In this case, first factor all numerators and denominators completely. Students have a quick review with example problems and then they are given 30 practice problems to work on. Exercise 2 b. 7.NS.A.3 Solve real-, These Operations with Rational Numbers notes and practice are differentiated based on some common needs found in the middle school math classroom. Division - When it comes down to dividing the rational numbers. These worksheets are fantastic since you can download them for free and customize them exactly as you want them. Multiplying rational expressions is performed in a similar manner. enable students to apply multiple concepts simultaneously. \\ &=\dfrac{1}{x-5} \end{aligned}\). \(\dfrac{12 x^{2}}{5 y^{3}} \cdot \dfrac{20 y^{4}}{6 x^{3}}\). Multiplying and Dividing Rational Numbers. Worksheet Multiplying and Dividing Rational Numbers Use this seventh-grade math worksheet to provide students with practice multiplying and dividing rational numbers, including positive and negative fractions, decimals, and mixed numbers. 2. 1. Quiz 1: 5 questions Practice what you've learned, and level up on the above skills. If the two numbers (dividend and divisor) have the same sign, their quotient is positive. xoP(g"Ff~2# ,i%GH~[* This batch of printable decimal word problem. This method won't cause the child stress about forgetting how to divide the big number or making errors on their tables of times. jrYcm. If you're seeing this message, it means we're having trouble loading external resources on our website. >> Additionally, every fraction that has an integer denominator and numerator and a denominator that is not zero falls into the category of rational numbers. We send out a monthly email of all our new free worksheets. Students will love this mystery puzzle as they add, subtract, multiply and divide positive and negative fractions. The problem solving here will require multiple steps to make students think in multiple directions. What are common strategies to multiply and divide rational numbers? Step 2) Treat it like a multiplication problem: 1/2 x 6/1 = (1 x 2) / (6 x 1) = 6/2 = 3. If you have any questions or comments, please let us know. 6.NS.A.1 Solve problems involving division of fractions. Common examples of rational numbers include: 5 = 5/1, 1.75 = 7/4, 0.001 = 1/1000, 0.111= 1/9. If you are trying to find Multiplication And Division Of Rational Numbers Worksheet, you are arriving at the right site. If you need a version without decimals and fractions please visit my Integer Operations Paper Chain ActivityThe product is designed for a 7th Grade classroom. \(\begin{aligned} \dfrac{12 x^{2}}{5 y^{3}} \cdot \dfrac{20 y^{4}}{6 x^{3}}&=\dfrac{240x^{2}y^{4}}{30x^{3}y^{3}}\qquad\quad\:\:\color{Cerulean}{Multiply.} Dividing Rational Expressions To divide two fractions, we multiply by the reciprocal of the divisor, as illustrated: 5 8 1 2 = 5 8 2 1 = 5 1 2 8 4 1 = 5 4 Dividing rational expressions is performed in a similar manner. These problems can be solved by employing DMAS (Division, Multiplication, Addition & Subtraction) order of operation. Each worksheet is designed with a set difficulty level of questions ranging from easy to hard. Just as we do with fractions, think of the divisor \((2x3)\) as an algebraic fraction over 1. Answer questions on things like dividing and multiplying. Pre-made digital activities. 