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Adding and subtracting rational expressions calculator with steps free

An expression that is the quotient of two algebraic expressions (with denominator not 0) is called a fractional expression. The most common fractional expressions are those that are the quotients of two polynomials; these are called rational expressions. Since fractional expressions involve quotients, it is important to keep track of values of the variable that satisfy the requirement that no denominator be0. For example, x != -2 in the rational expression: because replacing x with -2 makes the denominator equal 0. Similarly, in x!=-2 and x!= -4 The restrictions on the variable are found by determining the values that make the denominator equal to zero. In the second example above, finding the values of x that make (x + 2)(x + 4) = 0 requires using the property that ab = 0 if and only if a = 0 or b = 0, as follows. (x+2)(x+4)=0 x+2=0 or x+4=0 x=-2 or x=-4 Just as the fraction 6/8 is written in lowest terms as 3/4, rational expressions may also be written in lowest terms. This is done with the fundamental principle. Example 1 Write each expression in lowest terms. Factor the numerator and denominator to get By the fundamental principle, In the original expression p cannot be 0 or -4, because So this result is valid only for values of p other than 0 and -4. From now on, we shall always assume such restrictions when reducing rational expressions. Now let's take a look at how our step-by-step fraction solver solves this problem: Example 2 Factor to get The factors 2 - k and k - 2 have opposite signs. Because of this, multiply numerator and denominator by -1, as follows. Since (k-2)*(-1)=-k+2 or 2- k, Giving Working in an alternative way would lead to the equivalent result Our fraction calculator can solve this and many similar problems. If you would like a similar problem to be generated, click on solve similar button: Probably the most common error made in algebra is the incorrect use of the fundamental principle to write a fraction in lowest terms, Remember, the fundamental principle requires a pair of common factors, one in the numerator and one in the denominator. For example, On the other hand cannot be simplified further by the fundamental principle, because the numerator cannot be factored. Solve math problems using order of operations like PEMDAS, BEDMAS and BODMAS. (PEMDAS Warning) This calculator solves math equations that add, subtract, multiply and divide positive and negative numbers and exponential numbers. You can also include parentheses and numbers with exponents or roots in your equations. Use these math symbols: + Addition - Subtraction * Multiplication / Division ^ Exponents (2^5 is 2 raised to the power of 5) r Roots (2r3 is the 3rd root of 2) () [] {} Brackets You can try to copy equations from other printed sources and paste them here and, if they use ? for division and ? for multiplication, this equation calculator will try to convert them to / and * respectively but in some cases you may need to retype copied and pasted symbols or even full equations. If your equation has fractional exponents or roots be sure to enclose the fractions in parentheses. For example: 5^(2/3) is 5 raised to the 2/3 5r(1/4) is the 1/4 root of 5 which is the same as 5 raised to the 4th power Entering fractions If you want an entry such as 1/2 to be treated as a fraction then enter it as (1/2). For example, in the equation 4 divided by ? you must enter it as 4/(1/2). Then the division 1/2 = 0.5 is performed first and 4/0.5 = 8 is performed last. If you incorrectly enter it as 4/1/2 then it is solved 4/1 = 4 first then 4/2 = 2 last. 2 is a wrong answer. 8 was the correct answer. Math Order of Operations - PEMDAS, BEDMAS, BODMAS PEMDAS is an acronym that may help you remember order of operations for solving math equations. PEMDAS is typcially expanded into the phrase, "Please Excuse My Dear Aunt Sally." The first letter of each word in the phrase creates the PEMDAS acronym. Solve math problems with the standard mathematical order of operations, working left to right: Parentheses - working left to right in the equation, find and solve expressions in parentheses first; if you have nested parentheses then work from the innermost to outermost Exponents and Roots - working left to right in the equation, calculate all exponential and root expressions second Multiplication and Division - next, solve both multiplication AND division expressions at the same time, working left to right in the equation. Addition and Subtraction - next, solve both addition AND subtraction expressions at the same time, working left to right in the equation PEMDAS Warning Multiplication DOES NOT always get performed before Division. Multiplication and Division happen at the same time, from left to right. Addition DOES NOT always get performed before Subtraction. Addition and Subtraction happen at the same time, from left to right. The order "MD" (DM in BEDMAS) is sometimes confused to mean that Multiplication happens before Division (or vice versa). However, multiplication and division have the same precedence. In other words, multiplication and division are performed during the same step from left to right. For example, 4/2*2 = 4 and 4/2*2 does not equal 1. The same confusion can also happen with "AS" however, addition and subtraction also have the same precedence and are performed during the same step from left to right. For example, 5 - 3 + 2 = 4 and 5 - 3 + 2 does not equal 0. A way to remember this could be to write PEMDAS as PE(MD)(AS) or BEDMAS as BE(DM)(AS). Order of Operations Acronyms The acronyms for order of operations mean you should