Simplifying Algebraic Expressions (10ab + 7ab² - 5ab² + 13) - (-7ab² + 3ab + 8 - 4ab)

Hey guys! Ever feel like algebraic expressions are just a jumble of letters and numbers? Don't worry, you're not alone! But trust me, once you understand the basics, they can actually be pretty fun to work with. Today, we're going to break down a specific expression: (10ab + 7ab² - 5ab² + 13) - (-7ab² + 3ab + 8 - 4ab). We'll walk through each step, so you can confidently tackle similar problems in the future.

Understanding the Expression

Before we dive into solving it, let's understand what this expression actually means. At its core, it's a subtraction problem involving two algebraic expressions enclosed in parentheses. Each expression contains terms with variables (like 'ab' and 'ab²') and constants (just numbers, like 13 and 8). The goal is to simplify this expression by combining like terms. To do this effectively, we need to understand the concept of like terms and the rules for addition and subtraction within algebraic expressions. Like terms are terms that have the same variables raised to the same powers. For example, 7ab² and -5ab² are like terms because they both contain 'ab²'. However, 10ab and 7ab² are not like terms because the powers of 'b' are different (b¹ in 10ab and b² in 7ab²). Recognizing like terms is crucial for simplifying expressions because only like terms can be combined. When we combine like terms, we are essentially adding or subtracting their coefficients (the numbers in front of the variables). This is similar to combining apples and oranges – you can’t combine them directly, but you can combine the numbers of each fruit if they are the same type. To simplify expressions, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). We first address any operations within parentheses, then exponents, followed by multiplication and division, and finally addition and subtraction. In the expression we are dealing with today, our primary focus will be on handling parentheses and then combining like terms through addition and subtraction.

Breaking Down the Parentheses

The first step is to get rid of those parentheses. But remember, there's a minus sign in front of the second set of parentheses, which means we need to distribute that negative sign to every term inside. Think of it like multiplying each term inside the parentheses by -1. This changes the sign of each term: positive becomes negative, and negative becomes positive. So, let's rewrite the expression by distributing the negative sign: (10ab + 7ab² - 5ab² + 13) + (7ab² - 3ab - 8 + 4ab). Notice how the signs of the terms inside the second set of parentheses have flipped. This is a crucial step because forgetting to distribute the negative sign is a common mistake that can lead to an incorrect answer. Now that we've eliminated the parentheses, our expression looks much cleaner and easier to work with. We've transformed the original subtraction problem into an addition problem, which simplifies the next steps. With the parentheses out of the way, we can now focus on identifying and combining like terms. This involves grouping together the terms that have the same variables raised to the same powers. By accurately distributing the negative sign, we’ve set the stage for the next phase of simplification, which will bring us closer to the final, simplified form of the algebraic expression. The importance of this initial step cannot be overstated, as it ensures that the subsequent steps are based on a correct foundation.

Identifying and Grouping Like Terms

Now comes the fun part: identifying and grouping those like terms! Remember, like terms have the same variables raised to the same powers. Let's go through our expression 10ab + 7ab² - 5ab² + 13 + 7ab² - 3ab - 8 + 4ab and highlight the like terms. We have 'ab' terms: 10ab, -3ab, and 4ab. These all have the same variables, 'a' and 'b', each raised to the power of 1. Next, we have 'ab²' terms: 7ab², -5ab², and 7ab². These terms have the same variables, 'a' and 'b', with 'b' raised to the power of 2. Finally, we have our constant terms: 13 and -8. These are just plain numbers without any variables. Grouping these terms together helps us visualize the next step, which is combining them. We can rewrite the expression as: (10ab - 3ab + 4ab) + (7ab² - 5ab² + 7ab²) + (13 - 8). This grouping makes it clear which terms we can combine. By arranging the like terms together, we reduce the likelihood of making errors when adding or subtracting. This step is not just about organization; it's about making the simplification process more intuitive and less prone to mistakes. Properly identifying and grouping like terms is a fundamental skill in algebra that allows us to tackle more complex expressions with confidence. This methodical approach ensures that we are only combining terms that are mathematically compatible, leading to an accurate simplification of the original expression.

Combining Like Terms and Simplifying

Alright, let's combine those like terms we've grouped together! We'll start with the 'ab' terms: 10ab - 3ab + 4ab. To combine these, we simply add or subtract their coefficients (the numbers in front of the variables). So, 10 - 3 + 4 equals 11. That means our 'ab' term simplifies to 11ab. Next up are the 'ab²' terms: 7ab² - 5ab² + 7ab². Again, we add or subtract the coefficients: 7 - 5 + 7 equals 9. So, our 'ab²' term simplifies to 9ab². Finally, we have our constant terms: 13 - 8. This is a simple subtraction, and 13 - 8 equals 5. Now, let's put it all together. We have 11ab, 9ab², and 5. Combining these gives us our simplified expression: 9ab² + 11ab + 5. We typically write the term with the highest power of the variable first, which is why we have 9ab² first, followed by 11ab, and then the constant term 5. This simplified expression is much cleaner and easier to understand than the original one. By combining like terms, we've reduced the complexity of the expression without changing its value. This process of simplification is a key skill in algebra and is used in many different contexts. It allows us to work with expressions more efficiently and to solve equations more easily. The ability to accurately combine like terms is essential for success in more advanced algebraic topics.

Final Result

So, there you have it! The simplified form of the expression (10ab + 7ab² - 5ab² + 13) - (-7ab² + 3ab + 8 - 4ab) is 9ab² + 11ab + 5. We did it! We took a seemingly complex expression, broke it down step by step, and simplified it to its core components. This whole process highlights the beauty and logic of algebra. By following simple rules and techniques, we can unravel even the most intricate expressions. Remember, the key is to understand the basics: like terms, distributing negative signs, and combining coefficients. With practice, you'll be simplifying algebraic expressions like a pro! I hope this explanation has helped you understand how to tackle these types of problems. Feel free to try similar expressions on your own, and you'll see how quickly you improve. Keep practicing, and you'll master algebra in no time!

Keep rocking the math world!