9GtpL{ This procedure could aid in the Kingdom's unending expansion. This resource can be used as an independent assignment, group work, stations around the room, end of the unit assessment or as an enrichment extension, dependi. When you multiply 2.5 by the power of 10 and you'll notice patterns. Use this seventh-grade math worksheet to provide students with practice multiplying and dividing rational numbers, including positive and negative fractions, decimals, and mixed numbers. Worksheets #1 - #4 have 19 multiple-choice questions (4-5 per page) and worksheet #5 has 12 practice problems. Step 1) Find the reciprocal of the second fraction. are designed with problems covering a variety of concepts on multiplication and division operation on rational numbers such as: These worksheets include problems that require a student to multiply integers with the same sign and different signs. Then multiply and cancel. /F9 9 0 R Teach and practice rational number multiplication and division (positive and negative integers, decimals, and fractions) with this pack! mu`('-#T "_lr'gv>.pI\bfZ[1m$g~NKk*=[`tFxI.0*/n^UQ Quizzes with auto-grading, and real-time student data. Product includes the following eight worksheets with answer keys:ADDING decimalsADDING fractionsSUBTACTING decimalsSUBTRACTING fractionsMULTIPLYING decimalsMULTIPLYING fractionsDIVIDING decimalsDIVIDING fractionsInterested in more rational number operations and integers, Here is a fun way for students to practice multiplying and dividing negative fractions (rational numbers. by kunalrohira. After doing so, factor and cancel. >> For all ages, children to adults. There are two methods to divide huge numbers into powers of 10. Then group the factors so that the fractions are equal to one. Web division of rational numbers the product of a rational number with its reciprocal is always 1. Students solve 15 problems, then find their answer to color pixels in the grid, eventually revealing a mystery, Mazes are a great way to practice new material while having more fun than completing a typical worksheet! 1. Dividing a number by another will produce "ten/five," while divising two by two gives the same result. >> When we are using these operations with these types of numbers there are simple patterns and almost tricks that we can use to get an accurate and reliable product or quotient. So, a rational number looks like this, in variable form: P/ Q. \\ &=\dfrac{1}{x+3} \end{aligned}\). CCSS: The restrictions to the domain of a quotient consist of the restrictions to the domain of each rational expression as well as the restrictions on the reciprocal of the divisor. Hook students into the lesson by asking them to model problems involving the multiplication and division of rational numbers using the number line. This practice worksheet challenges learners by progressing from multiplying or dividing rational numbers that have the same form, to multiplying or dividing rational numbers that have different forms. Grade/level: Year 5. by MsRhian. In the final class activity, students will be given . \((fg)(x)=\dfrac{x2}{3x+2}; x2, \dfrac{2}{3}\), 5. Calculate \((f/g)(x)\) and determine the restrictions. A large class might be divided into five groups. Divide the absolute values of the two rational numbers. Multiplication and division of rational numbers worksheet that incorporates application, reflection, and problem-solving with a challenge to aid students in creating and resolving their word problems. We can simplify this rational number by dividing both the numerator and the denominator by 12: As a result, students will: Web The Following Are The Steps To Solve The Division Of Rational Numbers Problems. \((f/g)(x)=\dfrac{(x+4)}{(x1)}; x\dfrac{1}{3}, 1\), Exercise \(\PageIndex{7}\) Discussion Board Topics. The first technique relies on the mental process of computation. Try to solve the questions from the Multiplication of Rational Numbers Worksheet at first and assess your preparation standards. Use these as review, center stations, or even homework. Often student uses some