Simplifying Rational Algebraic Expressions With Step-by-Step Solutions

Hey guys! Let's dive into simplifying rational algebraic expressions. I know it might sound intimidating, but trust me, we'll break it down step-by-step so it's super easy to understand. We're going to tackle five different expressions, showing you the solution for each one. So, grab your pencils, and let's get started!

1. Simplifying ${\frac{10x}{5x}}$

In this first expression, simplifying rational expressions starts with recognizing that we have a fraction where both the numerator and the denominator contain terms with the variable x. Our goal is to reduce this fraction to its simplest form. To do this, we identify common factors in both the numerator and the denominator. In this case, we have 10x in the numerator and 5x in the denominator. Notice that both 10 and 5 are divisible by 5, and both terms have x as a common factor. When simplifying algebraic fractions, we can think of it like this:

• Factor out common terms: We can rewrite 10x as 2 * 5 * x and 5x as 1 * 5 * x. Now, it’s clear that 5 and x are common factors.
• Divide out common factors: We divide both the numerator and the denominator by the common factors (5 and x). So, (2 * 5 * x) / (5 * x) simplifies to 2/1.
• Express in simplest form: The simplified form is simply 2 because any number divided by 1 is itself.

This process of simplifying rational expressions involves identifying common factors and eliminating them to reduce the fraction to its most basic form. Remember, the key is to look for numbers and variables that appear in both the top and bottom of the fraction. This method is fundamental in algebra and is used extensively when dealing with more complex equations and expressions. By understanding this basic principle, we can move forward to tackle more challenging problems with confidence. Simplifying expressions like these not only makes them easier to work with but also helps in solving equations and understanding the relationships between variables. So, let's move on to the next example and continue building our skills!

Solution:

$$\frac{10x}{5x} = \frac{2 \cdot 5 \cdot x}{5 \cdot x} = 2$$

2. Simplifying ${\frac{12y^2}{6y}}$

Alright, let's move on to the second expression. In this case, we're simplifying rational expressions that involve exponents. Don't let that scare you! We'll approach it in the same friendly, step-by-step way. We have 12y² in the numerator and 6y in the denominator. The y² means y multiplied by itself (y * y), so we have a y term in both the numerator and the denominator. Just like before, our aim is to find and cancel out common factors.

• Break down the terms: Let's rewrite the expression. 12 can be written as 2 * 6, and y² can be written as y * y. So, our fraction becomes (2 * 6 * y * y) / (6 * y).
• Identify and cancel common factors: We can see that 6 is a common factor, and so is one y. We divide both the numerator and the denominator by 6 and by y.
• Simplify what’s left: After canceling out the 6 and one y, we're left with 2 * y in the numerator and 1 in the denominator. So, our simplified expression is 2y.

See, simplifying algebraic fractions with exponents isn't so bad once you break it down! The important thing to remember is that when you have variables raised to powers, you're just multiplying that variable by itself a certain number of times. This makes it easier to identify common factors. This kind of simplification is crucial in many areas of algebra, such as solving equations, graphing functions, and dealing with polynomial expressions. It’s all about making the expression as neat and manageable as possible. As we continue, we'll see even more ways these skills come in handy. So, let’s keep rolling and check out the next problem!

Solution:

$$\frac{12y^2}{6y} = \frac{2 \cdot 6 \cdot y \cdot y}{6 \cdot y} = 2y$$

3. Simplifying ${\frac{2x + 4}{x + 2}}$

Now, let's tackle an expression where we have addition involved. This one might look a little trickier at first, but don't worry, we've got this! Simplifying rational expressions with addition or subtraction requires an extra step: factoring. Factoring is like the reverse of distributing; we're trying to find common factors within the terms of the numerator or denominator.

• Factor the numerator: Look at the numerator, 2x + 4. Do you see a common factor in both terms? That's right, it's 2! We can factor out a 2 from both terms, which gives us 2(x + 2).
• Rewrite the expression: Now our expression looks like this: (2(x + 2)) / (x + 2).
• Identify and cancel common factors: Notice that we have (x + 2) in both the numerator and the denominator. This whole expression (x + 2) is a common factor, so we can cancel it out.
• Simplify: After canceling, we're left with 2 in the numerator and 1 in the denominator, so our simplified expression is just 2.

