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Simplifying Expressions

Simplifying expressions mean 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, and then in the simplified expression, we will be only left with unlike terms that cannot be reduced further. Let us learn more about simplifying expressions in this article.

1.	How to Simplify Expressions?
2.	Simplifying Expressions with Exponents
3.	Simplifying Expressions with Distributive Property
4.	Simplifying Expressions with Fractions
5.	FAQs on Simplifying Expressions

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. The general rule to simplify expressions is PEMDAS - stands for Parentheses, Exponents, Multiplication, Division, Addition, Subtraction. In this article, we will be focussing more on how to simplify algebraic expressions. Let's begin!

We need to learn how to simplify expressions as it allows us to work more efficiently with algebraic expressions and ease out our calculations. To simplify algebraic expressions, follow the steps given below:

Step 1: Solve parentheses by adding/subtracting like terms inside and by multiplying the terms inside the brackets with the factor written outside. For example, 2x (x + y) can be simplified as 2x² + 2xy.
Step 2: Use the exponent rules to simplify terms containing exponents.
Step 3: Add or subtract the like terms.
Step 4: At last, write the expression obtained in the standard form (from highest power to the lowest power).

Let us take an example for a better understanding. Simplify the expression: x (6 – x) – x (3 – x). Here, there are two parentheses both having 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.

Look at the image given below showing another simplifying expression example.

simplifying expressions

Rules for Simplifying Algebraic Expressions

The basic rule for simplifying expressions is to combine like terms together and write unlike terms as it is. 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 Exponents

To simplify expressions with exponents is done by applying the rules of exponents on the terms. For example, (3x²)(2x) can be simplified as 6x³. The exponent rules chart that can be used for simplifying algebraic expressions is given below:

Zero Exponent Rule	a⁰ = 1
Identity Exponent Rule	a¹ = a
Product Rule	a^m × aⁿ = a^m+n
Quotient Rule	a^m/aⁿ= a^m-n
Negative Exponents Rule	a^-m = 1/a^m; (a/b)^-m = (b/a)^m
Power of a Power Rule	(a^m)ⁿ = a^mn
Power of a Product Rule	(ab)^m = a^mb^m
Power of a Quotient Rule	(a/b)^m = a^m/b^m

Example: Simplify: 2ab + 4b (b² - 2a).

To simplify this expression, let us first open the bracket by multiplying 4b to both the terms written inside. This implies, 2ab + 4b (b²) - 4b (2a). By using the product rule of exponents, it can be written as 2ab + 4b³ - 8ab, which is equal to 4b³ - 6ab.

This is how we can simplify expressions with exponents using the rules of exponents.

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 while 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.

Simplifying expressions with 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 and only unlike terms should be present in the simplified expression. For an 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.

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Simplifying Expressions Examples

Example 1: Find the simplified form of the expression formed by the following statement: "Addition of k and 8 multiplied by the subtraction of k from 16".

Solution: From the given statement, the expression formed is (k + 8)(16 - k). To simplify this expression, we need to use the concept of multiplication of algebraic expressions. By using the distributive property of simplifying expression, it can be simplified as,

⇒ k (16 - k) + 8 (16 - k)

⇒ 16k - k² + 128 - 8k

⇒ - k² + 16k - 8k + 128

⇒ - k² + 8k + 128

Therefore, - k² + 8k + 128 is the simplified form of the given expression.
Example 2: Simplify the expression: 4ps - 2s - 3(ps +1) - 2s .

Solution: By using the rules of simplifying expressions, 4ps - 2s - 3(ps +1) - 2s can be simplified as,

⇒ 4ps - 2s - 3(ps +1) - 2s

⇒ 4ps - 2s - 3ps - 3 - 2s

⇒ 4ps - 3ps - 2s - 2s - 3

⇒ ps - 4s - 3

Therefore, 4ps - 2s - 3(ps +1) - 2s = ps - 4s - 3.
Example 3: Daniel bought 5 pencils each costing $x, and Victoria bought 6 pencils each costing $x. Find the total cost of buying pencils by both of them.

Solution: Given, Daniel bought 5 pencils each for $x. The cost of all 5 pencils = $5x. And, Victoria bought 6 pencils each for $x, so the cost of 6 pencils = $6x. Therefore, the total cost of pencils bought by them = $5x + $6x = $11x.

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Practice Questions on Simplifying Algebraic Expressions

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FAQs on Simplifying Expressions

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. Along with PEMDAS, exponent rules, and the knowledge about operations on expressions also need to be used while simplifying algebraic expressions.

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 write 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).

How do Simplifying Expressions and Solving Equations Differ?

Equations refer to those statements that have an equal to "=" sign between the term(s) written on the left side and the term(s) written on the right side. Solving equations mean finding the value of the unknown variable given. On the other hand, simplifying expressions mean only reducing the expression to its lowest form. It does not intend to find the value of an unknown quantity.

What is an Example of Simplifying Expressions?

Simplifying algebraic expressions refer to the process of reducing the expression to its lowest form. An example of simplifying algebraic expressions is given below:

2x + 6x (y - 7) - 8

= 2x + 6xy - 42x - 8

= 6xy - 40x - 8

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