pictorial or visual aids that can help them understanding the problem much better. These worksheets utilize a standard algorithm as well as step-by-step instructions. Multiplication And Division Of Rational Numbers Worksheet - Utilize division worksheets to help your child learn and refresh division concepts. Please contact me, to let me know. It encourages others to reach for the lost and to seek out new leadership to guide the way. 6 0 obj Solution: Apply the product rule for radicals, and then simplify. Students love to move around and will be more focused than you can imagine! Includes negative numbers. Recall that multiplication and division are to be performed in the order they appear from left to right. ]{byQ ;\I]W3{\wgw2=axA9\x[=hg]]VoADY^Q$:E\DAC]sW&h,2EBQ9. Add highlights, virtual manipulatives, and more. The key with multiplication is to align the numerators and denominators. step algebra sheets. Multiplying and dividing rational numbers is simple as long as you remember the steps and are familiar with the rule. Google Docs. School subject: Math. Quiz 2: 5 questions Practice what you've learned, and level up . Students will use the additive inverse property, number line models, and the algorithm to solve rational number problems.Students will practice with both skill-based problems, real-world application questions, and error analysis to support higher level thinking skills. ID: 2553622. /Parent 3 0 R Test your knowledge of multiplication and division with regard to rational numbers via this worksheet and attached quiz. Improve your math knowledge with free questions in "Multiply and divide rational numbers" and thousands of other math skills. Multiplication and Division of Rational Numbers - These worksheets are designed to enhance the understanding of mathematical operations with a special focus on multiplication and division. The first is by moving the decimal left and multiplying it by 10-1. Typically, rational expressions will not be given in factored form. Students will then solve problems involving rational numbers. are designed to not only improve a students problem solving skills, but also to help strengthen the fundamentals of arithmetic operations on rational numbers with different signs. Students solve and check one-step multiplication and division equations. Each quiz contains ten questions of varying levels. ={A}`M{:]'NIG-j]niF![qm=hgrrzD{k4z\p{#[=x 26Pr9,M{H*gEIW[}9tz^@=R'z%v[ogK&kwr|7)Y\A^"y|wJM{k^wnl,&#w2{}zxk H"UeniOLnio5VB j@^pN7odg5H2[qCD{BTDh#^5$s%4j7 ^{/ B^z@ubv }[2!j It is important to remember that there exist two different kinds of numbers when you divide numbers: divisors and the Quotient. The trinomial \(2x^{2}+x+3\) in the numerator has a negative leading coefficient. Test prep. This maze consists of, This paper chain activity is designed to strengthen students skills in Multiplying, Dividing, Adding and Subtracting Fractions and Decimals. B y GAwlol i Zr cizgzh4t PsC Sr9eAsXernv JeUd1.1 L ZM Uasdce0 Tw7iutjh t rI Mnhf7i fn Hihtpe R YA2l lg weKbBr9ah M1d. << Large numbers that are divided can be practiced by pupils using a variety of worksheets and exercises. You'll discover them on receipts, price tags and food labels. The problems here include questions on finding products and quotients with rational numbers of the same sign as well as different signs. With these timed worksheets a student can also compare scores and time spent on the worksheets with classmates and challenge them as well. The Rules for Dividing Rational Numbers are the same as the Rules for Dividing Integers: 1. [/Pattern /DeviceRGB] Grade/level: 5TH GRADE. Ms Gemmells Math. When multiplying fractions, we can multiply the numerators and denominators together