This example really highlights how important factoring is when simplifying algebraic fractions. Factoring allows us to see the common terms that we can cancel out, even when they are part of a larger expression. Think of factoring as unlocking the hidden simplifications within the expression. This skill is essential for more advanced algebra topics like solving rational equations and working with rational functions. By mastering factoring, you'll be able to handle a wide variety of algebraic problems with confidence. So, keep practicing, and let's move on to the next challenge!

Solution:

$$\frac{2x + 4}{x + 2} = \frac{2(x + 2)}{x + 2} = 2$$

4. Simplifying ${\frac{6x^2 - 3x}{3x}}$

Okay, let's jump into another one! This time, we've got a mix of terms, including a squared variable and subtraction. But don't sweat it, we’ll break it down just like before. The key to simplifying rational expressions like this is, again, factoring. We need to find the common factors in the numerator and see if we can cancel anything out with the denominator.

• Factor the numerator: Look at the numerator: 6x² - 3x. What's the greatest common factor here? Well, both terms are divisible by 3, and they both have at least one x. So, the greatest common factor is 3x. Let’s factor that out: 3x(2x - 1).
• Rewrite the expression: Now we have (3x(2x - 1)) / (3x).
• Identify and cancel common factors: We can see that 3x is a common factor in both the numerator and the denominator. So, we cancel them out.
• Simplify: After canceling, we're left with 2x - 1. That's our simplified expression!

This problem really emphasizes the power of factoring in simplifying algebraic fractions. When you see an expression with multiple terms and subtraction (or addition), factoring is often the first thing you should try. It allows you to rewrite the expression in a way that makes common factors more obvious. This skill is incredibly useful in higher-level math, especially when dealing with polynomials and rational functions. By getting comfortable with factoring, you're setting yourself up for success in more advanced topics. Now, let’s keep the momentum going and tackle our final expression!

Solution:

$$\frac{6x^2 - 3x}{3x} = \frac{3x(2x - 1)}{3x} = 2x - 1$$

5. Simplifying ${\frac{9m + 18}{3m + 6}}$

Alright, we've made it to the final expression! Let's finish strong. This one looks similar to some of the previous ones, so we already have a good idea of what to do. Remember, our main strategy for simplifying rational expressions with addition is to factor first. So, let’s dive in and see what common factors we can find.

• Factor the numerator: Look at the numerator: 9m + 18. What's the greatest common factor here? Both terms are divisible by 9, so we can factor out a 9: 9(m + 2).
• Factor the denominator: Now let's look at the denominator: 3m + 6. What's the greatest common factor here? Both terms are divisible by 3, so we factor out a 3: 3(m + 2).
• Rewrite the expression: Our expression now looks like this: (9(m + 2)) / (3(m + 2)).
• Identify and cancel common factors: We've got (m + 2) in both the numerator and the denominator, so we can cancel that out. Also, notice that 9 and 3 have a common factor of 3. We can divide 9 by 3 to get 3.
• Simplify: After canceling, we're left with 3 in the numerator and 1 in the denominator, so our simplified expression is just 3.

Fantastic job, guys! You've made it through all five expressions. This last example really ties together everything we've learned about simplifying algebraic fractions. We used factoring in both the numerator and the denominator, and we canceled out not just single terms, but also entire expressions. This is a powerful technique that you'll use again and again in algebra. Remember, the key is to always look for common factors and to simplify as much as possible. With practice, you'll become a pro at simplifying these expressions. Keep up the great work, and you'll be acing those math problems in no time!

Solution:

$$\frac{9m + 18}{3m + 6} = \frac{9(m + 2)}{3(m + 2)} = \frac{9}{3} = 3$$

Conclusion

And there you have it! We've successfully simplified five different rational algebraic expressions, and hopefully, you feel a lot more confident about how to tackle these types of problems. Remember, the key to simplifying rational expressions is to look for common factors, and factoring is your best friend when dealing with addition or subtraction in the numerator or denominator. Keep practicing, and you'll master this skill in no time. You've got this! If you have any questions or want to explore more examples, feel free to reach out. Happy simplifying!