3 Ways to Simplify Algebraic Expressions

2024 Author: Samantha Chapman | [email protected]. Last modified: 2024-01-15 13:47

Learning to simplify algebraic expressions is a key aspect of mastering basic algebra and is a valuable tool for all mathematicians. Simplification makes it possible to transform a long, complex or abstruse expression into another equivalent, more understandable expression. It is quite easy to acquire the basic skills of this process, even for those people who are not very inclined to mathematics. By following a few simple steps it is possible to rephrase several of the most common types of algebraic expressions more clearly, without the need for special mathematical knowledge. Read on to learn more!

Steps

Understanding the Fundamental Concepts

Step 1. Recognize "similar terms" by the variable and exponent

In algebra, "similar terms" are those that have the same configuration as regards the variable element raised to the same power. In other words, for two terms to be "similar", they must have the same or the same variables or none; moreover, the variable (if present) must have the same exponent. The order in which the various elements of the term are written is not important.

For example, 3x² and 4x² they are similar terms because they both contain the unknown x raised to the second power. However, x and x² they cannot be defined as similar, because each term has a different exponent. Likewise, -3yx and 5xz are not similar, because they have different unknown parts.

Step 2. Break down the numbers by writing them as products of two factors

The decomposition expects to represent a given number as the product of two factors multiplied together. Numbers can have more than a couple of factors; for example, 12 can be represented as 1 × 12, 2 × 6 and 3 × 4; you can therefore state that 1; 2; 3; 4; 6 and 12 are all factors of 12. Another way to look at this concept is to remember that the factors of a number are those by which the number itself is divisible.

• For example, if you want to break down the number 20, you can rewrite it as 4 × 5.
• Note that terms with variables can also be decomposed - for example 20x can be represented as 4 (5x).
• Prime numbers cannot be factored, because they are only divisible by one and themselves.

Step 3. Use the acronym PEMDAS to remember the order of operations

Sometimes, simplifying an expression means nothing more than doing the present operations until you can continue. In these cases, it is important to know the order of the operations, in order not to make arithmetic errors. The acronym PEMDAS helps you remember this, because each letter corresponds to the type of operations you should perform in the correct order. If there is both multiplication and division in a problem, you simply have to do them in order from left to right as soon as you reach that point. The same goes for addition and subtraction. The image related to this step shows you a wrong answer. In fact, in the last step it is not added and subtracted from left to right, but the addition is carried out first. Actually, the correct order is 25-20 = 5, then 5 + 6 = 11.

• P.: brackets;
• AND: exponent;
• M.: multiplication;
• D.: division;
• TO: addition;
• S.: subtraction.

Method 1 of 3: Combine Similar Terms

Step 1. Write the equation

The simpler algebraic ones (which provide only a few variable terms with integer numerical coefficients and without fractions, radicals and so on) can be solved in a few steps. As with most math problems, the first step of simplification is to write the equation itself!

As an example problem for the next steps consider the expression: 1 + 2x - 3 + 4x.

Step 2. Recognize similar terms

The next step is to look at the expression to find these terms; remember that they must have the same variable (or variables) and exponent.

For example, find similar terms in the expression 1 + 2x - 3 + 4x. 2x and 4x both have the same unknown with identical exponent (which in this case is 1). Furthermore, 1 and -3 are similar terms, since they have no variables; accordingly, you can state that in the expression 2x and 4x And 1 and -3 are similar terms.

Step 3. Join similar terms

Now that you have identified them, you can combine them together to simplify the expression. Add them (or subtract them in the case of negative ones) to reduce a series of terms with identical unknowns and exponents to a single element.

• Add the similar terms from the example expression.

  • 2x + 4x = 6x.
  • 1 + -3 = - 2.

  

  Step 4. Create a simplified expression using the terms you have reduced

  After combining the similar ones, build the expression using the new, smaller set of elements. You should get a more linear problem that has only one term for each type of variable and power present in the original one. This new expression is equivalent to the first.

  In the example under consideration, the simplified terms are 6x and -2; the new expression can then be rewritten as 6x - 2. This more basic version is equivalent to the original (1 + 2x - 3 + 4x), but is shorter and easier to manage. It also implies fewer difficulties if you want to factor it, another important skill for simplifying math problems.

  

  Step 5. Respect the order of operations when combining similar terms

  In the case of very simple expressions, such as the one considered in the previous example, it is not difficult to recognize similar terms. However, when the problem is more complex, such as those involving parentheses, fractions and radicals, the terms can be represented in such a way that their similarity does not appear obvious. In these cases, follow the order of the operations by performing them on the terms of the expression as necessary, until there are only additions and subtractions.

  • For example, consider the expression 5 (3x-1) + x ((2x) / (2)) + 8 - 3x. It would be wrong to immediately identify the terms 3x and 2x as similar and combine them, because there are brackets that impose a certain order of operations. First, do the arithmetic operations of the expression in the right order, so that you get some terms that you can use. Here's how to proceed:

    • 5 (3x-1) + x ((2x) / (2)) + 8 - 3x.
    • 15x - 5 + x (x) + 8 - 3x.
    • 15x - 5 + x² + 8 - 3x. At this point, since the only operations left are just add and subtract, you can combine similar terms.
    • x² + (15x - 3x) + (8 - 5).
    • x² + 12x + 3.

    Method 2 of 3: Factor Down

    

    Step 1. Find the greatest common divisor within the expression

    Decomposition is a method that allows you to simplify expressions by eliminating the common factors present in all terms. To begin, find the greatest common divisor of all elements of the problem - in other words, the largest number that can divide all terms of the expression.