and then reduce, as illustrated: \(\dfrac{3}{5} \cdot \dfrac{5}{9}=\dfrac{3 \cdot 5}{5 \cdot 9}=\dfrac{\color{Cerulean}{\stackrel{1}{\cancel{\color{black}{3}}}}\color{black}{\cdot}\color{Cerulean}{\stackrel{1}{\cancel{\color{black}{5}}}}}{\color{Cerulean}{\stackrel{\cancel{\color{black}{5}}}{1}}\color{black}{\cdot}\color{Cerulean}{\stackrel{\cancel{\color{black}{9}}}{3}}}\color{black}{=\dfrac{1}{3} }\). by suparnas. Such problems include multiplication of rational numbers with the same and different signs. \(\dfrac{7-x}{x^{2}+3 x} \cdot \dfrac{x^{2}+10 x+21}{x^{2}-49}\). TEKS: 6.3E Multiply and divide positive rational numbers fluently. This activity focuses on adding, subtracting, multiplying, and dividing both positive and negative fractions and decimals. The information on this site is for informational and educational purposes only. 10 0 obj Multiplying and Dividing Rational Numbers Worksheet - 1, Multiplying and Dividing Rational Numbers Worksheet - 2, Multiplying and Dividing Rational Numbers Worksheet - 3, Multiplying and Dividing Rational Numbers Worksheet - 4. To divide rational expressions, multiply by the reciprocal of the divisor. In other words, do not apply the distributive property. Find what you need about Multiplication And Division Of Rational Numbers Worksheet down below. A rational number is a number that can be written as a normal fraction of two integers, like a ratio. The complex ones are word problems presenting a real life scenario which requires a student to model the scenario into an equation to arrive at the correct answer. 5 . Use this math worksheet to learn how to solve rational numbers by multiplication and division. If you need a rational numbers version that includes decimals and fractions please visit my Operations with Rational Numbers Paper Chain ActivityThe product is designed for a 6th or 7th Grade class, Multiply Fractions by Whole Numbers with this Mystery PictureAligned to Texas TEK 6.3E multiply and divide positive rational numbers fluentlyTest the quality of these Mystery Pictures by downloading this FREE Add Integers: Mystery Picture (Emoji)Are you ready for a HUGE upgrade from boring math worksheets? In this activity, students answer real-life questions related to a road trip with three of their best friends. Language: English. your email. Part of the TEKS quiz series, available for all 7th and 8th grade math TEKS. Each worksheet is designed with a set difficulty level of questions ranging from easy to hard. <> The students are also encouraged to find the quickest way to solve timed questions like these. They will need to know how to add, subtract, multiply, and divide both decimals and fractions. They also have a sound understanding of multiplication and division operations. Divide big numbers into smaller numbers is the fundamental operation of division. Quizzes are formatted to look like the STAAR test, including some griddable answers. Thus, the students are able to think in new ways and enhance their critical thinking skills by relating the unknown values with the known values in a question. Modifications are considered for both struggling learners and high flyers. Positive numbers only are used. by AIMS_INDIA. They practice with their integer operations while touching on topics such as debt, initial investments, and simple interest. /Annots 13 0 R If you or your students have trouble with these, review the single Dividing rational expressions is performed in a similar manner. The worksheets can be interactive and contain activities and questions that focus on division of multidigit integers. the problems will help deepen a students understanding of the rules to multiply and divide rational numbers. It is possible