Adding And Simplifying Algebraic Fractions 9/(5x^2) + 2/(25x) A Step-by-Step Guide

Hey guys! Today, we're diving into the world of algebraic fractions. Don't worry, it's not as scary as it sounds! We're going to break down how to add and simplify the expression 9/(5x^2) + 2/(25x). This is a common type of problem you'll see in algebra, so let's get right to it!

Understanding the Basics of Algebraic Fractions

Before we jump into the problem, let's quickly recap what algebraic fractions are. Think of them as regular fractions, but instead of just numbers, they also include variables (like x). The same rules for adding, subtracting, multiplying, and dividing regular fractions apply to algebraic fractions, but we need to be a little extra careful with the variables. When dealing with fractions, the most important thing to remember when adding or subtracting is that you must have a common denominator. This is the golden rule, guys! If the denominators are different, you can't directly add or subtract the numerators. So, that's our first hurdle to tackle in this problem.

Identifying the Denominators

In our expression, 9/(5x^2) + 2/(25x), we have two fractions. The first fraction has a denominator of 5x^2, and the second fraction has a denominator of 25x. See? Pretty straightforward. Now, the fun part begins: finding that common denominator! Remember, the goal is to rewrite each fraction so that they both have the same denominator, allowing us to combine them easily. Finding the Least Common Denominator (LCD) will help simplify the process and prevent us from dealing with unnecessarily large numbers. So, let's figure out how to get these denominators playing nice together.

Finding the Least Common Denominator (LCD)

The Least Common Denominator (LCD) is the smallest expression that both denominators can divide into evenly. It's like finding the smallest multiple that both numbers share. There are a couple ways to find the LCD, but let's use the method of prime factorization. First, we look at the coefficients (the numbers in front of the x). We have 5 and 25. The smallest number that both 5 and 25 divide into is 25. So, that's the numerical part of our LCD. Next, we look at the variable part. We have x^2 and x. The highest power of x is x^2, so that's what we'll use in our LCD. Putting it all together, the LCD for 5x^2 and 25x is 25x^2. That wasn't too bad, right? Now that we have the LCD, we can move on to rewriting our fractions.

Rewriting the Fractions with the Common Denominator

Okay, we've got our LCD: 25x^2. Now, we need to rewrite each fraction in the original expression so that they both have this denominator. To do this, we'll multiply each fraction by a clever form of 1. What do I mean by "clever form of 1"? Well, we'll multiply the numerator and denominator of each fraction by the same expression, which doesn't change the value of the fraction, but it does change its appearance. Let's start with the first fraction, 9/(5x^2). We want to change the denominator from 5x^2 to 25x^2. What do we need to multiply 5x^2 by to get 25x^2? The answer is 5! So, we'll multiply both the numerator and the denominator of the first fraction by 5. This gives us (9 * 5) / (5x^2 * 5) which simplifies to 45/(25x^2). See how we're getting there? Now, let's tackle the second fraction.

Adjusting the Second Fraction

Now, let's look at the second fraction, 2/(25x). We want to get the denominator to be 25x^2 again. This time, we need to multiply 25x by something to get 25x^2. What is that something? You guessed it: x! So, we'll multiply both the numerator and the denominator of the second fraction by x. This gives us (2 * x) / (25x * x), which simplifies to 2x/(25x^2). Awesome! We've successfully rewritten both fractions with the common denominator of 25x^2. Now, they're ready for some addition action! Remember, the key here is to make sure you multiply both the top and bottom of the fraction by the same thing. This keeps the value of the fraction the same, but it gets us one step closer to our final answer. So, with both fractions sporting the same denominator, we're ready to add them together. Let's do it!

Adding the Fractions

Alright, this is the fun part! Now that both fractions have the same denominator, 25x^2, we can finally add them together. Remember, when you add fractions with a common denominator, you simply add the numerators and keep the denominator the same. So, we have 45/(25x^2) + 2x/(25x^2). Adding the numerators, we get 45 + 2x. So, our expression becomes (45 + 2x) / (25x^2). We're almost there, guys! We've successfully added the fractions, but we're not quite finished yet. The last step is to see if we can simplify our result. This means looking for any common factors in the numerator and denominator that we can cancel out. So, let's put on our simplifying hats and see what we can do!

Combining the Numerators

We've reached the point where we can add the numerators together, since our denominators match perfectly. Our expression now looks like this: (45 + 2x) / (25x^2). This step is pretty straightforward, but it's crucial for getting to the simplified answer. We're not just slapping numbers together; we're following the rules of fraction addition. And with that, we've combined the numerators into a single expression. Now, the question is, can we make this expression any simpler? That's where the simplification step comes into play. We need to examine our new fraction and see if there are any common factors that we can divide out of both the numerator and the denominator. This is like detective work for math, searching for clues that will lead us to the most simplified form of our answer.

Simplifying the Result

Okay, we've got (45 + 2x) / (25x^2). Now, we need to see if we can simplify this fraction. Simplifying means looking for common factors in the numerator (45 + 2x) and the denominator (25x^2) that we can cancel out. But hold on a sec! Take a close look at the numerator. We have 45 + 2x. These terms are not like terms, meaning we can't combine them any further. The 45 is a constant, and the 2x has a variable. They're like apples and oranges – you can't just add them together. Now, let's think about factors. Does the numerator share any common factors with the denominator? The denominator, 25x^2, has factors of 5 and x. But if you look at the numerator, 45 + 2x, you'll see that it doesn't have a factor of 5 (2 doesn't divide by 5), and it doesn't have a factor of x (the 45 doesn't have an x). So, bummer! There are no common factors we can cancel out. This means our fraction is already in its simplest form!

Checking for Common Factors

When we talk about simplifying fractions, we're really talking about finding the greatest common factor (GCF) between the numerator and the denominator. If they share a GCF other than 1, we can divide both by that GCF to make the fraction simpler. It's like reducing a fraction to its lowest terms. But in our case, (45 + 2x) / (25x^2), a careful look reveals that there are no shared factors. The constant term in the numerator (45) and the coefficient of x (2) don't share any factors with the coefficient in the denominator (25), and the numerator as a whole doesn't have a factor of x. This might feel like a letdown – all that work, and no simplification to be done! But it's a good reminder that not every fraction can be simplified, and that's perfectly okay. Our goal is always to present the answer in its simplest form, and in this case, what we have is already as simple as it gets.

Final Answer

So, after all that, our final simplified answer is (45 + 2x) / (25x^2). We found the least common denominator, rewrote the fractions, added them together, and then checked for any simplifications. And guess what? We nailed it! Algebraic fractions might seem tricky at first, but with a little practice, you'll be adding and simplifying them like a pro. Remember the key steps: find the LCD, rewrite the fractions, add the numerators, and simplify if possible. You got this! Keep practicing, and you'll become a master of algebraic fractions in no time. And that's a wrap, folks! I hope this breakdown helped you understand how to add and simplify algebraic fractions. Until next time, keep those fractions in line!