Simplifying Algebraic Expressions: Unraveling (y)(xy)

Hey guys! Let's dive into a fun little math problem. We're going to simplify the expression (y)(xy). Don't worry, it's not as scary as it looks! We'll break it down step by step, and by the end, you'll be a pro at simplifying algebraic expressions. This type of problem often pops up in algebra, so understanding it is super important. We'll explore the different options and figure out which one is the correct simplified form. Ready to get started? Let's go!

Understanding the Basics of Simplification

Alright, before we jump into the problem, let's quickly recap some basic rules of simplification. When we talk about simplifying expressions, we're basically trying to make them as concise and easy to read as possible. This usually involves combining like terms and applying the rules of exponents. Remember, the goal is always to rewrite the expression in an equivalent form that's less cluttered. So, what are like terms? They are terms that have the same variables raised to the same powers. For example, 3x and 5x are like terms because they both have the variable 'x' raised to the power of 1. You can add or subtract like terms, but you can't combine unlike terms. Now, let's talk about exponents. When you multiply terms with the same base, you add their exponents. For instance, x² * x³ = x^(2+3) = x⁵. Also, remember that any term raised to the power of 0 is equal to 1. Got it? Awesome! We'll be using these concepts throughout our simplification journey. These concepts are the bedrock of algebra, so a firm grasp will serve you well. It's like having a superpower that lets you conquer complex equations. The more comfortable you are with these fundamentals, the easier it will be to tackle more advanced problems. This simplification process is all about making things cleaner and easier to work with, allowing you to see the underlying structure of the expression more clearly. Therefore, understanding the basics is paramount to simplifying any algebraic expression. So, let's keep these rules in mind as we tackle our problem.

Breaking Down the Expression (y)(xy)

Okay, let's take a look at our expression: (y)(xy). The first thing to notice is that we're multiplying 'y' by 'xy'. When you see variables next to each other, like 'xy', it means they're being multiplied. So, 'xy' is the same as 'x * y'. Now, let's rewrite our original expression, explicitly showing the multiplication: y * x * y. Now, to simplify, we can rearrange the terms. Multiplication is commutative, which means we can change the order of the terms without changing the result. So, y * x * y is the same as x * y * y. Now, what's y * y? Well, that's y². So, our expression simplifies to x * y². Remember that when we multiply the same variable by itself, we raise it to the power of the number of times it appears. In this case, 'y' appears twice, hence y². So, the simplified form of (y)(xy) is xy². This is the most compact and easily understandable form of the original expression. Simplifying like this helps you quickly grasp the essence of the problem and makes further calculations easier. This also minimizes the chances of making errors when dealing with more complex expressions. Also, it’s a good practice to always double-check your work to ensure everything is correct. It helps in building your confidence in solving mathematical problems.

Analyzing the Answer Choices

Now, let's look at the answer choices provided. We need to figure out which one matches our simplified form of xy². Here are the options:

a. (xz)^-2 b. (xz³)^-2 c. (xz)² d. (xz³) e. (xz)²

Let's analyze each one. Notice that our simplified expression, xy², contains 'x' and 'y', but none of the answer choices directly have 'y' in the format. This looks tricky, but let's go back and reconsider our steps. Since we simplified the original expression to xy², there seems to be an error in the answer choice; perhaps it was intended to have y in the answers. However, let's check what the answer choices say. We need to find an answer choice that, when simplified, results in xy². Since none of the answers contain 'y', something seems incorrect. However, with the format provided, the correct answer should be: there is no correct answer since 'y' does not exist in the answer choices. But let's check anyway. A close match, if we had to choose, would be any choice if we changed the 'x' in the answer choices with 'y'. Therefore, based on the format provided, this question cannot be answered correctly. However, if we were to change each 'x' in the answer choice with 'y', then option 'c. (xz)²' would be the correct answer. The process of elimination is helpful because we know what the correct answer should be. Then we can evaluate the answers more effectively and avoid common traps that can lead to wrong answers. Always remember to check if your answer matches your expectations. So, it's essential to understand and apply these concepts to accurately assess the answer choices and choose the correct one. Remember to revisit your work when the answers don't look correct. This will improve your math skills!

Conclusion

So, to wrap things up, we've successfully simplified the expression (y)(xy). Although the answer choices do not align with our simplified version, the correct way to solve for the correct answer would be to change each 'x' with 'y'. We walked through the basic concepts of simplification, rearranged the terms, and combined like terms to arrive at our answer. Remember, practicing these types of problems is key to mastering algebra. Keep working at it, and you'll become a pro in no time! So, keep up the great work, and happy simplifying!