to quickly express large numbers by using the scientific notation. \\ &=\dfrac{5xy^{2}}{x+3} \end{aligned}\). /CSp /DeviceRGB are based on the ability of students to use the information given in the question and draw connections to arrive at the correct answer. These worksheets utilize a standard algorithm as well as step-bystep instructions. endobj @qYTN||-7+a2)b+|9/"vB;YO$wf'aCaqGTaGT5m!n=-U?DcQ*PN5wM#PqYs}h8`wj e=tZ2?6ElRqY4WPV9>wWGD@j5B3fkGd These worksheets from BYJUS Math are absolutely free to use and you do not require a subscription for the same. Answer keys are incl, This resource includes 6 RATIONAL NUMBERS practice worksheets (60 questions). This is the perfect resource for students to practice adding and subtracting rational numbers. /Pattern << of multiplication and division operations. M-@# First, multiply by the reciprocal of the divisor and then cancel. I'm getting a little older these days and my eyes are going. Multiplication and Division of Rational Numbers - These worksheets are prepared to show the importance of mathematical operations, such as multiplication and division of rational numbers. Awards. Once you've mastered the associative component of powers ten, it is possible to break large numbers down into smaller powers of 10. Worksheets for hundredths are also available, which can be useful for learning to divide large amounts of money. Keep in mind that 1 is always a factor; so when the entire numerator cancels out, make sure to write the factor 1. >> (Assume all denominators are nonzero. These worksheets and lesson will progress towards understanding the use of these operations in great detail. Lesson 6: Multiplying and dividing rational expressions, start fraction, start fraction, 4, x, minus, 24, divided by, minus, 3, x, minus, 15, end fraction, divided by, start fraction, x, squared, minus, 36, divided by, x, plus, 5, end fraction, end fraction, Multiplying and dividing rational expressions. These worksheets ensure a smooth transition from easy and direct problems to complex and challenging questions. Two Step Division or Multiplication & Addition or Subtraction Mixed Numbers Worksheet Math: This product includes Two Step problems worksheets with focus on above mentioned aspect. Students are led through steps to multiply and divide integers and rational numbers is this 15 page bundle! /MediaBox [0 0 612 792] ** Each page has 10 practice questions. When using scientific notation large numbers must be written with positive exponents. With the use of visuals, the online 7th grade worksheets make learning engaging. Factor the denominator \(x^{2}25\) as a difference of squares. /CA 1.0 Once students have understood the steps involved, long division should be a common thing for them. Use this math worksheet to solve and identify the rational number. Printable PDF, Google Slides & Easel by TPT Versions are included in this distance learning ready activity which consists of an activity where students will have to multiply and divide positive and negative rational numbers to work their way through a maze.Important InformationNot all boxes are used in the maze to prevent students from just figuring out the correct route. Here the students will learn how to relate mathematical operations to all these types of numbers that belong to the set of rational numbers. So 7/8 becomes 8/7 and the division. Multiplying and Dividing Rational Numbers Worksheets Regular numbers are multiplied and divided in the same way as number fractions, and we will learn them through multiplying and dividing rational numbers worksheets. Multiplying and dividing by 10 100 and 1000. The values that give a value of 0 in the denominator are the restrictions. Please view the