Properties Used in Simplifying the Expression

Oct 13, 2025 by windailys.com 33 views

Introduction

Algebraic expressions are a fundamental concept in mathematics, and simplifying them is a crucial skill to master. In this article, we will focus on simplifying the given expression $6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2$ . We will break down the process into manageable steps and identify the properties used in each step.

Step 1: Rearranging the Terms

The first step in simplifying the given expression is to rearrange the terms in a way that makes it easier to combine like terms. We can start by grouping the terms with the same variables together.

$\begin{align*} 6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2 & = 4a^2b + 3a^2b - 3ab^2 + 6ab^2 - 2a^2b^2 \\ & = (4 + 3)a^2b + (-3 + 6)ab^2 - 2a^2b^2 \end{align*}$

In this step, we used the commutative property of addition, which states that the order of the terms does not change the result.

Step 2: Combining Like Terms

The next step is to combine like terms, which involves adding or subtracting terms with the same variables.

$\begin{align*} (4 + 3)a^2b + (-3 + 6)ab^2 - 2a^2b^2 & = 7a^2b + 3ab^2 - 2a^2b^2 \end{align*}$

In this step, we used the associative property of addition, which states that the order in which we add or subtract terms does not change the result.

Properties Used in Simplifying the Expression

In the above steps, we used the following properties:

Commutative property of addition: The order of the terms does not change the result.
Associative property of addition: The order in which we add or subtract terms does not change the result.

Conclusion

Simplifying algebraic expressions is an essential skill in mathematics, and understanding the properties used in each step is crucial. In this article, we identified the properties used in the first two steps of simplifying the given expression $6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2$ . We will continue to explore more properties and techniques in future articles.

Future Articles

In our next article, we will explore the following topics:

Distributive property: How to use the distributive property to simplify expressions.
Order of operations: How to apply the order of operations to simplify expressions.
Simplifying expressions with exponents: How to simplify expressions with exponents.

Stay tuned for more articles on simplifying algebraic expressions and other mathematical topics.

References

Glossary

Algebraic expression: An expression that consists of variables, constants, and mathematical operations.
Like terms: Terms that have the same variables and exponents.
Commutative property of addition: The order of the terms does not change the result.
Associative property of addition: The order in which we add or subtract terms does not change the result.
Simplifying Algebraic Expressions: A Q&A Guide =====================================================

Introduction

In our previous article, we explored the first two steps in simplifying the given expression $6ab^2 + 4a^2b - 3ab^2 + 3a^2b - 2a^2b^2$ . We identified the properties used in each step and provided a step-by-step guide on how to simplify the expression. In this article, we will answer some frequently asked questions (FAQs) related to simplifying algebraic expressions.

Q&A

Q: What is the difference between like terms and unlike terms?

A: Like terms are terms that have the same variables and exponents, while unlike terms are terms that have different variables or exponents.

Q: How do I identify like terms in an expression?

A: To identify like terms, look for terms that have the same variables and exponents. For example, in the expression $2x^2 + 3x^2 + 4x$ , the like terms are $2x^2$ and $3x^2$ .

Q: What is the commutative property of addition?

A: The commutative property of addition states that the order of the terms does not change the result. For example, $a + b = b + a$ .

Q: What is the associative property of addition?

A: The associative property of addition states that the order in which we add or subtract terms does not change the result. For example, $(a + b) + c = a + (b + c)$ .

Q: How do I simplify an expression with exponents?

A: To simplify an expression with exponents, use the rules of exponents, such as $a^m \cdot a^n = a^{m+n}$ and $(a^m)^n = a^{mn}$ .

Q: What is the order of operations?

A: The order of operations is a set of rules that tells us which operations to perform first when simplifying an expression. The order of operations is:

Parentheses: Evaluate expressions inside parentheses first.
Exponents: Evaluate expressions with exponents next.
Multiplication and Division: Evaluate multiplication and division operations from left to right.
Addition and Subtraction: Finally, evaluate addition and subtraction operations from left to right.

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, use the rules of fractions, such as $\frac{a}{b} \cdot \frac{c}{d} = \frac{ac}{bd}$ and $\frac{a}{b} + \frac{c}{d} = \frac{ad + bc}{bd}$ .

Q: What are some common mistakes to avoid when simplifying algebraic expressions?

A: Some common mistakes to avoid when simplifying algebraic expressions include:

Not identifying like terms correctly.
Not using the correct order of operations.
Not simplifying expressions with exponents correctly.
Not simplifying expressions with fractions correctly.