Simplifying Algebraic Expressions Matching Expressions With Their Simplified Forms

Jul 9, 2025 by ADMIN 83 views

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It enables us to reduce complex expressions into more manageable and understandable forms. This article delves into the process of matching expressions with their simplified forms, providing a step-by-step guide to ensure clarity and accuracy. We will specifically address the expressions: $3a^5 - (-a^5)$ , $6a^7 - 2a^7$ , $-3a^5 - (-2a^5)$ , and $7a^7 - 8a^7$ . Through this detailed exploration, you will gain a solid understanding of how to simplify and match algebraic expressions effectively.

Understanding the Basics of Simplifying Algebraic Expressions

Before we dive into the specific examples, let’s first establish a strong foundation by understanding the core principles of simplifying algebraic expressions. At its heart, simplification involves combining like terms. Like terms are terms that have the same variable raised to the same power. For instance, $3a^5$ and $-a^5$ are like terms because they both contain the variable 'a' raised to the power of 5. Similarly, $6a^7$ and $-2a^7$ are like terms as they both have 'a' raised to the power of 7. However, $3a^5$ and $6a^7$ are not like terms because the powers of 'a' are different.

To simplify an expression, we perform arithmetic operations (addition, subtraction, multiplication, and division) on the coefficients of like terms. The coefficient is the numerical factor that multiplies the variable. For example, in the term $3a^5$ , the coefficient is 3. When combining like terms, we simply add or subtract the coefficients while keeping the variable and its exponent the same. This process streamlines the expression, making it easier to analyze and manipulate. Understanding these basics is crucial for tackling more complex algebraic problems and forms the backbone of our discussion today.

Detailed Simplification of Expressions

Expression 1: $3a^5 - (-a^5)$

Our initial focus is on the expression $3a^5 - (-a^5)$ . This expression involves subtracting a negative term, which, as many might recall, is equivalent to addition. To simplify, we transform the subtraction of a negative into addition: $3a^5 + a^5$ . Now we have two like terms, $3a^5$ and $a^5$ , which can be combined by adding their coefficients. The coefficient of the first term is 3, and the coefficient of the second term is implicitly 1 (since $a^5$ is the same as $1a^5$ ). Thus, we add the coefficients: $3 + 1 = 4$ . Therefore, the simplified form of the expression is $4a^5$ . This straightforward transformation highlights the importance of recognizing and applying the rules of arithmetic operations within algebraic contexts. The ability to correctly handle negative signs and coefficients is essential for simplifying any algebraic expression, ensuring accurate results and laying the groundwork for more complex manipulations.

Expression 2: $6a^7 - 2a^7$

The second expression we will simplify is $6a^7 - 2a^7$ . In this case, we are subtracting two like terms. Both terms contain the variable 'a' raised to the power of 7, making them directly combinable. To simplify, we subtract the coefficients of the terms. The coefficient of the first term is 6, and the coefficient of the second term is 2. Performing the subtraction, we have $6 - 2 = 4$ . Thus, the simplified form of the expression is $4a^7$ . This example clearly demonstrates the straightforward process of combining like terms by subtracting their coefficients. It underscores the principle that simplifying algebraic expressions often involves basic arithmetic operations applied to the coefficients of like terms, making the process both efficient and mathematically sound. Recognizing and correctly handling these operations is crucial for mastering algebraic simplification.

Expression 3: $-3a^5 - (-2a^5)$

Now, let's tackle the expression $-3a^5 - (-2a^5)$ . This expression, similar to the first one, involves subtracting a negative term. As we've established, subtracting a negative is the same as adding the positive counterpart. Therefore, we can rewrite the expression as $-3a^5 + 2a^5$ . Here, we have two like terms with coefficients -3 and 2. To simplify, we add these coefficients: $-3 + 2 = -1$ . Thus, the simplified form of the expression is $-1a^5$ , which is more commonly written as $-a^5$ . This simplification illustrates how crucial it is to correctly handle negative numbers in algebraic manipulations. By converting the subtraction of a negative to addition and then combining the coefficients, we arrive at the simplified form. This process highlights the importance of paying close attention to signs and their impact on the overall simplification.

Expression 4: $7a^7 - 8a^7$

Finally, we consider the expression $7a^7 - 8a^7$ . This expression involves subtracting two like terms, both containing the variable 'a' raised to the power of 7. The coefficients of the terms are 7 and -8, respectively. To simplify, we subtract the coefficients: $7 - 8 = -1$ . Therefore, the simplified form of the expression is $-1a^7$ , which is typically written as $-a^7$ . This final simplification underscores the importance of careful arithmetic when dealing with coefficients, especially when subtracting a larger number from a smaller one. The result is a negative coefficient, which correctly reflects the relationship between the terms in the original expression. Understanding and executing these subtractions accurately is essential for confidently simplifying algebraic expressions.

Matching Simplified Forms

Having simplified each expression, we can now match them with their corresponding simplified forms:

$3a^5 - (-a^5)$ simplifies to $4a^5$
$6a^7 - 2a^7$ simplifies to $4a^7$
$-3a^5 - (-2a^5)$ simplifies to $-a^5$
$7a^7 - 8a^7$ simplifies to $-a^7$

This matching process reinforces the importance of accurate simplification. By correctly performing each step, we can confidently pair the original expressions with their simplified counterparts. This exercise not only solidifies our understanding of algebraic simplification but also sharpens our ability to recognize and apply these skills in various mathematical contexts. The precision and attention to detail required in this process are crucial for success in algebra and beyond.

Conclusion: Mastering Simplification

In conclusion, simplifying algebraic expressions is a crucial skill in mathematics. By understanding the principles of combining like terms and accurately applying arithmetic operations, we can transform complex expressions into simpler, more manageable forms. The examples discussed in this article— $3a^5 - (-a^5)$ , $6a^7 - 2a^7$ , $-3a^5 - (-2a^5)$ , and $7a^7 - 8a^7$ —demonstrate the step-by-step process of simplification, emphasizing the importance of careful attention to signs and coefficients. Matching the original expressions with their simplified forms reinforces the need for accuracy and precision in algebraic manipulations. Mastering these skills will not only enhance your mathematical abilities but also provide a solid foundation for tackling more advanced algebraic concepts. The ability to simplify expressions effectively is a cornerstone of mathematical proficiency, enabling you to approach complex problems with confidence and clarity. Continue practicing and applying these techniques to further solidify your understanding and expertise in algebra.