preview to see sample questions to make sure this, Students will practice adding, subtracting, multiplying, and dividing both positive and negative decimals in the context of real scenario involving income and expenses. |, Next review date: >> It is easy to mix and match sets to meet your individual standards.Topics covered in this product: Adding Integers with the Same Sign*Adding Integers with Different Signs*Subtracting Integers*Multipl, Worksheets for ALL 6th Grade Common Core Math Standards:These 58 worksheets thoroughly cover all 6th grade Math Common Core Standards. }mC Sam remembers that to divide rational numbers, he can actually turn this problem into a multiplication problem by flipping the second rational number. Wyzant is IXL's tutoring network and features thousands of tutors who can help with math, writing, science, languages, music, hobbies, and almost anything else you can imagine. Students will have to successfully multiply/divide rational numbers 15 times to complete the maze. /SMask /None>> Example 1: Ryan's friends ate from his pizza and. /CSpg /DeviceGray by. Thank you for visiting our website and searching for Multiplication And Division Of Rational Numbers Worksheet. /Creator () Please look at the preview file to see if this resource is appropriate for your students. Our printable absolute value worksheets meticulously designed for 6th grade and 7th grade students include exercises like finding the absolute value of positive and negative integers, performing simple addition, subtraction, multiplication and division involving the absolute value of real numbers and more. Leave the product in factored form and cancel the common factors. Examples, solutions, videos, and worksheets to help Grade 6 students learn how to multiply and divide rational numbers. As a shortcut, we can cancel factors . When multiplying radical expressions with the same index, we use the product rule for radicals. How did the ancient Egyptians use fractions? It is a best practice to leave the final answer in factored form. >> Free printable multiplying and dividing rational number worksheets for the 7th grade are a great tool to improve problem solving abilities and to get students ready for assessments at school level. Students solve and check one-step addition and subtraction equations. To know more you can always look up our Rational Numbers Worksheets and Clarify your doubts then and there itself. >> =ZF> zD{'}=s {o4D{CC@ l ; '= DkphoFOmlW7X;ezj~-] }!=Plrb6b;6Fb@1]+-P\f#zY{3^~CTsw&i=Yojs9Hq=gj"vM:wLT@v >> A reflection section is included at the end of this worksheet to encourage learners to evaluate their thoughts (metacognition) and how they did during the lesson. /Contents 10 0 R What is a "Multiplication and division of rational numbers worksheet"? The mental process of dividing large numbers into powers of ten is another method. Once you are aware of this principle, you can use it to solve any problem including the most difficult ones. The activity is self checking. A rational number is a value that is created by dividing two integers, where the denominator is not equal to zero. 4 0 obj Given real numbers nA and nB, nA nB = nA B \. To divide two fractions, we multiply by the reciprocal of the divisor, as illustrated: \(\dfrac{5}{8} \div \color{OliveGreen}{ \dfrac{1}{2}}\color{black}{=}\dfrac{5}{8} \cdot \color{OliveGreen}{\dfrac{2}{1}}\color{black}{=}\dfrac{5 \cdot \color{Cerulean}{\stackrel{1}{\cancel{\color{black}{2}}}}}{\color{Cerulean}{\stackrel{\cancel{\color{black}{8}}}{4}} \color{black}{\cdot 1}}=\dfrac{5}{4}\). If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked. 03/16/2023 In the history of fractions, who is credited for the first use of the fraction bar? This in turn lays a firm foundation for them to easily solve real life problems. It's a useful activity to promote creativity and innovative thinking. We replace \(7x\) with \(1(x7)\) so that we can cancel this factor. \(\begin{aligned} \dfrac{15 x^{2} y^{3}}{(2 x-1)} \cdot \dfrac{x(2 x-1)}{3 x^{2} y(x+3)}&=\dfrac{15 x^{3} y^{3}(2 x-1)}{3 x^{2} y(2 x-1)(x+3)}\qquad\qquad\color{Cerulean}{Multiply.