how to simplify expressions with fractions

To create this article, 15 people, some anonymous, worked to edit and improve it over time. Example 1Write as a single fraction in reduced (simplified) form if possible. In these examples, we apply the above properties to reduce the given fractions in order to explain the use of these properties.a)Given: \( \dfrac{5}{3} + \dfrac{1}{5} - \dfrac{1}{3} - \dfrac{3}{5} \)Use commutativity of addition to write\( \quad\quad \dfrac{5}{3} + \dfrac{1}{5} - \dfrac{1}{3} - \dfrac{3}{5} - \dfrac{3}{4} = \dfrac{5}{3} - \dfrac{1}{3} + \dfrac{1}{5} - \dfrac{3}{5} \)Use associativity to write the above as\( \quad\quad = (\dfrac{5}{3} - \dfrac{1}{3} ) + (\dfrac{1}{5} - \dfrac{3}{5}) \)Add and subtract the fractions inside the brackets\( \quad\quad = \dfrac{4}{3} - \dfrac{2}{5} \)Rewrite with common denominator\( \quad\quad = \dfrac{4}{3} \times \dfrac{5}{5} - \dfrac{2}{5} \times \dfrac{3}{3} \)Simplify\( \quad\quad = \dfrac{20}{15} - \dfrac{6}{15} \)Subtract the fractions\( \quad\quad = \dfrac{14}{15} \)b)Given: \( \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} \)Use distributivity to write\( \quad\quad \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} = \dfrac{1}{3} \times \dfrac{x}{2} + \dfrac{1}{3} \times \dfrac{1}{8} - \dfrac{x}{6} \)Simplify\( \quad\quad = \dfrac{x}{6} + \dfrac{1}{24} - \dfrac{x}{6} \)Use commutativity to write the above as\( \quad\quad = \dfrac{x}{6} - \dfrac{x}{6} + \dfrac{1}{24} \)Use associativity to write the above as\( \quad\quad = (\dfrac{x}{6} - \dfrac{x}{6}) + \dfrac{1}{24} \)Simplify\( \quad\quad = 0 + \dfrac{1}{24} \)Simplify\( \quad\quad = \dfrac{1}{24} \)c)Given: \( \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \dfrac{1}{3} \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \)Use distributivity ( from right to left) to factor out the fraction \( \dfrac{1}{3} \).\( \quad\quad = \dfrac{1}{3} \left( \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \right) \)Use distibutivity to write the above as\( \quad\quad = \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} + \dfrac{3}{8} - \dfrac{5}{2} \right) \)Use commutativity to write the above as\( \quad\quad = \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{5}{2} + \dfrac{3}{8} - \dfrac{3}{8} \right) \)Use associativity to write the above as\( \quad\quad = \dfrac{1}{3} \left( (\dfrac{9}{2} - \dfrac{5}{2}) + (\dfrac{3}{8} - \dfrac{3}{8}) \right) \)Subtract fractions inside brackets\( \quad\quad = \dfrac{1}{3} (\dfrac{4}{2} + 0) \)Reduce the fraction \( \dfrac{4}{2} \) to \( \dfrac{2}{1} \)\( \quad\quad = \dfrac{1}{3} \times \dfrac{2}{1} \)Multiply fractions and simplify\( = \dfrac{2}{3} \)d)Given: \( \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) \)Use distributivity to write\( \quad\quad = \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) = \dfrac{x}{2} \times \dfrac{1}{x} + \dfrac{x}{2} \times \dfrac{3}{2x} \)Multiply the fractions in the above expression\( \quad\quad = \dfrac{x}{2 x} + \dfrac{3 x}{4 x} \)\( x \) is a common factor to both numerator and denominator and therefor the fractions may be reduced\( \quad\quad = \dfrac{1}{2} + \dfrac{3 }{4 } \)Rewrite the fraction \( \dfrac{1}{2} \) with denominator \( 4 \) as follows\( \quad\quad = \dfrac{1}{2} \times \dfrac{2}{2} + \dfrac{3 }{4 } \)Simplify\( \quad\quad = \dfrac{2}{4} + \dfrac{3 }{4 } \)Add the fractions and simplify\( \quad\quad = \dfrac{5}{4} \), Example 2Expand and simplify the following expressions.a) \( \quad \dfrac{1}{3} ( \dfrac{x}{2} - \dfrac{1}{2} ) + \dfrac{1}{2} ( \dfrac{2 x}{3} - \dfrac{4}{3} ) \)b) \( \quad - \dfrac{1}{2} ( \dfrac{1}{5} - \dfrac{x}{5} ) + \dfrac{1}{5} ( \dfrac{3 x}{2} - \dfrac{3}{2} ) \), Solution to Example 2a)Given: \( \quad \dfrac{1}{3} ( \dfrac{x}{2} - \dfrac{1}{2} ) + \dfrac{1}{2} ( \dfrac{2 x}{3} - \dfrac{4}{3} ) \)Use distributivity to expand the given expressions\( \quad = \dfrac{1}{3} \times \dfrac{x}{2} - \dfrac{1}{3} \times \dfrac{1}{2} + \dfrac{1}{2} \times \dfrac{2 x}{3} - \dfrac{1}{2} \times \dfrac{4}{3} \)Multiply fractions and simplify\( \quad = \dfrac{x}{6} - \dfrac{1}{6} + \dfrac{2x}{6} - \dfrac{4}{6} \)The fractions have a common denominator and therefore the above may be written as\( \quad = \dfrac{x + 2x - 1 - 4}{6} \)Simplify\( \quad = \dfrac{3x - 5}{6} \)b)Given: \( \quad - \dfrac{1}{2} ( \dfrac{1}{5} - \dfrac{x}{5} ) + \dfrac{1}{5} ( \dfrac{3 x}{2} - \dfrac{3}{2}) \)Use distributivity to expand the given expressions\( \quad = - \dfrac{1}{2} \times \dfrac{1}{5} - \dfrac{1}{2} \times (- \dfrac{x}{5}) + \dfrac{1}{5} \times \dfrac{3 x}{2} + \dfrac{1}{5} \times (- \dfrac{3}{2} ) \)Multiply fractions and simplify\( \quad = - \dfrac{1}{10} + \dfrac{x}{10} + \dfrac{3x }{10} - \dfrac{3}{10} \)The fractions in the above expression have a common denominator and therefore the above may be written as\( \quad = \dfrac{-1 + x + 3x - 3}{6} \)Simplify\( \quad = \dfrac{4x - 4}{10} \)Factor numerator and denominator\( \quad = \dfrac{2(2x - 2)}{2 \times 5} \)Reduce\( = \dfrac{2x - 2}{5} \). To simplify a complex fraction, turn it into a division problem first. Because the properties of and operations with numbers never change. Well, once again, we can view this as negative 16/9 divided by 3/7. By using our site, you agree to our. Simplifying Expressions with Exponents, Further Examples (2.1) a) Simplify 3a 2 b 4 × 2ab 2. Learn more... Algebraic fractions look incredibly difficult at first, and can seem daunting to tackle for the untrained student. Denominator: The bottom part of the fraction (ie. Simplifying fractions. Purplemath. \quad\quad = \dfrac {x} {2 x} + \dfrac {3 x} {4 x} x is a common factor to both numerator and denominator and therefor the fractions may be reduced. With a mixture of variables, numbers, and even exponents, it is hard to know where to begin. The two (y-1)'s cancel each other, and the final fraction is (y+1) / (y+2). If you really can’t stand to see another ad again, then please consider supporting our work with a contribution to wikiHow. Step 1 : If you have radical sign for the entire fraction, you have to take radical sign … SIMPLIFYING RADICAL EXPRESSIONS INVOLVING FRACTIONS. Collecting like terms means to simplify terms in expressions in which the variables are the same. The expressions above and below the fraction bar should be treated as if they were in parentheses. Then, take any terms that are in both the numerator and the denominator and remove them. The (-1)'s will cancel each other, and you're left with (y²-1) / (y²+y-2). If there are fractions in the expression, split them into the square root of the numerator and square root of the denominator. The final, simplified expression is (4x + 5) / (2x - 3). Simplifying (or reducing) fractions means to make the fraction as simple as possible. For example, [latex]\frac{4+8}{5 - 3}[/latex] means [latex]\left(4+8\right)\div \left(5 - 3\right)[/latex]. What is an imaginary number anyway? These types of expressions can be daunting, especially when they are algebraic expressions including variables. Basic Simplifying With Neg. This Pre-Algebra video tutorial explains the process of simplifying algebraic fractions with exponents and variables. 3 × 2 × a 2 a × b 4 b 2 = 6 × a 3 × b 6 = 6a 3 b 6 b) Simplify ( 2a 3 b 2) 2. Simplify any Algebraic Expression. Thinking back to when you were dealing with whole-number fractions, one of the first things you did was simplify them: You "cancelled off" factors which were in common between the numerator and denominator. For this rational expression (this polynomial fraction), I can similarly cancel off any common numerical or variable factors. We've simplified expressions with integers. Solution A good first step in simplifying expressions with exponents such as this, is to look to group like terms