Simplifying Algebraic Expressions and Combining Like Terms

Dec 10, · Simplification of an algebraic expression can be defined as the process of writing an expression in the most efficient and compact form without affecting the value of the original expression. The process entails collecting like terms, which implies adding or Ratings: One of the big jobs we do in Algebra is simplification. You will often be asked to put something "in simplest form" What is the Simplest Form? In general, it is simpler when it is easier to use.

Last Updated: September 12, References. To create this article, 48 people, some anonymous, worked to edit and how to watch movies on playstation 3 it over time. There are 11 references cited in this article, which can be found at the bottom of the page. This article has been viewedtimes. Learn more Learning how to simplify algebraic expressions is a key part of mastering basic algebra and an extremely valuable tool for all mathematicians to have under their belt.

Basic simplification skills are fairly easy to learn - even for the math-averse. By following a few simple steps it is possible to simplify many of the most common types of algebraic expressions without any sort of special mathematical knowledge at all. See Step 1 below to begin! To simplify algebraic expressions, start by identifying the like terms, which are terms that have the same variables and exponents. Then, combine the like terms by adding them together to get the simplified expression.

You can also simplify the expression further by finding the greatest common factor and then dividing all of the terms in the expression by that number. After you've done that, put the expression in parentheses with the greatest common factor on the outside. To learn more ways you what is a system flow diagram simplify algebraic expressions, scroll down! Did this summary help you? Yes No. Log in Social login does not work in incognito and private browsers.

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Related Articles. Article Summary. Method 1 how do i get my kids to listen Define "like terms" by their variables and powers. In algebra, "like terms" have the same configuration of variables, raised to the same powers.

In other words, for two terms to be "like", they must have the same variable or variables, or none at all, and each variable must be raised to the same power, or no power at all. The order of variables within the term does not matter. However, x and x 2 are not like terms because each term has x raised to a different power. Similarly, -3yx and 5xz are not like terms because each term has a different set of variables.

Factor by writing numbers as the product of two factors. Factoring is the concept of representing a given number as the product of two factors multiplied together. Another way of thinking of this is that a number's factors are the numbers by which it is evenly divisible. Note that variable terms can also be factored - 20x, for instance, can be written as 4 5x. Prime numbers can't be factored because they are only evenly divisible by themselves and 1.

Sometimes, simplifying an expression means nothing more than performing the operations in the expression until no more can be done.

In these cases, it's important to remember the order of operations so that no arithmetic errors are made. The acronym PEMDAS can help you remember the order of operations - the letters correspond to the types of operations you should perform, in order. If there is multiplication and division in the same problem, you must complete those operations from left to right when you get to that point.

The same goes for addition and subtraction. The image above gives the incorrect answer. The last step did not work the addition and subtraction from left to right. It did the addition first. Method 2 of Write your equation.

The simplest algebraic equations, those involving just a few variable terms with whole number coefficients and no fractions, radicals, etc. As with most math problems, the first step to simplifying your equation is to write it out!

Identify like terms. Next, search your equation for like terms. Remember that like terms have both the same variable s and exponent s. In addition, 1 and -3 are like terms, as neither has any variables. So, in our equation, 2x and 4x and 1 and -3 are like terms. Combine like terms. Now that you've identified like terms, you can combine them to simplify your equation. Add terms together or subtract in the case of negative terms to reduce each set of terms with the same variables and exponents to one singular term.

Create a simplified expression from your simplified terms. After combining your like terms, construct an expression from your new, smaller set of terms. You should get a simpler expression that has one term for each different set of variables and exponents in the original expression.

This new expression is equal to the first. In our example, our simplified terms are 6x and -2, so our new expression is 6x - 2. It's also easier to factor, which, as we'll see below, is another important simplifying skill. Obey the order of operation when combining like terms.

In extremely simple expressions like the one dealt with in the example problems above, identifying like terms is simple. However, in more complex expressions, like ones that involve terms in parentheses, fractions, and radicals, like terms which can be how to make quaker oatmeal taste better may not be immediately apparent.

In these cases, follow the order of operations, performing operations on the terms in your expression as necessary until only addition and subtraction operations remain. It would be incorrect to immediately identify 3x and 2x as like terms and combine them because the parentheses in the expression dictate that we're supposed to do other operations first.

First, let's perform the arithmetic operations in the expression in accordance with the order of operations to obtain terms we can use. Nowsince the only operations left are addition and subtraction, we can combine like terms. Method 3 of Identify the greatest common factor in the expression. Factoring is a way to simplify expressions by removing factors that are common across all the terms in the expression.