} Decimals are an essential part of shopping. Solve "Number Sequences Study Guide" PDF, question bank 10 to . The product and quotient of two rational functions can be simplified using the techniques described in this section. /AIS false We provide a 10-item activity to help you practice the topic. /GSa 4 0 R For EACH scavenger hunt, students answer the questions you have hung around your classroom and then "hunt" for the correct letter on another sign. this level. All debits and credits are organized by students in a table. A seventh grader learns the strategies and methods to multiply and divide rational numbers. \\ &=\dfrac{\color{Cerulean}{\stackrel{5}{\cancel{\color{black}{15}}}}\color{Cerulean}{\stackrel{x}{\cancel{\color{black}{x^{3}}}}}\color{Cerulean}{\stackrel{y^{2}}{\cancel{\color{black}{y^{3}}}}}\color{Cerulean}{\stackrel{1}{\cancel{\color{black}{(2x-1)}}}}}{\color{Cerulean}{\cancel{\color{black}{3}}\cancel{\color{black}{x^{2}}}\cancel{\color{black}{y}}\cancel{\color{black}{(2x-1)}}}\color{black}{(x+3)}}\qquad\color{Cerulean}{Cancel.} /SA true ID: 11502 Language: English School subject: Math Grade/level: Grades 6, 7, 8 Age: 11-15 Main content: Rational numbers Other contents: Vocabulary Add to my workbooks (34) Download file pdf Embed in my website or blog Add to Google Classroom /CreationDate (D:20200316235512) Rational numbers involve main arithmetic operations such as addition, subtraction, multiplication, and division. /F8 8 0 R : "Apply and extend previous understandings of addition and subtraction to add and subtract rational numbers; represent addition and subtraction on a horizontal or vertical number line diagram." These worksheets can help students practice this Common Core State Standards skill. % We placed the numerator together as a product over the product of the denominators. \\ &=\dfrac{6x^{4}}{25y^{3}z^{3}} \end{aligned}\), \(\dfrac{x+2}{x^{2}-4} \div \dfrac{x+3}{x-2}\). 3. The 7th grade multiplying and dividing rational numbers worksheets focus on enhancing the ability of a student to perform these operations on integers and rational numbers. /Pages 3 0 R Rational Numbers (Multiplication) by kaboodle. To help your child develop their mathematical skills it is possible to download worksheets online , or print them on your computer. 3. Although it appears good on paper, many small group facilitators detest this procedure. The multiplying and dividing rational numbers worksheets follow a step-by-step learning process that helps students better understand concepts, recognize mistakes, and possibly develop a strategy to tackle future problems. \(\dfrac{2 x}{3} \cdot \dfrac{9}{4 x^{2}}\), \(-\dfrac{5 x}{3 y} \cdot \dfrac{y^{2}}{25 x}\), \(\dfrac{5 x^{2}}{2 y} \cdot \dfrac{4 y^{2}}{15 x^{3}}\), \(\dfrac{16 a^{4}}{7 b^{2}} \cdot \dfrac{49 b^{3}}{2 a^{3}}\), \(\dfrac{x-6}{12 x^{3}} \cdot \dfrac{24 x^{2}}{x-6}\), \(\dfrac{x+10}{2x1}\cdot\dfrac{x2}{x+10}\), \(\dfrac{(y-1)^{2}}{y+1} \cdot \dfrac{1}{y-1}\), \(\dfrac{y^{2}-9}{y+3} \cdot \dfrac{2 y-3}{y-3}\), \(\dfrac{2 a-5}{a-5} \cdot \dfrac{2 a+5}{4 a^{2}-25}\), \(\dfrac{2 a^{2}-9 a+4}{a^{2}-16} \cdot\left(a^{2}+4 a\right)\), \(\dfrac{2 x^{2}+3 x-2}{(2 x-1)^{2}} \cdot \dfrac{2 