together, then proceed. $$ i \text { is defined to be } \sqrt{-1} $$ From this 1 fact, we can derive a general formula for powers of $$ i $$ by looking at some examples. Home. Can you simplify this expression over here? The expressions above and below the fraction bar should be treated as if they were in parentheses. This video shows how to simplify a couple of algebraic expressions by combining like terms by adding, subtracting, and using distribution. Fractions follow the same rules as any other kind of term in algebra. We use cookies to make wikiHow great. Simplify trigonometric expressions Calculator Get detailed solutions to your math problems with our Simplify trigonometric expressions step-by-step calculator. You may need to review the, of commutativity, associativity and distributivity and the different. Quotient Property of Radicals. Check your work when factoring by multiplying the factor back into the equation -- you will get the same number you started with. For example: (The "1 's" in the simplifications above are for clarity's sake, in case it's been a while since you last worked with negative powers. Combine the like terms by addition or subtraction Simplifying Radical Expressions Involving Fractions - Concept - Solved Examples. To simplify any algebraic expression, the following are the basic rules and steps: Remove any grouping symbol such as brackets and parentheses by multiplying factors. start by finding the inverse of the denominator, which you can do by simply flipping the fraction. SIMPLIFYING RADICAL EXPRESSIONS INVOLVING FRACTIONS. Now it is time to focus on fractions - positive and negative. By signing up you are agreeing to receive emails according to our privacy policy. Simplify Basic Expressions in Fraction Form. Before taking a look at simplifying algebraic fractions, let's remind ourselves how to simplify numerical fractions. If you have some tough algebraic expression to simplify, this page will try everything this … The expressions above and below the fraction bar should be treated as if they were in parentheses. The following terms will be used throughout the examples, and are common in problems involving algebraic fractions: Numerator: The top part of a fraction (ie. To simplify a complex fraction, remember that the fraction bar means division. (x+5)/(2x+3)). Multiply the fractions in the above expression. Please help us continue to provide you with our trusted how-to guides and videos for free by whitelisting wikiHow on your ad blocker. Know the vocabulary for algebraic fractions. All tip submissions are carefully reviewed before being published. Simplify the following expression: To simplify a numerical fraction, I would cancel off any common numerical factors. This article has been viewed 79,235 times. Simplify numerical fractions by dividing or "canceling out" factors. Fraction bars act as grouping symbols. Check out all of our online calculators here! Forget the laws of indices, and you're dead meat. Justify your steps.a) \( \quad \dfrac{5}{3} + \dfrac{1}{5} - \dfrac{1}{3} - \dfrac{3}{5} \)b) \( \quad \dfrac{1}{3} \left( \dfrac{x}{2} + \dfrac{1}{8} \right) - \dfrac{x}{6} \)c) \( \quad \dfrac{1}{3} \left( \dfrac{9}{2} - \dfrac{3}{8} \right) - \dfrac{1}{3} \left( - \dfrac{3}{8} + \dfrac{5}{2} \right) \)d) \( \quad \dfrac{x}{2} \left( \dfrac{1}{x} + \dfrac{3}{2x} \right) \) for \( x \ne 0 \), Solution to Example 1 There may be several ways to reduce (or simplify) the given fractions. They do not change the general procedures for how to simplify an algebraic expression using the distributive property. Simplify … \sqrt[3] 8 = 8 ^ {\red { \frac 1 3} } Simplify the following expression: (–5x –2 y)(–2x –3 y 2) Again, I can work either of two ways: multiply first and then handle the negative exponents, or else handle the exponents and then multiply the resulting fractions. Help With Your Math Homework. Demystifies