Simplify $19 A^7+(-30 A^7)$ Combining Like Terms In Algebra

Jul 17, 2025 by ADMIN 62 views

Simplify Algebraic Expressions: A Comprehensive Guide to Combining Like Terms

In the realm of mathematics, simplifying algebraic expressions is a fundamental skill. It allows us to take complex expressions and distill them into a more manageable form, making them easier to understand and work with. One common technique for simplifying expressions involves combining like terms. This process hinges on identifying terms that share the same variable and exponent, and then adding or subtracting their coefficients. Guys, let's dive into how we can simplify the expression $19 a^7+(-30 a^7)$ using this method. This article provides a detailed explanation and explores the underlying principles, offering valuable insights for anyone looking to master this essential algebraic skill. You'll be amazed at how straightforward it becomes once you grasp the core concepts. Let’s get started!

Understanding Like Terms

Before we jump into simplifying our expression, it's crucial to understand what like terms actually are. Think of them as the building blocks of algebraic simplification. Like terms are terms that have the same variable raised to the same power. The coefficient, which is the number multiplying the variable, can be different. For instance, in the expression $5x^2 + 3x - 2x^2 + 7$, the terms $5x^2$ and $-2x^2$ are like terms because they both have the variable $x$ raised to the power of 2. The term $3x$ is not a like term because it has $x$ raised to the power of 1, and 7 is a constant term, meaning it doesn't have any variable at all. Identifying like terms is the first step in simplifying any algebraic expression. It’s like sorting your laundry – you group similar items together to make the process easier. In our expression, $19 a^7$ and $(-30 a^7)$, we can see that both terms have the same variable, $a$, raised to the same power, 7. This makes them like terms, and we can proceed with combining them. Recognizing this similarity is the key to unlocking the simplified form of the expression. By understanding the concept of like terms, you're setting yourself up for success in simplifying a wide variety of algebraic expressions. This foundation will be invaluable as you tackle more complex problems in algebra and beyond. Remember, it’s all about identifying what’s similar and grouping it together – a principle that applies not only in math but also in many aspects of life.

Combining Like Terms: The Process

Now that we know what like terms are, let's get down to the nitty-gritty of combining them. The process is actually quite straightforward. It involves adding or subtracting the coefficients of the like terms while keeping the variable and exponent the same. Think of it as adding apples to apples or oranges to oranges – you're only combining things that are the same. The variable and exponent act as the unit of measurement, and the coefficient tells you how many of that unit you have. So, if you have 5 apples and you add 3 more apples, you end up with 8 apples. Similarly, in algebra, if you have $5x^2$ and you add $3x^2$, you get $8x^2$. The key here is to focus on the coefficients. In our expression, $19 a^7+(-30 a^7)$, the coefficients are 19 and -30. We need to add these two numbers together. This is where your basic arithmetic skills come into play. Adding a negative number is the same as subtracting the positive version of that number. So, $19 + (-30)$ is the same as $19 - 30$. This calculation gives us $-11$. Now that we have the new coefficient, we simply attach it to the variable and exponent, which remain unchanged. This gives us the simplified term $-11a^7$. It's like putting the apples back into the basket – you've counted them, and now you know you have 11 apples in total. The process of combining like terms is a fundamental skill in algebra, and it's used extensively in solving equations and simplifying more complex expressions. By mastering this technique, you'll be well-equipped to tackle a wide range of mathematical problems. Remember, the key is to identify the like terms, add or subtract their coefficients, and keep the variable and exponent the same. With a little practice, it will become second nature!

Applying the Process to Our Expression

Alright, let's put our newfound knowledge into action and simplify the expression $19 a^7+(-30 a^7)$. We've already identified that $19 a^7$ and $(-30 a^7)$ are like terms because they both have the variable $a$ raised to the power of 7. Now, it's time to combine them. As we discussed, this involves adding the coefficients. In this case, we need to add 19 and -30. Think of it as starting at 19 on a number line and moving 30 units to the left. This will take you into the negative numbers. The calculation is as follows: $19 + (-30) = 19 - 30 = -11$. So, the new coefficient is -11. Now, we simply attach this coefficient to the variable and exponent, which remain as $a^7$. This gives us the simplified term $-11a^7$. And that's it! We've successfully simplified the expression. The original expression, $19 a^7+(-30 a^7)$, is equivalent to the simplified expression $-11a^7$. Isn't that neat? By combining like terms, we've taken a slightly more complex expression and reduced it to its simplest form. This makes it easier to understand and work with. You can think of it as tidying up your mathematical workspace – you're getting rid of the clutter and making things clear and organized. This process is not only useful for simplifying expressions in textbooks but also has practical applications in various fields, such as physics, engineering, and computer science. By mastering this skill, you're not just learning math; you're developing a valuable problem-solving tool that can be applied in many different contexts. So, keep practicing, and you'll become a pro at simplifying algebraic expressions in no time!

The Simplified Expression

After carefully combining the like terms in the expression $19 a^7+(-30 a^7)$, we've arrived at our final answer. The simplified expression is $-11a^7$. This result represents the most concise form of the original expression, making it easier to understand and use in further calculations. The process of simplification not only reduces the complexity of the expression but also highlights its essential components. In this case, we've shown that the combination of $19 a^7$ and $-30 a^7$ ultimately results in a single term, $-11a^7$. This simplified form clearly illustrates the relationship between the variable $a$, its exponent 7, and the coefficient -11. It's like taking a complex machine and breaking it down into its core components – you gain a better understanding of how the whole thing works. Moreover, simplified expressions are crucial for solving equations and inequalities. When you encounter a complicated expression in an equation, simplifying it is often the first step towards finding the solution. By reducing the expression to its simplest form, you make the equation easier to manipulate and solve. This skill is invaluable in algebra and beyond. So, remember, the goal of simplifying expressions is not just to get a shorter answer; it's to gain a deeper understanding of the mathematical relationships and to make problem-solving more manageable. The simplified expression $-11a^7$ is a testament to the power of this process. It's a clear, concise, and easily understandable representation of the original, more complex expression. Keep this principle in mind as you tackle more challenging mathematical problems, and you'll find that simplification is a key tool in your arsenal.

Conclusion: Mastering Simplification

In conclusion, simplifying algebraic expressions is a fundamental skill in mathematics, and we've successfully simplified the expression $19 a^7+(-30 a^7)$ to $-11a^7$. The key to this process lies in understanding and combining like terms. By identifying terms with the same variable and exponent, and then adding or subtracting their coefficients, we can reduce complex expressions to their simplest forms. This not only makes the expressions easier to understand but also facilitates further calculations and problem-solving. Throughout this article, we've walked through the process step-by-step, from defining like terms to applying the process to our specific expression. We've emphasized the importance of understanding the underlying principles and the practical applications of simplification in various fields. Mastering this skill is not just about getting the right answer; it's about developing a deeper understanding of mathematical relationships and building a solid foundation for future studies in algebra and beyond. Think of it as building a house – a strong foundation is essential for the structure to stand tall. Similarly, a solid understanding of simplification is crucial for tackling more advanced mathematical concepts. So, keep practicing, keep exploring, and keep simplifying! The more you work with algebraic expressions, the more comfortable and confident you'll become. And remember, simplification is not just a mathematical technique; it's a mindset. It's about finding the most efficient and elegant way to solve a problem. By embracing this mindset, you'll not only excel in math but also develop valuable problem-solving skills that can be applied in all areas of your life. So, go forth and simplify, guys! The world of mathematics awaits your mastery.