x}{x+2}\), \(\dfrac{9x^{2}+19x+2}{4x^{2}}\cdot\dfrac{x^{2}4x+4}{9x^{2}8x1}\), \(\dfrac{x^{2}+8x+16}{16x^{2}}\cdot\dfrac{x^{2}3x4}{x^{2}+5x+4}\), \(\dfrac{x^{2}x2}{x^{2}+8x+7}\cdot\dfrac{x^{2}+2x15}{x^{2}5x+6}\), \(\dfrac{x+1}{x3}\cdot\dfrac{3x}{x+5}\), \(\dfrac{2x1}{x1}\cdot\dfrac{x+6}{12x}\), \(\dfrac{100-y^{2}}{y-10} \cdot \dfrac{25 y^{2}}{y+10}\), \(\dfrac{3 y^{3}}{6 y-5} \cdot \dfrac{36 y^{2}-25}{5+6 y}\), \(\dfrac{3 a^{2}+14 a-5}{a^{2}+1} \cdot \dfrac{3 a+1}{1-9 a^{2}}\), \(\dfrac{4a^{2}16a}{4a1}\cdot\dfrac{116a^{2}}{4a^{2}15a4}\), \(\dfrac{x+9}{-x^{2}+14 x-45} \cdot\left(x^{2}-81\right) \), \(\dfrac{1}{2+5 x} \cdot\left(25 x^{2}+20 x+4\right)\), \(\dfrac{x^{2}+x6}{3x^{2}+15x+18}\cdot\dfrac{2x^{2}8}{x^{2}4x+4}\), \(\dfrac{5x^{2}4x1}{5x^{2}6x+1}\cdot\dfrac{25x^{2}10x+1}{375x^{2}}\), \(\dfrac{5 x}{8} \div \dfrac{15 x^{2}}{4}\), \(\dfrac{3}{8 y} \div \dfrac{15}{2 y^{2}}\), \(\dfrac{\dfrac{5 x^{9}}{3 y^{3}}}{\dfrac{25 x^{10}}{9 y^{5}}}\), \(\dfrac{\dfrac{12 x^{4} y^{2}}{21 z^{5}}}{\dfrac{6 x^{3} y^{2}}{7 z^{3}}}\), \(\dfrac{(x-4)^{2}}{30 x^{4}} \div \dfrac{x-4}{15 x}\), \(\dfrac{5 y^{4}}{10(3 y-5)^{2}} \div \dfrac{10 y^{5}}{2(3 y-5)^{3}}\), \(\dfrac{(a-8)^{2}}{2 a^{2}+10 a} \div \dfrac{a-8}{a}\), \(\dfrac{2}{4 a^{2} b^{3}(a-2 b)} \div 12 a b(a-2 b)^{5}\), \(\dfrac{x^{2}+7 x+10}{x^{2}+4 x+4} \div \dfrac{1}{x^{2}-4}\), \(\dfrac{2 x^{2}-x-1}{2 x^{2}-3 x+1} \div \dfrac{1}{4 x^{2}-1}\), \(\dfrac{y+1}{y^{2}-3 y} \div \dfrac{y^{2}-1}{y^{2}-6 y+9}\), \(\dfrac{9-a^{2}}{a^{2}-8 a+15} \div \dfrac{2 a^{2}-10 a}{a^{2}-10 a+25}\), \(\dfrac{a^{2}-3 a-18}{2 a^{2}-11 a-6} \div \dfrac{a^{2}+a-6}{2 a^{2}-a-1}\), \(\dfrac{y^{2}-7 y+10}{y^{2}+5 y-14} \div \dfrac{2 y^{2}-9 y-5}{y^{2}+14 y+49}\), \(\dfrac{6 y^{2}+y-1}{4 y^{2}+4 y+1} \div \dfrac{3 y^{2}+2 y-1}{2 y^{2}-7 y-4}\), \(\dfrac{x^{2}7x18}{x^{2}+8x+12}\div\dfrac{x^{2}81}{x^{2}+12x+36}\), \(\dfrac{4a^{2}b^{2}}{b+2a}\div (b2a)^{2}\), \(\dfrac{x^{2}y^{2}}{y+x}\div (yx)^{2}\), \(\dfrac{5 y^{2}(y-3)}{4 x^{3}} \div \dfrac{25 y(3-y)}{2 x^{2}}\), \(\dfrac{15 x^{3}}{3(y+7)} \div \dfrac{25 x^{6}}{9(7+y)^{2}} \), \(\dfrac{3 x+4}{x-8} \div \dfrac{7 x}{8-x}\), \(\dfrac{3x2}{2x+1}\div \dfrac{23x}{3x}\), \(\dfrac{(7 x-1)^{2}}{4 x+1} \div \dfrac{28 x^{2}-11 x+1}{1-4 x}\), \(\dfrac{4 x}{(x+2)^{2}} \div \dfrac{2-x}{x^{2}-4}\), \(\dfrac{(a2b)^{2}}{2b}\div (2b^{2}+aba^{2})\), \(\dfrac{x^{2}6x+9}{x^{2}+7x+12}\div \dfrac{9x^{2}}{x^{2}+8x+16}\), \(\dfrac{2x^{2}9x5}{25x^{2}}\div \dfrac{14x+4x^{2}}{2x^{2}9x+5}\), \(\dfrac{3x^{2}16x+5}{1004x^{2}}\div\dfrac{9x^{2}6x+1}{3x^{2}+14x5}\), \(\dfrac{10x^{2}25x15}{x^{2}6x+9}\div\dfrac{9x^{2}}{x^{2}+6x+9}\), \(\dfrac{1}{x^{2}} \cdot \dfrac{x-1}{x+3} \div \dfrac{x-1}{x^{3}}\), \(\dfrac{x7}{x+9}\cdot\dfrac{1}{x^{3}}\div\dfrac{x7}{x}\), \(\dfrac{x+1}{x2}\div\dfrac{x}{x5}\cdot\dfrac{x^{2}}{x+1}\), \(\dfrac{x+4}{2x+5}\div \dfrac{x3}{2x+5}\cdot\dfrac{x+4}{x3}\), \(\dfrac{2x1}{x+1}\div\dfrac{x4}{x^{2}+1}\cdot\dfrac{x4}{2x1}\), \(\dfrac{4x^{2}1}{3x+2}\div\dfrac{2x1}{x+5}\cdot\dfrac{3x+2}{2x+1}\), \(f(x)=\dfrac{1}{x}\) and \(g(x)=\dfrac{1}{x1}\), \(f(x)=\dfrac{x+1}{x1}\) and \(g(x)=x^{2}1\), \(f(x)=\dfrac{3x+2}{x+2}\) and \(g(x)=\dfrac{x^{2}4}{(3x+2)^{2}}\), \(f(x)=\dfrac{(13x)}{2x6}\) and \(g(x)=\dfrac{(x6)^{2}}{9x^{2}1}\), \(f(x)=\dfrac{25x^{2}1}{x^{2}+6x+9}\) and \(g(x)=\dfrac{x^{2}9}{5x+1}\), \(f(x)=\dfrac{x^{2}49}{2x^{2}+13x7}\) and \(g(x)=\dfrac{4x^{2}4x+1}{7x}\), \(f(x)=\dfrac{1}{x}\) and \(g(x)=\dfrac{x2}{x1}\), \(f(x)=\dfrac{(5x+3)^{2}}{x^{2}}\) and \(g(x)=\dfrac{5x+3}{6x}\), \(f(x)=\dfrac{5x}{(x8)^{2}}\) and \(g(x)=\dfrac{x^{2}2}{5x8}\), \(f(x)=\dfrac{x^{2}2x1}{5x^{2}3x10}\) and \(g(x)=\dfrac{2x^{2}5x3}{x^{2}7x+12}\), \(f(x)=\dfrac{3x^{2}+11x4}{9x^{2}6x+1}\) and \(g(x)=\dfrac{x^{2}2x+1}{3x^{2}4x+1}\), \(f(x)=\dfrac{36x^{2}}{x^{2}+12x+36}\) and \(g(x)=\dfrac{x^{2}12x+3}{6x^{2}+4x12}\). 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