the exponent rules, and explains how to think one's way through exercises to reliably obtain the correct results. To simplify algebraic fractions, start by factoring out as many numbers as you can for the numerator, which is the top part of the fraction. The expressions above and below the fraction bar should be treated as if they were in parentheses. They can be written as a fraction with both the numerator and denominator as integers. Use factoring to simplify fractions. The order of operations tells us to simplify the numerator and the denominator first—as if there were … Simplifying hairy expression with fractional exponents. \( \newcommand\ccancel[2][black]{\color{#1}{\xcancel{\color{black}{#2}}}}\) LESSON 17: Simplify Expressions Containing Fractions by Combining Like TermsLESSON 18: Simplify Rational Number Expressions Using the Distributive PropertyLESSON 19: Writing Algebraic Expressions to Solve Perimeter ProblemsLESSON 20: An Introduction To Programming in SCRATCH. (x+5)/(2x+3)). Math for Everyone. A rational expression is considered simplified if the numerator and denominator have no factors in common. Imaginary numbers are based on the mathematical number $$ i $$. Plots & Geometry. Next, find a common factor in the denominator, which is the bottom part of the fraction, by looking for a number that can divide into both parts. Fraction bars act as grouping symbols. Simplify any Algebraic Expression - powered by WebMath. before you start the examples and questions below. Thanks to all authors for creating a page that has been read 79,235 times. wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. You may now be wondering why factoring is useful if, after removing the greatest common factor, the new expression must be multiplied by it again. Powers Complex Examples. Welcome to Simplifying Fractions Step by Step with Mr. J! Plan your 60-minute lesson in Math or Simplifying Equations and Expressions … References. You could do this because dividing any number by itself gives you just " 1 ", and you can ignore factors of " 1 ". Simplifying an Expression With a Fraction Bar. We can simplify rational expressions in much the same way as we simplify numerical fractions. Evaluate [latex]x+\frac{1}{3}[/latex] when K-8 Math. Radicals (miscellaneous videos) Simplifying square-root expressions: no variables. Fractions with variables are also included.Do NOT use the calculator to answer the questions. \( \) \( \)\( \)\( \) Fractions that have only numbers (and no variables) in both the numerator and denominator can be simplified in several ways. % of people told us that this article helped them. When there are no more common factors in the top or the bottom, the fraction is simplified! In each case the signs were chosen so that multiplying the factors together would result in the original expression. Simplifying Fractions 1 - Cool Math has free online cool math lessons, cool math games and fun math activities. So, this could be rewritten as, and I can write it either as negative 16 over 9 or I could rewrite it as negative 16/9. Demonstrates how to simplify exponent expressions. Demystifies the exponent rules, and explains how to think one's way through exercises to reliably obtain the correct results. For example, [latex]\Large\frac{4+8}{5 - 3}[/latex] means [latex]\left(4+8\right)\div\left(5 - 3\right)[/latex]. Need help with how to simplify fractions? Simplifying square roots of fractions. Email . For example, \(\frac{4+8}{5-3}\) means \(\left(4+8\right)÷\left(5-3\right).\) The order of operations tells us to simplify the numerator and the denominator first—as if there were parentheses—before we divide. Guided Problem Solving. Really clear math lessons (pre-algebra, algebra, precalculus), cool math games, online graphing calculators, geometry art, fractals, polyhedra, parents and teachers areas too. Simplifying Radical Expressions Involving Fractions - Concept - Solved Examples. Use factoring to simplify fractions. For more tips on factoring, read on! So the complex fraction 3 4 5 8 3 4 5 8 can be written as 3 4 ÷ 5 8 3 4 ÷ 5 8. Trig. To simplify a radical expression, simplify any perfect squares or cubes, fractional exponents, or negative exponents, and combine any like terms that result. 