Simplify The Expression 9r(r + 2s - 9)
Simplifying algebraic expressions is a fundamental skill in mathematics. It allows us to rewrite complex expressions in a more manageable and understandable form. In this comprehensive guide, we'll break down the process of simplifying the expression 9r(r + 2s - 9) step by step. By understanding the underlying principles and techniques, you'll gain the confidence to tackle various algebraic simplifications.
Understanding the Distributive Property
At the heart of simplifying this expression lies the distributive property. The distributive property states that for any numbers a, b, and c:
a(b + c) = ab + ac
This property essentially allows us to multiply a term by a group of terms inside parentheses. In our case, we need to distribute the 9r across the terms inside the parentheses (r + 2s - 9). Let's break this down further:
To effectively apply the distributive property in this scenario, it is vital to first grasp its core concept. Essentially, this property provides a pathway for multiplying a single term by a group of terms enclosed within parentheses. It is a cornerstone of algebraic manipulation and simplification. In the given expression, 9r(r + 2s - 9), we have a term, 9r, that needs to be distributed across the terms inside the parentheses, which are r, 2s, and -9. This process involves multiplying 9r individually by each term within the parentheses and then combining the results. The distributive property not only simplifies calculations but also reveals the underlying structure of algebraic expressions, making them easier to understand and work with. By applying this property correctly, we can transform the original expression into a more expanded form, which can then be further simplified by combining like terms, if any exist. Therefore, a firm understanding of the distributive property is crucial for mastering algebraic simplification and solving more complex mathematical problems.
Step 1: Distribute 9r
We begin by multiplying 9r by each term inside the parentheses:
- 9r * r = 9r²
- 9r * 2s = 18rs
- 9r * (-9) = -81r
This step is crucial because it expands the original expression, making it easier to combine like terms and simplify further. Each term inside the parentheses is multiplied by 9r according to the distributive property, which is a fundamental concept in algebra. This process transforms the expression from a compact form to an expanded form, which allows us to see all the individual components. For instance, multiplying 9r by r results in 9r², which represents the square of the variable r multiplied by the coefficient 9. Similarly, multiplying 9r by 2s gives us 18rs, which represents the product of the variables r and s multiplied by the coefficient 18. Lastly, multiplying 9r by -9 yields -81r, which is a linear term in r. These individual products are then combined to form the expanded expression, setting the stage for the next step of simplification. Understanding how each term is derived and how the distributive property is applied is key to mastering algebraic manipulation.
Step 2: Combine the Terms
Now, we combine the results from the distribution:
9r² + 18rs - 81r
In this step, we simply write down the results of the distribution performed in the previous step. Each term that was obtained by multiplying 9r with the terms inside the parentheses is now placed together in a single expression. This combination is a direct application of the distributive property, which allows us to rewrite the original expression in an expanded form. At this stage, we have successfully removed the parentheses, which is a significant part of simplifying the expression. The terms 9r², 18rs, and -81r represent different components of the expanded expression. 9r² is a quadratic term, 18rs is a mixed term involving both r and s, and -81r is a linear term. Each of these terms plays a unique role in the overall expression. Combining them is not just a mechanical step but a representation of how the multiplication has transformed the original form. This step prepares us for the next stage of simplification, where we look for like terms that can be combined to further reduce the complexity of the expression. Understanding the nature and role of each term is crucial for further algebraic manipulations and problem-solving.
Step 3: Check for Like Terms
In the expression 9r² + 18rs - 81r, we need to identify if there are any like terms that can be combined to simplify the expression further. Like terms are terms that have the same variable raised to the same power. For instance, 3x² and 5x² are like terms because they both contain the variable x raised to the power of 2. However, 3x² and 5x are not like terms because, although they have the same variable x, the powers are different (2 and 1, respectively). Similarly, 3x² and 5y² are not like terms because they involve different variables, x and y. In the expression 9r² + 18rs - 81r, we have three terms: 9r², 18rs, and -81r. The term 9r² involves the variable r raised to the power of 2, making it a quadratic term. The term 18rs involves both variables r and s, making it a mixed term. The term -81r involves the variable r raised to the power of 1, making it a linear term. Comparing these terms, we can see that there are no like terms because each term has a unique combination of variables and powers. Therefore, we cannot combine any terms in this expression. This means that the expression is already in its simplest form, and no further simplification is possible. Recognizing and combining like terms is a fundamental step in simplifying algebraic expressions, as it reduces the number of terms and makes the expression easier to work with.
In our expression, 9r², 18rs, and -81r do not have any like terms. Therefore, we cannot simplify further.
Final Simplified Expression
Therefore, the simplified expression is:
9r² + 18rs - 81r
This is the most simplified form of the original expression. We have successfully applied the distributive property, combined terms, and checked for any further simplifications. This final expression represents the same mathematical relationship as the original but in a more concise and understandable format. Simplifying expressions is a crucial skill in algebra because it allows us to work with mathematical equations and formulas more efficiently. A simplified expression is easier to evaluate, differentiate, integrate, and use in further calculations. It also helps in identifying the underlying structure and properties of the expression. For instance, the simplified expression 9r² + 18rs - 81r reveals that the original expression is a quadratic function in terms of r, with an additional term involving the variable s. Understanding the final form of the expression allows us to analyze its behavior, such as finding its roots, determining its maximum or minimum values, and graphing it. Therefore, mastering the techniques of simplification is essential for success in algebra and related fields. This skill not only aids in solving specific problems but also enhances the overall understanding of mathematical concepts and principles.
Conclusion
Simplifying algebraic expressions like 9r(r + 2s - 9) involves a systematic approach that includes applying the distributive property and combining like terms. By following these steps, you can effectively reduce complex expressions to their simplest forms. Remember, practice is key to mastering these techniques. The more you work with algebraic expressions, the more comfortable and confident you will become in simplifying them. This ability is not only crucial for success in mathematics courses but also for various applications in science, engineering, and other fields. Simplifying expressions is a fundamental skill that enhances problem-solving capabilities and provides a deeper understanding of mathematical relationships. Therefore, make sure to review and practice these steps regularly to build a strong foundation in algebra. This guide has provided a detailed walkthrough of the process, highlighting the importance of each step and explaining the underlying principles. With consistent effort and practice, you will be able to tackle a wide range of algebraic simplifications with ease and accuracy.