Simplifying Algebraic Expressions: A Step-by-Step Guide

by Rajiv Sharma 56 views

Hey everyone! Let's break down this math problem together. It looks like we're dealing with algebraic expressions and need to simplify them. Don't worry, it's not as scary as it seems! We'll go through each step, making sure everything is crystal clear. So, let’s dive into understanding and simplifying algebraic expressions. This topic is super important in mathematics, and mastering it will help you ace not just your current test but also future math challenges. Think of algebraic expressions as a kind of mathematical puzzle where we combine numbers, variables, and operations. Variables are just letters (like x and y) that stand in for unknown numbers. Our job is often to simplify these expressions, making them easier to work with. This usually means combining like terms – terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms because they both have x to the power of 1. But 3x and 3x^2 are not like terms because the powers of x are different. Operations like addition, subtraction, multiplication, and division are the glue that holds these terms together. Remember the order of operations (PEMDAS/BODMAS) – Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This order is crucial when simplifying expressions to ensure we get the correct answer. Now, let's get into the nitty-gritty of simplifying the expressions you've provided. We'll take it one step at a time, so you feel confident tackling similar problems on your own. Remember, practice makes perfect, so the more you work with these concepts, the easier they'll become.

Breaking Down the Expressions

Okay, let's take a closer look at the expressions we have: 3/4xy, -5xy^2, 6x^2y, 2xy^2, and xy^2y - 2x^2y. The key here is to identify like terms. Remember, like terms have the same variables raised to the same powers. Let's break each term down to make it easier. First up, we have 3/4xy. This term has x to the power of 1 and y to the power of 1. It's a straightforward term and we'll keep it as is for now. Next, we've got -5xy^2. Here, x is to the power of 1 and y is to the power of 2. Keep this in mind as we look for other terms with the same variable powers. Then comes 6x^2y. Notice that x is now squared (to the power of 2) and y is to the power of 1. This is different from our previous term, so it's not a like term with -5xy^2. Following that, we have 2xy^2. Ah, this looks familiar! x is to the power of 1 and y is to the power of 2, just like -5xy^2. This means 2xy^2 and -5xy^2 are like terms, and we'll be able to combine them later. Now, let's tackle xy^2y - 2x^2y. This one looks a bit more complicated, but let's simplify it. xy^2y can be rewritten as xy^(2+1) which simplifies to xy^3. So, this term has x to the power of 1 and y to the power of 3. The second part, -2x^2y, is the same as the 6x^2y term we saw earlier – x is squared and y is to the power of 1. Now that we've broken down each term, it's much easier to see which ones are like terms and can be combined. This is a crucial step in simplifying algebraic expressions, so make sure you're comfortable identifying like terms before moving on. We're well on our way to simplifying this expression!

Combining Like Terms: Step-by-Step

Alright, now for the fun part – combining those like terms! This is where we actually simplify the expression and make it look cleaner. Remember, we identified the like terms in the previous section. We had 3/4xy, -5xy^2, 6x^2y, 2xy^2, and xy^3 - 2x^2y. Let's focus on the like terms first. We noticed that -5xy^2 and 2xy^2 are like terms because they both have x to the power of 1 and y to the power of 2. To combine them, we simply add their coefficients (the numbers in front of the variables). So, -5xy^2 + 2xy^2 becomes (-5 + 2)xy^2, which simplifies to -3xy^2. Great! We've combined our first set of like terms. Next, we have 6x^2y and -2x^2y (from the xy^2y - 2x^2y term). These are also like terms because they both have x squared and y to the power of 1. Again, we add their coefficients: 6x^2y + (-2x^2y) becomes (6 - 2)x^2y, which simplifies to 4x^2y. Awesome, another set of like terms combined! Now, let's look at the remaining terms. We have 3/4xy and xy^3 (from the simplified xy^2y term). These terms are not like any other terms in the expression, so we can't combine them further. They'll just stay as they are in our simplified expression. So, putting it all together, our simplified expression looks like this: 3/4xy - 3xy^2 + 4x^2y + xy^3. We've taken the original, somewhat messy expression and made it much simpler and easier to understand. This is the power of combining like terms! It's like tidying up a messy room – everything is in its place and easier to find. Remember, the key is to carefully identify the like terms and then combine their coefficients. With practice, you'll become a pro at this. Let's move on to the final simplified form and recap what we've learned.

The Final Simplified Form and Key Takeaways

Okay, guys, after all that simplifying, we've arrived at the final form of our expression. Remember, we started with 3/4xy, -5xy^2, 6x^2y, 2xy^2, and xy^2y - 2x^2y. After carefully breaking down the terms, identifying like terms, and combining them, we ended up with: 3/4xy - 3xy^2 + 4x^2y + xy^3. This is our simplified expression! It's much cleaner and easier to work with than the original. You can see how combining like terms really helps to make algebraic expressions more manageable. Now, let's take a moment to recap the key takeaways from this exercise. First and foremost, the most important thing is to identify like terms. Remember, like terms have the same variables raised to the same powers. This is the foundation of simplifying algebraic expressions. Without correctly identifying like terms, you won't be able to combine them properly. Second, once you've identified the like terms, you combine them by adding or subtracting their coefficients. The coefficients are the numbers in front of the variables. Just remember to pay attention to the signs (positive or negative) when you're adding or subtracting. Third, don't forget to simplify any terms before you start combining like terms. In our example, we had to simplify xy^2y to xy^3 before we could determine if it was a like term with anything else. Fourth, practice makes perfect! The more you work with algebraic expressions, the easier it will become to identify like terms and simplify them. Try working through different examples and challenging yourself with more complex expressions. Finally, always double-check your work to make sure you haven't made any mistakes. It's easy to make a small error, especially when dealing with multiple terms and variables. By double-checking, you can catch any errors and ensure you have the correct simplified expression. So, there you have it! We've successfully simplified a complex algebraic expression and learned some valuable tips along the way. I hope this has helped you feel more confident in your ability to tackle similar problems. Remember, math is like a puzzle, and simplifying expressions is just one piece of the puzzle. Keep practicing, keep learning, and you'll be amazed at what you can achieve!

I hope this explanation helps you ace your test! Good luck, and remember to take it one step at a time. You've got this!