6x² + 5x - 21 factors into (3x + 7) and (2x - 3). In fact, factoring allows a mathematician to perform a variety of tricks to simplify an expression. This algebra video tutorial explains how to simplify complex fractions especially those with variables and exponents - positive and negative exponents. In the example cited, 12x² + 43x +35 factors into (3x + 7) and (4x + 5). This is known as "solving by inspection." When factoring algebraic fractions, how do I know which one will have a plus or minus sign? The expressions above and below the fraction bar should be treated as if they were in parentheses. Therefore, stuff them into your brain at all costs. \quad\quad = \dfrac {1} {2} + \dfrac {3 } {4 } Rewrite the fraction \dfrac {1} {2} with denominator 4 … Rational numbers include integers and terminating and repeating decimals. Simplify an Algebraic Expression by Combining Like Terms. General Math. We present examples on how to use the properties of commutativity, associativity and distributivity and the different rules of fractions to simplify expressions including fractions. First simplify a little by factoring (-1) from both numerator and denominator: -(y²-1) / -(y²+y-2). Why does it work exactly like this all the time? Example: Simplify a) 4x 3 + x 2 - 2x 3 + 5 b) 10x 5 + 3(2x 5 - 4b 2) Show Video Lesson In this case there is a factor of (3x + 7) in both the numerator and denominator, so that they cancel each other. Common Denominator: This is a number that you can divide out of both the top and bottom of a fraction. More questions and their answers are also included. Free simplify calculator - simplify algebraic expressions step-by-step This website uses cookies to ensure you get the best experience. Free simplify calculator - simplify algebraic expressions step-by-step This website uses cookies to ensure you get the best experience. Includes worked examples of fractional exponent expressions. Fraction bars act as grouping symbols. Simplifying an Expression With a Fraction Bar. Remember, to evaluate an expression, we substitute the value of the variable into the expression and then simplify. Visit Cosmeo for explanations and help with your homework problems! Algebra. Introduction. Simplifying rational exponent expressions: mixed exponents and radicals. We know ads can be annoying, but they’re what allow us to make all of wikiHow available for free. In other words, you just look at ("inspect") the expression you're factoring, and choose plus and minus signs so that the factors, when multiplied together, result in the original expression. For example, the simplified version of \dfrac 68 86 Simplifying Expressions with Negative Exponents. What if there's a question like this: (1–y²) / (–y² – y + 2)? Why say four-eighths (48 ) when we really mean half (12) ? By using this website, you agree to our Cookie Policy. Luckily however, the same rules needed to simplify regular fractions, like 15/25, still apply to algebraic fractions. Factor numerator and denominator: [(y+1)(y-1)] / [(y+2)(y-1)]. Example. & Calculus. Sometimes you'll run across more complicated rational expressions to simplify - fractions within fractions, or even fractions within fractions within fractions. Recall that negative exponents indicates that we need to move the base to the other side of the fraction line. wikiHow is where trusted research and expert knowledge come together. Includes worked examples of fractional exponent expressions. Always factor out the largest numbers you can to simplify your equation