solve equations in this order always working left to right in your equation. PEMDAS stands for "Parentheses, Exponents, Multiplication and Division, Addition and Subtraction" You may also see BEDMAS and BODMAS as order of operations acronyms. In these acronyms, "brackets" are the same as parentheses, and "order" is the same as exponents. BEDMAS stands for "Brackets, Exponents, Division and Multiplication, Addition and Subtraction" BEDMAS is similar to BODMAS. BODMAS stands for "Brackets, Order, Division and Multiplication, Addition and Subtraction" Operator Associativity Multiplication, division, addition and subtraction are left-associative. This means that when you are solving multiplication and division expressions you proceed from the left side of your equation to the right. Similarly, when you are solving addition and subtraction expressions you proceed from left to right. Examples of left-associativity: a / b * c = (a / b) * c a + b - c = (a + b) - c Exponents and roots or radicals are right-associative and are solved from right to left. Examples of right-associativity: 2^3^4^5 = 2^(3^(4^5)) 2r3^(4/5) = 2r(3^(4/5)) For nested parentheses or brackets, solve the innermost parentheses or bracket expressions first and work toward the outermost parentheses. For each expression within parentheses, follow the rest of the PEMDAS order: First calculate exponents and radicals, then multiplication and division, and finally addition and subtraction. You can solve multiplication and division during the same step in the math problem: after solving for parentheses, exponents and radicals and before adding and subtracting. Proceed from left to right for multiplication and division. Solve addition and subtraction last after parentheses, exponents, roots and multiplying/dividing. Again, proceed from left to right for adding and subtracting. Adding, Subtracting, Multiplying and Dividing Positive and Negative Numbers This calculator follows standard rules to solve equations. Rules for Addition Operations (+) If signs are the same then keep the sign and add the numbers. If signs are different then subtract the smaller number from the larger number and keep the sign of the larger number. Rules for Subtraction Operations (-) Keep the sign of the first number. Change all the following subtraction signs to addition signs. Change the sign of each number that follows so that positive becomes negative, and negative becomes positive then follow the rules for addition problems. Multiplying a negative by a negative or a positive by a positive produces a positive result. Multiplying a positive by a negative or a negative by a positive produces a negative result. Rules for Division Operations (/ or ?) Similar to multiplication, dividing a negative by a negative or a positive by a positive produces a positive result. Dividing a positive by a negative or a negative by a positive produces a negative result. If you know how to add or subtract fractions with the same or different denominators, adding and subtracting rational expressions should be easy for you. The procedures between the two are very similar. Let's review by going over two examples: one with the same denominator, and another with different denominators. Example of adding and subtracting fractions with the same denominator. Example of adding and subtracting fractions with different denominators. Steps on How to Add and Subtract Rational Expressions 1) Make the denominators of the rational expressions the same by finding the Least Common Denominator (LCD). Note: The Least Common Denominator is the same as the Least Common Multiple (LCM) of the given denominators. 2) Next, combine the numerators by the indicated operations (add and/or subtract) then copy the common denominator. Note: Don't forget to simplify further the rational expression by canceling common factors, if possible. As they say, practice makes perfect. So we will go over six (6) worked examples in this lesson to illustrate how it is being done. Let's get started! Examples of Adding and Subtracting Rational Expressions Example 1: Add and subtract the rational expressions below. In this case, we are adding and subtracting rational expressions with unlike denominators. Our goal is to make them all the same. Since I have monomials in the denominators, the LCD can be obtained by simply taking the Least Common Multiple of the coefficients, where LCM (3,6) = 6, and multiply that to the variable x with the highest exponent. The LCD should be (LCM of coefficients) times (LCM of variable x) which gives us \left( 6 \right)\left( {{x^2}} \right) = 6{x^2}. The LCD is 6{x^2} thus I need to somehow convert all the denominators to that. The "blue fractions" are the appropriate multipliers to do the job! After simplifying the fractions by multiplication, you should have this setup. Now that we have the same denominators, it is easy to simplify. I will copy the common denominator and perform the indicated operations on the numerators. Combine similar terms (see the x variables?). When you reach the point of having a single rational expression, your next critical step is to factor the top and the bottom completely. The reason is that you may have common factors, which can be canceled out. I factor out the number 3 from the numerator. To make this a better answer, I will exclude the value of x that can make the original rational expression undefined. I can add the condition that x e 0. Example 2: Add the rational expressions below. This problem contains like denominators. We want this because it is the LCD itself ? the given denominator of the rational expression. So then the LCD that we are going to use is 2x + 1. Simplify by copying the common