To start, find the greatest common factor that all of the terms in the expression share - in other words, the largest number by which all the terms in the expression are evenly divisible. Notice that every term in this equation is divisible by 3. Since the terms aren't all evenly divisible by any larger number, we can say that 3 is our expression's greatest common factor.

Divide the terms in the expression by the greatest common factor. Next, divide every term in your equation by the greatest common factor you've just found. The resulting terms will all have what makes an allele dominant recessive or codominant coefficients than in the original expression.

To do so, we'll divide each term by 3. Represent your expression as the product of the greatest common factor and the remaining terms. How to lower ping on cs new expression isn't equal to your old one, how to build a paid membership site it isn't accurate to say that it's simplified.

To make our new expression equal to the old one, we'll need to account for the fact that it has been divided by the greatest common factor. Enclose your new expression in parentheses and set the greatest common factor of the original equation as a coefficient for the expression in parentheses. Use factoring to simplify fractions. 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.

In fact, factoring allows a mathematician to perform a variety of tricks to simplify an expression. One of the easiest of these involves taking advantage of the fact that multiplying a fraction's numerator and denominator by the same number gives an equivalent fraction. We can use factoring to simplify this fraction. Method 4 of Simplify fractions by dividing through by common factors. As noted above, if the numerator and denominator of an expression share factors, these factors can be removed from the fraction entirely.

Sometimes this will require factoring the numerator, denominator, or both as was the case in the example problem above while other times the shared factors are immediately apparent.

Note that it is also possible to divide the numerator terms by the expression in the denominator individually to obtain a simplified expression. Use square factors to simplify radicals.

What Are Like Terms?

In algebra, simplifying and factoring expressions are opposite processes. Simplifying an expression often means removing a pair of parentheses; factoring an expression often means applying them. Suppose you begin with the expression 5x(2x2 – 3x + 7). To simplify this expression, you remove the parentheses by multiplying 5x by each of the three terms inside [ ]. Mar 15, · Simplify (1 + 2) * 3. These are called integer expressions because they are mathematical expressions with integers or numbers. Integers are our positive and negative whole numbers. Order of Operations. Enter the expression you want to simplify into the editor. The simplification calculator allows you to take a simple or complex expression and simplify and reduce the expression to it's simplest form. The calculator works for both numbers and expressions containing variables. Step 2: Click the blue arrow to submit and see the result!

Last Updated: September 29, References. To create this article, 16 people, some anonymous, worked to edit and improve it over time. This article has been viewed , times. Learn more Math students are often asked to give their answer in "the simplest terms"—in other words, to write answers as small as possible.

Though a long, ungainly expression and a short, elegant one may technically equal the same thing, often, a math problem isn't considered "done" until the answer has been reduced to simplest terms. In addition, answers in simplest terms are almost always the easiest expressions to work with. For these reasons, learning how to simplify expressions is a crucial skill for aspiring mathematicians. To simplify math expressions using the order of operations, start by solving all of the terms in parentheses.

Next, solve the exponents, then perform any necessary multiplication. Move on to solving division, then finish up with adding and lastly, subtracting.

Since the order of operations is parentheses, exponents, multiplication, division, addition, and subtraction, use the handy acronym "Please excuse my Dear Aunt Sally" or "PEMDAS" to help you remember! For tips on solving more complex expressions, read on!

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Learn why people trust wikiHow. Download Article Explore this Article methods. Related Articles. Article Summary. Method 1 of All rights reserved. This image may not be used by other entities without the express written consent of wikiHow, Inc. Know the order of operations. When simplifying math expressions, you can't simply proceed from left to right, multiplying, adding, subtracting, and so on as you go. Some math operations take precedence over others and must be done first.

In fact, doing operations out of order can give you the wrong answer. The order of operations is: terms in parentheses, exponents, multiplication, division, addition, and, finally, subtraction. Note that, while basic knowledge of the order of operations makes it possible to simplify most basic expressions, specialized techniques are needed to simplify many variable expressions, including nearly all polynomials.

See Method Two below for more information. Start by solving all of the terms in parentheses. In math, parentheses indicate that the terms inside should be calculated separately from the surrounding expression. Regardless of the operations being performed within them, be sure to tackle the terms in parentheses as your first act when you attempt to simplify an expression.

