Simplify Algebraic Fractions: X+y/xy * 2x^2y/x^2-y^2

Alright guys, let's dive into the awesome world of simplifying algebraic fractions! Today, we've got a doozy that looks a bit intimidating at first glance: rac{x+y}{x y} ullet rac{2 x^2 y}{x^2-y^2}. Don't sweat it, though! We're going to break it down step-by-step, and by the end, you'll be a pro at tackling these kinds of problems. So, grab your notebooks, get comfy, and let's get this math party started! First off, let's remember what simplifying algebraic fractions is all about. It's basically like reducing regular fractions, where we look for common factors in the numerator and the denominator and cancel them out. The goal is to get the expression into its simplest form, where no further cancellation is possible. This is super useful in higher-level math, like calculus or even when you're just trying to solve complex equations. It helps make things tidier and easier to work with. Our expression here involves multiplication of two fractions. When multiplying fractions, we multiply the numerators together and the denominators together. But before we do that, it's always a good idea to see if we can simplify before multiplying. This often saves us a ton of work and reduces the chance of making errors. So, let's take a good look at our expression: rac{x+y}{x y} ullet rac{2 x^2 y}{x^2-y^2}. We have $(x+y)$ in the numerator of the first fraction, and $xy$ in its denominator. Then we have $2x^2y$ in the numerator of the second fraction and $x^2-y^2$ in its denominator. Now, here's where the magic happens. We need to look for any factors that appear in both the numerator and the denominator across the entire expression. That means we can look at the numerator of the first fraction and the denominator of the second, or the numerator of the second and the denominator of the first, or even within the numerators and denominators of each fraction individually if they weren't already simplified. This is the beauty of multiplication – we can cross-cancel! Our first clue is that $x^2-y^2$ in the denominator of the second fraction. Does that look familiar? Yes, it's the difference of squares! And we know that the difference of squares can be factored. Specifically, $x^2-y^2 = (x-y)(x+y)$. This is a crucial step, guys, so make sure you've got this factorization down pat. It's one of those fundamental algebra skills that will serve you well. So, let's rewrite our expression with this factorization in mind. Our expression now looks like this: rac{x+y}{x y} ullet rac{2 x^2 y}{(x-y)(x+y)}. See what's happening? We have an $(x+y)$ term in the numerator of the first fraction, and we also have an $(x+y)$ term in the denominator of the second fraction. Bingo! These are common factors. Since we are multiplying, we can cancel these out. Think of it like this: we have rac{ ext{something} ullet (x+y)}{ ext{something else}} multiplied by rac{ ext{another thing}}{(x-y) ullet (x+y)}. The $(x+y)$ in the numerator and the $(x+y)$ in the denominator cancel each other out, leaving us with 1 in their place. So, after canceling out the $(x+y)$ terms, our expression simplifies significantly. It becomes: rac{1}{x y} ullet rac{2 x^2 y}{(x-y)}. We've made some serious progress, but we're not done yet! Let's keep our eyes peeled for more common factors. We have $xy$ in the denominator of the first fraction and $2x^2y$ in the numerator of the second. Let's break down $2x^2y$. It can be written as 2 ullet x ullet x ullet y. Our denominator has x ullet y. So, we have $x$'s and $y$'s that we can cancel. Specifically, we have an $x$ in the denominator and $x^2$ (which is x ullet x) in the numerator. We can cancel one $x$ from the denominator with one $x$ from the numerator. Similarly, we have a $y$ in the denominator and a $y$ in the numerator. We can cancel these out as well. After canceling one $x$ and one $y$, the term $2x^2y$ in the numerator becomes $2xy$, and the term $xy$ in the denominator becomes just 1. Let's show this clearly. We have rac{1}{x y} ullet rac{2 x^2 y}{(x-y)}. The numerator is 1 ullet 2x^2y = 2x^2y. The denominator is xy ullet (x-y). So, we are looking at rac{2x^2y}{xy(x-y)}. Now, we can see the common factors between the numerator $2x^2y$ and the denominator $xy(x-y)$. The factors in the numerator are $2$, $x$, $x$, and $y$. The factors in the denominator are $x$, $y$, and $(x-y)$. We can cancel one $x$ from the top with one $x$ from the bottom. We can also cancel the $y$ from the top with the $y$ from the bottom. So, what's left? In the numerator, we have $2$ and one $x$. That gives us $2x$. In the denominator, we have $(x-y)$ remaining. Therefore, the simplified expression is rac{2x}{x-y}. And voilà! We've successfully simplified the given algebraic expression. This process highlights the importance of factoring, especially recognizing patterns like the difference of squares, and then systematically canceling out common factors. Remember, guys, practice makes perfect! The more you work through these problems, the quicker you'll spot the common factors and the more confident you'll become. Keep at it, and you'll master these algebraic manipulations in no time!