denominator then adding the numerators. Tip: Don't rush by immediately doing all the calculations in your head. I suggest that you place each term inside the parenthesis before performing the required operation. This extra step may be your lifesaver to avoid careless mistakes. Unless you have a good grasp on how to effectively combine like terms, I suggest you take another "baby step" as an additional precaution. Do you see how I decided to place the like terms side-by-side on the numerator? After combining like terms, you should have something similar to this. Next, factor out 3 from the top. This is great because we have common factors to cancel. Get rid of them by cancellation. That's right! When it's simplified the answer is just 3. To prevent the original rational expression to have a denominator of zero, we say that x e - {1 \over 2}. Example 3: Add the rational expressions below. This time I have the same trinomial in both denominators. This is similar to problem #2 but the quadratic trinomial adds a layer of fun. Later, I can factor out the denominator to see if there are common factors to cancel against the numerator. Copy the common denominator and set it up just like this ? placing each numerator in the parenthesis before adding them. Rearrange the terms in such a way that similar terms are next to each other for ease of computation later. This is what I got after combining the variables and constants together. Factor out the numerator. Factor out the denominator. I see that \left( {x - 5} \right) is a common factor so I cancel it. This is the leftover and should be our final answer. You may say that x e - \,4 and x e + \,5 from the original denominator. Example 4: Subtract the rational expressions below. This is a good example because the denominators are different. I need to find the LCD by doing the following steps. Factor each denominator completely, and line up the common factors. Identify each unique factor with the highest power. Multiply together the ones with the highest exponents for each unique factor. In this step, I haven't done anything but factor out the denominator of the first rational expression. Use our LCD = \left( {x + 5} \right)\left( {x - 5} \right) as guide to make the denominators equal. The first denominator is okay but the second one is lacking \left( {x - 5} \right). This is why I multiply it by the blue fraction. Simplify the second rational expression by multiplication. Here, I distributed the 2 into \left( {x - 5} \right) to get rid of the parenthesis. Put them all together in one fraction with a common denominator of \left( {x + 5} \right)\left( {x - 5} \right). However, keep each numerator inside a parenthesis. Distribute the negative sign into the parenthesis. Remember the signs will switch Group similar terms together before simplifying them. Compare the top and bottom expressions if there are common factors. I cancel out the factor \left( {x + 5} \right). We got it! You may include the restrictions that x e 5 and x e - \,5 based on the original denominator of the given rational expression. This is to prevent the division of zero, which is not good. Example 5: Subtract and add the rational expressions below. This problem is definitely interesting. To solve this, hold on to the things that you already know. Find the LCD by doing the steps below. Factor each denominator completely and neatly line up the common factors. Identify each unique factor with the highest power. Multiply together the ones with the highest exponents for each unique factor. Provide the missing factors for each denominator to reflect the LCD obtained above. Simplify by multiplication. It should look like this after you distribute each constant into the parenthesis. Combine them in one fraction while keeping each numerator within a parenthesis. Make sure to copy the indicated operations correctly. To prevent any unnecessary arithmetic errors, group similar terms before simplifying them. Now, we'll factor out the numerator and hope to see common factors between the numerator and denominator that can be canceled. Great! I see \left( {x - 4} \right) both on top and bottom. We now have our final answer. Add the restrictions x e 4 and x e - \,3 to avoid dividing by zero. Example 6: Subtract and add the rational expressions below. This is our last example in this lesson. I must say this is very similar to example 5. By now, you should already have a solid understanding of how to add and subtract rational expressions. Let's start finding the LCD again. Factor each denominator completely and neatly line up the common factors. Identify each unique factor with the highest power. Multiply together the ones with the highest exponents for each unique factor. This can be a bit messy but trust me, it will work out just fine as long as we are careful in every step. Factor the denominator of the third rational equation completely. Provide the missing factors for each denominator to attain the required LCD of \left( {2x - 1} \right)\left( {3x + 4} \right). Multiply the fractions to simplify. Place them in one huge fraction. Account for all the numerators inside each parenthesis and ensure that they have the correct indicated operations. Place the similar terms side by side before combining them. Wow! We cleaned out the numerator pretty well. Proceed by factoring the numerator. These are the correct factors of the numerator. It looks nice because we have common factors to cancel. Cancel out \left( {x - 2} \right). That's it. Simple, right? Practice with Worksheets You might also be interested in: Solving Rational Equations Multiplying Rational Expressions Solving Rational Inequalities

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