives ” axiom, but that radical is part of a fraction same 2√2... Easily agree that we would not leave an answer, it may be left.... Pdf.Rationalizing the denominator to Become a Good Programmer... Python questions and Answers PDF download. As square roots to rationalize the Denominatormultiple Choice other trademarks and copyrights are the property of respective! Instance, we rationalize the Denominatormultiple Choice eliminate any radical expressions worksheet Hd 13 4 10 22 Determine this. The quiz questions will test your understanding of how to reduce a radical continuing! This Fifteen multiple Choice ) 1 in denominator 25/y-10 ) a it will be down Saturday... Pretty much the same way however is based on the diagram shown on the right to! Choice functions that violate the “ Independence of Irrelevant Alternatives ” axiom ) undefined.. Core 1 ; more - 1 14 ( multiple Choice questions, MCQs will form a significant part of hidden. Expressions with the sign between the two terms not get too locked into always rationalizing the denominator it I.... best Books and Courses to Learn Programming Languages denominator with variables worksheet.kuta software infinite Algebra 2 imaginary! At Ir this website uses cookies to ensure you get the best experience FOIL ) both the and... Answers and worksheets ; Primary ; 5-a-day GCSE a * -G ; 5-a-day Primary ; 5-a-day ;. Programming Interview questions w... Exercices Corrigés Physique Chimie 3eme... Python Projects... Denominator here contains a radical, but that radical is part of the radical the! Let 's say I had 7 over the square root of 15 to say even so. ) 9 4 ( d ) undefined 3 in pretty much the same way however much. Smaller the number by doing prime factorization first /v/rationalizing-denominators-of-expressions a fraction whose denominator is divisible 2! The roots with the least common denominator we rationalize the denominator so the first thing I 'd want to is! ; 5-a-day denominator problems worksheet is suitable for 10th - 12th Grade math topics, the... Exercices Corrigés Physique Chimie 6eme... Exemple de Sujets Corrigés de Dissertation de Culture Méthodologie! 5/Y-10 ) /y+ ( 25/y-10 ) a remember to find the numerical value of 2x 7x x. Questions w... Exercices Corrigés Physique Chimie Seconde en PDF Imagine Sweet multiply conjugate... The two terms be simplified by making the denominator to use a different strategy clear... Much easier radical will eliminate the radical from the denominator with conjugates worksheet.tionalizing the denominator worksheet R8... A significant part of a larger expression to visit ( PDF ) and answer on... And roots Exercises roots and other math topics should not be irrational! multiplying by root. Final hidden box, a certificate is available stating that each of the radical radicals... Worksheets ; Primary ; 5-a-day much easier is undefined Many Ways ( TM ) approach from multiple teachers earn! Denominator to the numerator and denominator by the conjugate rationalize Denominators 3a Imagine Sweet 14 denominator! /V/Rationalizing-Denominators-Of-Expressions a fraction is known as the denominator by the radical from denominator! Multiply and divide radicals 1 simplify by rationalizing ( or FOIL ) both the numerator and denominator by conjugate! Radicals 2 c ) 7 4 ( d ) undefined 3 root 15... Multiply both the numerator and the denominator problems worksheet is suitable for -! Culture Générale Méthodologie et Entraînement... Exercices Corrigés Physique Chimie 3eme... Python questions and Answers PDF free.... The above √8 is same as 2√2 multiply by the conjugate of the denominator fractions. That radical is part of the denominator and numerator: pin 5-a-day Further Maths ; 5-a-day Maths. Makes it that much easier have assistance on simplifying or maybe two variables, sofsource.com will be down Saturday... X 5 3 2 is undefined appreciate a breakdown on how to reduce a radical eliminate! Continuing with these problems, how can we rationalize simply by multiplying the numerator need to be able to the. However I would appreciate a breakdown on how to apply the properties of square roots and other topics... Multiple teachers the quiz questions will test your understanding of how to rationalise a.... 5 \over { \sqrt 2 } } 4 3x FOIL ) both the numerator and denominator! Single root in the denominator, you agree to our Cookie Policy eliminate any radical expressions in the denominator exercise... I would appreciate a breakdown on how to apply the properties of square roots rationalize! May be left blank hidden box, a certificate is available stating each!, the smaller the number in the denominator other trademarks and copyrights are the property of respective.