    • Consider the expression 9x² + 27x - 3. Notice how each present term is divisible by 3. Since none of them is divisible by a greater number, you can say that

      Step 3. is the greatest common divisor of the expression.

    

    Step 2. Divide the terms of the expression by the greatest common factor

    The next step is to divide the whole expression by the common factor, thus rewriting it with smaller coefficients.

    • Break down the example expression by dividing it by the greatest common factor, which is the number 3. To do this, divide all terms by 3.

      • 9x²/ 3 = 3x².
      • 27x / 3 = 9x.
      • -3/3 = -1.
      • At this point, you can rephrase the expression as: 3x² + 9x - 1.

      

      Step 3. Represent the expression as the product of the greatest common factor and the remaining terms

      The new problem is not equivalent to the original one, so it would be imprecise to say that it has been simplified. To make the new expression equivalent to the previous one, you have to take into account the fact that the terms have been divided by the greatest common factor. Enclose the expression in parentheses and put the greatest common factor as the outer coefficient.

      Considering the example expression, 3x² + 9x - 1, you should enclose it in parentheses, multiply everything by the greatest common divisor and rewrite: 3 (3x² + 9x - 1). This way, the expression you get is equivalent to the original: 9x² + 27x - 3.

      

      Step 4. Use decomposition to simplify fractions

      At this point, you may be wondering what the usefulness of decomposition is, if after dividing it you have to multiply the expression again. This technique actually allows the mathematician to perform a series of "tricks" to simplify an expression. One of the simplest is to take advantage of the fact that by multiplying the numerator and denominator of a fraction by the same number, an equivalent fraction is obtained. Here's how to proceed:

      • Suppose the example expression: 9x² + 27x - 3 represents the numerator of a large fraction with a denominator of 3. The fraction would look like this: (9x² + 27x - 3) / 3. You can use the decomposition to simplify the fraction.

        • Replace the original expression, which is in the numerator, with the decomposed and equivalent one: (3 (3x² + 9x - 1)) / 3.
        • Notice how, at this point, both the numerator and the denominator share the same coefficient 3. Dividing both by 3 you get: (3x² + 9x - 1) / 1.
        • Since any fraction with a denominator equal to "1" is equal to the terms present in the numerator, you can say that the original fraction can be simplified to: 3x² + 9x - 1.

        Method 3 of 3: Use Additional Simplification Skills

        

        Step 1. Simplify the fractions by dividing them by the common factors

        As described above, if the numerator and denominator of an expression share some identical factors, they can be eliminated. Sometimes, it is necessary to break down the numerator, the denominator, or both (as in the example described above), while in other circumstances the common factors are obvious. Note that it is also possible to divide the terms of the numerator individually by the expression in the denominator, to obtain a simplified one.

        • Take on an example that doesn't necessarily require a long breakdown. For the fraction (5x² + 10x + 20) / 10, you can divide each term of the numerator by the number 10 present in the denominator, even if the coefficient "5" of 5x² it is less than 10 and therefore does not count it among its factors.

          Proceeding in this way you get: ((5x²) / 10) + x + 2. If you wish, you can rewrite the first term as (1/2) x² to get the expression (1/2) x² + x + 2.

          

          Step 2. Use square factors to simplify radicals

          Expressions under the square root sign are called radical expressions. You can simplify them by detecting square factors (those that are the square of an integer), doing the square root operation on them separately, and removing them from the root sign.

          • Solve this simple example: √ (90). If you think of the number 90 as the product of two of its factors, 9 and 10, you can calculate the square root of 9 to get 3 and extract it from the radical. In other words:

            • √(90).
            • √(9 × 10).
            • (√(9) × √(10)).
            • 3 × √(10).
            • 3√(10).

            

            Step 3. Add the exponents when you need to multiply two powers and subtract them when you divide them

            Some algebraic expressions require you to multiply or divide exponential terms. Instead of calculating the value of each power individually and then multiplying or dividing it, you can simply add the exponents when you are faced with a multiplication of powers and subtract them when you need to perform a division; in this way you save time. The same concept can be applied to simplify expressions with variables.

            • Consider, for example, the expression 6x³ × 8x⁴ + (x¹⁷/ x¹⁵). Whenever you need to multiply or divide powers, you can respectively add or subtract the exponents to quickly find a simplified term. Here's how to do it:

              • 6x³ × 8x⁴ + (x¹⁷/ x¹⁵).
              • (6 × 8) x^{3 + 4} + (x^{17 – 15}).
              • 48x⁷ + x².

            • To understand how this "trick" works, consider that:

              • The multiplication of exponential terms is essentially equivalent to the multiplication of a long series of non-exponential terms. For example, since x³ = x × x × x and x ⁵ = x × x × x × x × x, it follows that x³ × x⁵ = (x × x × x) × (x × x × x × x × x), i.e. x⁸.
              • Similarly, the division of exponential terms is equivalent to the division of a long series of non-exponential terms. x⁵/ x³ = (x × x × x × x × x) / (x × x × x). Since any term in the numerator can be elided with the corresponding term in the numerator, the solution is x².

              Advice

              • Always remember that you must consider the numbers complete with positive and negative sign. Many people get stuck thinking what sign they should match a value.
              • Get help if you need it!
              • It is not easy to simplify algebraic expressions; however, once you have mastered the method, you are able to use it forever.

              Warnings

              • Check that you have not accidentally added any extra numbers, powers, or operations that do not belong to the expression.
              • Always look for similar terms and don't be misled by the powers that be.