Note that, however, within each pair of parentheses, the order of operations still applies. For instance, within parentheses, you should multiply before you add, subtract, etc. Note - if there are multiple parentheses nested inside one another, solve the innermost terms first, than the second-innermost, and so on.

Solve the exponents. After tackling parentheses, next, solve your expression's exponents. This is easy to remember because, in exponents, the base number and the power are positioned right next to each other. Find the answer to each exponent problem, then substitute the answers back into your equation in place of the exponents themselves.

The only exponent in our example is 3 2 , which equals 9. Solve the multiplication problems in your expression. Next, perform any necessary multiplication in your expression. Remember that multiplication can be written several ways. However, a number hugging a parentheses or a variable like 4 x also denotes multiplication. We don't know the value of x, so let's leave 2x as it is.. Move on to division. As you search for division problems in your expression, keep in mind that, like multiplication, division can be written multiple ways.

This brings up an important point - you don't have to perform every operation in the PEMDAS acronym when simplifying an expression, just the ones that are present in your problem. Next, perform any addition problems in your expression. You can simply proceed from left to right through your expression, but you may find it easiest to add numbers that combine in simple, manageable ways first.

Now, we must add what we can - let's look at each addition problem from left to right. We can't add 2x and 28 because we don't know the value of x, so let's skip it.

Proceed through your problem, solving any remaining subtraction problems. You may address the addition of negative numbers in this step, or in the same step as the normal addition problems - it won't effect your answer.. Review your expression. After proceeding through the order of operations, you should be left with your expression in simplest terms. However, if your expression contains one or more variables, understand that the variable terms will remain largely untouched.

Simplifying variable expressions requires you to find the values of your variables or to use specialized techniques to simplify the expression see below.

We can't address this final addition problem until we know the value of x, but when we do, this expression will be much easier to solve than our initial lengthy expression. Method 2 of Add like variable terms. When dealing with variable expressions, it's important to remember that terms with the same variable and exponent or "like terms" can be added and subtracted like normal numbers.

The terms must not only have the same variable, but also the same exponent. For example, 7x and 5x can be added to each other, but 7x and 5x 2 can not. For instance, 2xy 2 can be added to -3xy 2 , but not -3x 2 y or -3y 2. In this expression, we can add the 3x and -8x terms because they are like terms.

Simplify numerical fractions by dividing or "canceling out" factors. Fractions that have only numbers and no variables in both the numerator and denominator can be simplified in several ways. First, and perhaps easiest, is to simply treat the fraction as a division problem and divide the numerator by the denominator.

In addition, any multiplicative factors that appear both in the numerator and denominator can be "canceled" because they divide to give the number 1. In other words, if both the numerator and denominator share a factor, this factor can be removed from the fraction, leaving a simplified answer. If we have a calculator handy, we can divide to get an answer of. If we don't, however, we can still simplify by removing common factors. However, we're not done yet - both 6 and 10 share the factor 2.

In variable fractions, cancel out variable factors. Variable expressions in the form of fractions offer unique opportunities for simplification. Like normal fractions, variable fractions allow you to remove factors that are shared by both the numerator and denominator. However, in variable fractions, these factors can be both numbers and actual variable expressions.

Note that you can't cancel just any term - you can only cancel multiplicative factors that appear both in the numerator and denominator. Multiply parenthetical terms by their constants.

When dealing with variable terms in parentheses with an adjacent constant, sometimes, multiplying each term in the parentheses by the constant can result in a simpler expression.

This holds true for purely numeric constants and for constants that include variables. Note that, in some cases, such as in variable fractions, the constant adjacent to the parentheses gives an opportunity for cancellation and thus shouldn't be multiplied through the parentheses. Simplify by factoring. Factoring is a technique by which some variable expressions, including polynomials, can be simplified. Think of factoring as the opposite of the "multiplying through parentheses" step above - sometimes, an expression can be rendered more simply as two terms multiplied by each other, rather than as one unified expression.

This is especially true if factoring an expression allows you to cancel part of it as you would in a fraction. In special cases often with quadratic equations , factoring even allows you to find answers to the equation. This expression can factor to x - 3 x - 2. As hinted at above, another reason you may want to factor your expression has to do with the fact that factoring can reveal answers to certain equations, especially when those equations are written as expressions equal to 0.

Since any number times zero equals zero, we know that if we can get either of the terms of parentheses to equal zero, the whole expression on the left side of the equals sign will equal zero as well.

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