Math students are often asked to give their answer in “simplest terms,” that is, write them as elegantly as possible. While a long, lanky expression and a short, elegant one may technically look the same, a math problem is generally not considered “solved” until the answer has been reduced to its simplest form. Also, it is almost always easier to work with answers in simpler terms. That is why learning to simplify expressions is an essential skill for aspiring mathematicians.
Method 1 of 2: Employ the order of operations
Step 1. Know the order of operations
When simplifying mathematical expressions, you can't just proceed from left to right, multiplying, adding, subtracting, etc. Some math operations can take precedence over others and must be solved first. In fact, solving operations in the wrong order can give you the wrong answer. The order of operations is: terms in parentheses, exponents, multiplication, division, addition (or addition), and lastly, subtraction (or subtraction). An acronym that may help you remember this order is "To Understand Mathematics, I Must Learn To Add" or "PEMDAS".
Note that while a basic understanding of the order of operations makes it possible to simplify most basic expressions, specialized techniques are required to simplify many variable expressions, including almost all polynomials. Read method two for more information
Step 2. Begin by solving all the terms in parentheses
In mathematics, the parentheses indicate that the terms within must be calculated separately from the rest of the expression. When trying to simplify an expression, regardless of what operations are performed within it, be sure to solve the terms in parentheses first. However, note that within each parenthesis, the order of operations must still apply. For example, you need to solve multiplication before addition or subtraction.
As an example, let's try to simplify this expression: 2x + 4 (5 + 2) + 32 - (3 + 4/2). In this expression, we will first solve the terms in parentheses, 5 + 2 and 3 + 4/2. 5 + 2 =
Step 7.. 3 + 4/2 = 3 + 2
The second term in parentheses simplifies to 5 because due to the order of operations, we divide 4/2 first. If we were just going left to right, we would add 3 and 4 and then divide by 2, giving the answer to 7/2, which is incorrect
- Note: If there are multiple parentheses placed one inside the other, solve the ones inside first and continue outside.
Step 3. Solve the exponents
After figuring out what's inside the parentheses, move on to the exponents of the expressions. This is easy to remember because, in exponents, the base number and the power are located next to each other. Solve for each exponent and then substitute them for the answers in the equation.
After solving what is inside the parentheses, our expression looks like this: 2x + 4 (7) + 32 - 5. The only exponent in our example is 32, which is equivalent to
Step 9.. Put this number in the equation instead of 32 to get 2x + 4 (7) + 9 - 5.
Step 4. Solve the multiplication problems in the expression
You then do all the necessary multiplication operations on the expression. An × symbol, a period, or an asterisk are ways of expressing the multiplication operation. However, a number enclosed in parentheses or a variable (such as 4 (x)) also denotes this operation.
There are two examples of multiplication in our problem: 2x (2x is 2 × x) and 4 (7). We don't know the value of x, so we leave it as is (2x). 4 (7) = 4 × 7 =
Step 28.. We can rewrite our equation as 2x + 28 + 9 - 5.
Step 5. Continue with the division
As you search for division problems in the expression, keep in mind that, like multiplication, division can be written in a variety of ways. The ÷ symbol is one of them, but remember that diagonals and bars in a fraction (like 3/4, for example) also mean division.
Because we already solved a division (4/2) problem when we addressed the terms in parentheses, our example no longer has any other operations of this type, so we will skip this step. This brings us to an important point; You don't need to perform all the operations mentioned in the PEMDAS acronym when simplifying an expression, just do the ones that are present in the problem
Step 6. Add
Now, solve all the addition problems that you find in the expression. In this case, you can simply proceed from left to right, but it may be easier for you to add the matching numbers first in a simple and manageable way. For example, in the expression 49 + 29 + 51 +71, it is easier to add 49 + 51 = 100, 29 + 71 = 100 and 100 + 100 = 200, instead of 49 + 29 = 78, 78 + 51 = 129 and 129 + 71 = 200.
In our example, we have partially simplified the expression to "2x + 28 + 9 - 5". Now, we need to add what we can, taking a look at each addition problem from left to right. We can't add 2x to 28 because we don't know the value of x, so we omit it. 28 + 9 =
Step 37., so that when rewriting the expression, it remains as "2x + 37 - 5".
Step 7. Subtract
The last step in PEMDAS is subtraction. Proceed with the problem by solving all the remaining subtraction problems. In this step, you could solve the sum of negative numbers or you could also have done it in the previous one; Either way, it won't affect the answer.
In our expression: "2x + 37 - 5", there is only one subtraction problem. 37 - 5 =
Step 8. Check the expression
After following the order of operations, the expression should remain in simplest terms. However, if the expression contains one or more variables, keep in mind that the terms of the variables will not be altered. To simplify expressions with variables, you need to find the values of your variables or use specialized techniques to simplify the expression (see below).
