Simplifying expressions means rewriting the same algebraic expression with no like terms and in a compact manner. To simplify expressions, we combine all the like terms and solve all the given brackets, if any. In the simplified expression, we will only be left with unlike terms that cannot be reduced further. Learn more about simplifying expressions in this article.
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How to Simplify Expressions?
Before learning about simplifying expressions, let us quickly go through the meaning of expressions in math. Expressions refer to mathematical statements having a minimum of two terms containing either numbers, variables, or both connected through an addition/subtraction operator in between. PEMDAS – stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction – is the general rule for simplifying expressions. In this article, we will focus more on how to simplify algebraic expressions. Let’s begin!
We must learn how to simplify expressions so that we can work more efficiently with algebraic expressions and simplify our calculations. To simplify algebraic expressions, follow the steps given below:
- In parentheses, add/subtract like terms and multiply the terms inside the brackets with the factor outside. For example, 2x (x + y) can be simplified as 2x2 + 2xy.
- Use the exponent rules to simplify terms containing exponents.
- Subtractors add like terms.
- At last, write the expression obtained in the standard form (from highest to lowest power).
Let us take an example for a better understanding. Simplify the expression: x (6 – x) – x (3 – x). Both of these parentheses have two, unlike terms. So, we will be solving the brackets first by multiplying x to the terms written inside. x(6 – x) can be simplified as 6x – x2, and -x(3 – x) can be simplified as -3x + x2. Now, combining all the terms will result in 6x – x2 – 3x + x2. In this expression, 6x and -3x are like terms, and -x2 and x2 are like terms. So, adding these two pairs of like terms will result in (6x – 3x) + (-x2 + x2). By simplifying it further, we will get 3x, which will be the final answer. Therefore, x (6 – x) – x (3 – x) = 3x.
Rules for Simplifying Algebraic Expressions
As a general rule, simplifying expressions involves combining like terms together and writing unlike terms as they are. Some of the rules for simplifying expressions are listed below:
- To add two or more like terms, add their coefficients and write the common variable with it.
- Use the distributive property to open up brackets in the expression which says that a (b + c) = ab + ac.
- If there is a negative sign just outside parentheses, change the sign of all the terms written inside that bracket to simplify it.
- If there is a ‘plus’ or a positive sign outside the bracket, just remove the bracket and write the terms as it is, retaining their original signs.
Simplifying Expressions with Distributive Property
Distributive property states that an expression given in the form of x (y + z) can be simplified as xy + xz. It can be very useful for simplifying expressions. Look at the above examples, and see whether and how we have used this property for the simplification of expressions. Let us take another example of simplifying 4(2a + 3a + 4) + 6b using the distributive property.
Therefore, 4(2a + 3a + 4) + 6b is simplified as 20a + 6b + 16. Now, let us learn how to use the distributive property to simplify expressions with fractions.
Simplifying Expressions with Fractions
When fractions are given in an expression, then we can use the distributive property and the exponent rules to simplify such expression. For example, 1/2 (x + 4) can be simplified as x/2 + 2. Let us take one more example to understand it.
Example: Simplify the expression: 3/4x + y/2 (4x + 7).
By using the distributive property, the given expression can be written as 3/4x + y/2 (4x) + y/2 (7). Now, to multiply fractions, we multiply the numerators and the denominators separately. So, y/2 × 4x/1 = (y × 4x)/2 = 4xy/2 = 2xy. And, y/2 × 7/1 = 7y/2. Therefore, 3/4x + y/2 (4x + 7) = 3/4x + 2xy + 7y/2. All three are unlike terms, so it is the simplified form of the given expression.
While simplifying expressions with fractions, we have to make sure that the fractions should be in the simplest form. Only unlike terms should be present in the simplified expression. For instance, (2/4)x + 3/6y is not the simplified expression, as fractions are not reduced to their lowest form. On the other hand, x/2 + 1/2y is in a simplified form as fractions are in the reduced form and both are unlike terms.
What is Simplifying Expressions in Math?
In math, simplifying expressions is a way to write an expression in its lowest form by combining all like terms together. It requires one to be familiar with the concepts of arithmetic operations on algebraic expressions, fractions, and exponents. We follow the same PEMDAS rule to simplify algebraic expressions as we do for simple arithmetic expressions. When simplifying algebraic expressions, exponent rules and the knowledge of operations on expressions are also useful.
What Mathematical Concepts are Important in Simplifying Expressions?
The mathematical concepts that are important in simplifying algebraic expressions are given below:
- Familiarity with like and unlike algebraic terms.
- Basic knowledge of algebraic expressions is required.
- Addition and subtraction of algebraic expressions.
- Multiplication and division of expressions.
- Understanding of terms with exponents and exponent rules.
- Algebraic identities and properties.
What are the Rules for Simplifying Expressions?
The rules for simplifying expressions are given below:
- Follow the PEMDAS rule to determine the order of terms to be simplified in an expression.
- Distributive property can be used to simplify the multiplication of two terms in an algebraic expression.
- Exponent rules can be used to simplify terms with exponents.
- First, we open the brackets, if any. Then we simplify the terms containing exponents.
- After that, combine all the like terms.
- The simplified expression will only have unlike terms connected by addition/subtraction operators that cannot be simplified further.
How to do Simplifying Expressions?
Follow the steps given below to learn how to simplify expressions:
- Open up brackets, if any. If there is a positive sign outside the bracket, then remove the bracket and type all the terms retaining their original signs. If there is a negative sign outside the bracket, then remove the bracket and change the signs of all the terms written inside from + to -, and – to +. And if there is a number or variable written just outside the bracket, then multiply it with all the terms inside using the distributive property.
- Use exponent rules to simplify terms with exponents, if any.
- Add/subtract all like terms.
- Write the simplified expression in the standard form (from the highest power term to the lowest power term).