fully. Evaluate Variable Expressions with Fractions. Quotient Property of Radicals . Can you simplify this complex fraction? Fraction bars act as grouping symbols. Practice your math skills and learn step by step with our math solver. This article has been viewed 79,235 times. To create this article, 15 people, some anonymous, worked to edit and improve it over time. wikiHow is a “wiki,” similar to Wikipedia, which means that many of our articles are co-written by multiple authors. Independent Problem Solving. Warmup . Write as a single fraction in reduced (simplified) form if possible. Solution Demonstrates how to simplify exponent expressions. We have evaluated expressions before, but now we can also evaluate expressions with fractions. Other Stuff. in Physics and Engineering, Exercises de Mathematiques Utilisant les Applets, Trigonometry Tutorials and Problems for Self Tests, Elementary Statistics and Probability Tutorials and Problems, Free Practice for SAT, ACT and Compass Math tests, \( \quad \dfrac{11}{6} + \dfrac{3}{10} - \dfrac{5}{6} + \dfrac{3}{10} \), \( \quad \dfrac{2}{5} \left( \dfrac{x}{2} + \dfrac{1}{5} \right) - \dfrac{x}{5} \), \( \quad \dfrac{1}{7} \left( \dfrac{1}{2} - \dfrac{3}{5} \right) - \dfrac{1}{7} \left( - \dfrac{3}{5} + \dfrac{5}{2} \right) \), \( \quad \dfrac{- x}{3} \left( \dfrac{1}{2 x} + \dfrac{1}{5x} \right) \) for \( x \ne 0 \), \( \quad \dfrac{1}{5} ( \dfrac{x}{4} - \dfrac{1}{4} ) - \dfrac{3}{4} ( \dfrac{2 x}{5} - \dfrac{4}{5} ) \), \( \quad - \dfrac{1}{9} ( \dfrac{1}{3} - \dfrac{x}{3} ) + \dfrac{1}{3} ( \dfrac{3 x}{9} - \dfrac{1}{3} ) \), \( \quad \dfrac{11}{6} + \dfrac{3}{10} - \dfrac{5}{6} + \dfrac{3}{10} = \dfrac{8}{5}\), \( \quad \dfrac{2}{5} \left( \dfrac{x}{2} + \dfrac{1}{5} \right) - \dfrac{x}{5} = \dfrac{2}{25}\), \( \quad \dfrac{1}{7} \left( \dfrac{1}{2} - \dfrac{3}{5} \right) - \dfrac{1}{7} \left( - \dfrac{3}{5} + \dfrac{5}{2} \right) = \dfrac{-2}{7} \), \( \quad \dfrac{- x}{3} \left( \dfrac{1}{2 x} + \dfrac{1}{5x} \right) \) for \( x \ne 0 = \dfrac{-7}{30}\), \( \quad \dfrac{1}{5} ( \dfrac{x}{4} - \dfrac{1}{4} ) - \dfrac{3}{4} ( \dfrac{2 x}{5} - \dfrac{4}{5} ) = \dfrac{-5x+11}{20}\), \( \quad - \dfrac{1}{9} ( \dfrac{1}{3} - \dfrac{x}{3} ) + \dfrac{1}{3} ( \dfrac{3 x}{9} - \dfrac{1}{3} ) = \dfrac{4x-4}{27}\). For example, [latex]\Large\frac{4+8}{5 - 3}[/latex] means [latex]\left(4+8\right)\div\left(5 - 3\right)[/latex]. By using this website, you agree to our Cookie Policy. Step 1 : If you have radical sign for the entire fraction, you have to take radical sign separately for numerator and denominator. When learning how to simplify surds students need to understand the difference between a rational and irrational number. For example, [latex]\frac{4+8}{5 - 3}[/latex] means [latex]\left(4+8\right)\div \left(5 - 3\right)[/latex]. You may now be wondering why factoring is useful if, after removing the greatest common factor, the new expression must be multiplied by it again. First, and perhaps easiest, is to simply treat the fraction as a division problem and divide the … Objective. Simplifying an Expression With a Fraction Bar. Simplifying an Expression With a Fraction Bar. A fraction containing a fraction in the numerator and denominator is a called a complex fraction. Simplifying them becomes easier when you remember that a fraction bar is the same thing as a division sign. Step 2 : We have to simplify the radical term according to its power. {"smallUrl":"https:\/\/www.wikihow.com\/images\/thumb\/d\/d6\/Simplify-Algebraic-Fractions-Step-1-Version-3.jpg\/v4-460px-Simplify-Algebraic-Fractions-Step-1-Version-3.jpg","bigUrl":"\/images\/thumb\/d\/d6\/Simplify-Algebraic-Fractions-Step-1-Version-3.jpg\/aid1424021-v4-728px-Simplify-Algebraic-Fractions-Step-1-Version-3.jpg","smallWidth":460,"smallHeight":345,"bigWidth":"728","bigHeight":"546","licensing":"

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