Our final answer is "2x + 32". We can't solve this addition problem until we know the value of x, but when we do, the expression will be much easier to solve than the original, which was larger
Method 2 of 2: Simplify Complex Expressions
Step 1. Add the terms with like variables
When dealing with expressions with variables, it is important to remember that terms with the same variable and exponent (or “like terms”) can be added or subtracted as normal numbers. The terms must not only have the same variable, but also the same exponent. For example, it is possible to add 7x and 5x, but not 7x and 5x2.
- This rule also applies to terms with multiple variables. For example, 2xy2 can be added with -3xy2, but not with -3x2and or -3y2.
- Let's take a look at the expression x2 + 3x + 6 - 8x. In this expression, we can add the terms 3x and -8x because they are similar. Simplified, our expression is x2 - 5x + 6.
Step 2. Simplify number fractions by dividing or "canceling" factors
Fractions that only have numbers (and not variables) in both the numerator and denominator can be simplified in many ways. The first (and perhaps simplest) is to simply treat the fraction as a division problem by dividing the denominator by the numerator. Likewise, any multiplying factor found in both the numerator and the denominator can be “canceled” because the result of their division is 1. In other words, if the numbered and the denominator share a factor, it can be canceled to obtain an answer simplified.
- For example, let's think about the fraction 36/60. If we have a calculator at hand, we can do a division to get an answer of 0, 6. On the contrary, if we do not have one, we can still simplify the fraction by eliminating common factors. Another way to think of 36/60 is (6 × 6) / (6 × 10). You can rewrite this as 6/6 × 6/10. 6/6 = 1, so our expression is actually 1 × 6/10 = 6/10. However, it is not over yet; both 6 and 10 share factor 2. By repeating the above procedure, we are left with 3/5.
Step 3. In fractions with variables, the factors that have variables are canceled
Variable expressions in the form of fractions offer unique opportunities for simplification. As with normal fractions, fractions with variables allow you to eliminate factors present in the numerator and denominator. However, in fractions with variables, these factors can be numbers and real variable expressions.
- Consider the expression (3x2 + 3x) / (- 3x2 + 15x) This fraction can be rewritten as (x + 1) (3x) / (3x) (5 - x), where 3x appears in both the numerator and denominator. Eliminating these factors in the equation leaves us with (x + 1) / (5 - x). Similarly, in the expression (2x2 + 4x + 6) / 2, since each term is divisible by 2, we can write the expression as (2 (x2 + 2x + 3)) / 2 and simplify it to x2 + 2x + 3.
- Please note that you cannot cancel any term; You can only cancel the multiplying factors that appear in the numerator and denominator. For example, in the expression (x (x + 2)) / x, the "x" cancels both in the numerator and in the denominator, resulting in (x + 2) / 1 = (x + 2). However, (x + 2) / x does not cancel at 2/1 = 2.
Step 4. Multiply the terms in parentheses by their constants
Sometimes when dealing with terms that have variables in parentheses with an adjacent constant, multiplying each term in the parentheses by the constant can result in a simpler expression. This applies to purely numeric constants and to those that include variables.
- For example, the expression 3 (x2 + 8) can be simplified to 3x2 + 24, while 3x (x2 + 8) can be simplified to 3x3 + 24x.
- Note that in some cases, such as fractions with variables, the constant adjacent to the parentheses can be canceled, so it should not be multiplied with the terms within the parentheses. For example, in the fraction (3 (x2 + 8)) / 3x, the factor 3 appears in both the numerator and denominator, so it is possible to cancel it and simplify the expression to (x2 + 8) / x. This expression is easier to handle than (3x3 + 24x) / 3x, which would be the answer we would get if we had done the multiplication.
Step 5. Simplify by factoring
Factoring is a technique with which you can simplify some expressions with variables, including polynomials. Think of factoring as the opposite of “multiplying through parentheses” mentioned in the previous step. Sometimes an expression can be represented more simply as two terms multiplied by each other rather than as a single, unified expression. This applies particularly to cases where an expression allows you to cancel out a part of it (as you would in a fraction). In special cases (usually with quadratic equations), factoring even allows you to find the answers to the equation.
- Consider once again the following expression: x2 - 5x + 6. This expression can be factored to (x - 3) (x - 2). So if x2 - 5x + 6 is the numerator of a certain expression with one of these terms in the denominator, as in the case of the expression (x2 - 5x + 6) / (2 (x - 2)), we may need to write it in factored form so that we can cancel it with the denominator. In other words, with (x - 3) (x - 2) / (2 (x - 2)), the term (x - 2) cancels, leaving us with (x - 3) / 2.
As indicated above, another reason why it would be necessary to factor the expression has to do with the fact that this operation can reveal answers to certain equations, especially when those equations are written as expressions equal to 0. For example, let's think about the equation x2 - 5x + 6 = 0. By factoring it we obtain (x - 3) (x - 2) = 0. Since any number multiplied by zero gives us zero, we know that, if we can make any of the terms within the parentheses be equal to zero, the entire expression to the left of the equals sign will also result in zero. Therefore,
Step 2. are two answers to the equation.