Simplifying Rational Algebraic Expressions With Solutions

Hey guys! Let's dive into the world of simplifying rational algebraic expressions. It might sound intimidating, but trust me, it's like solving a puzzle – super satisfying once you get the hang of it. We're going to break down five examples step-by-step, so you'll be simplifying like a pro in no time. So, let's buckle up and get started!

Understanding Rational Algebraic Expressions

Before we jump into the examples, let's quickly recap what rational algebraic expressions are. Think of them as fractions where the numerator and denominator are polynomials. A polynomial is an expression made up of variables and coefficients, combined using addition, subtraction, and non-negative exponents. For instance, 3x² + 2x - 1 is a polynomial. When we have one polynomial divided by another, we've got ourselves a rational algebraic expression. Simplifying these expressions is like reducing fractions – we want to find the simplest form possible by canceling out common factors. This not only makes the expression easier to work with but also provides a clearer understanding of its underlying structure and behavior. Simplification is crucial in various mathematical contexts, such as solving equations, graphing functions, and performing calculus operations. By mastering the techniques of simplifying rational expressions, you'll build a solid foundation for more advanced mathematical concepts and problem-solving scenarios.

Example 1: Simplifying ${\frac{10x}{5x}}$

Our first example is ${\frac{10x}{5x}}$. When simplifying rational expressions, always look for common factors in both the numerator and the denominator. Common factors are terms that divide evenly into both parts of the fraction. In this case, we see that both 10x and 5x have a common factor of 5 and x. Let's break it down: 10x can be written as 5 * 2 * x, and 5x is simply 5 * x. Now, we can rewrite the expression as ${\frac{5 * 2 * x}{5 * x}}$. See those matching 5s and xs? They can be canceled out because any number (except 0) divided by itself is 1. So, ${\frac{5}{5}}$ becomes 1 and ${\frac{x}{x}}$ becomes 1. After canceling the common factors, we're left with just 2 in the numerator. There's nothing left in the denominator, which means it's essentially 1 (we don't usually write it, but it's there). Therefore, the simplified form of ${\frac{10x}{5x}}$ is simply 2. Remember, the key is to identify and cancel out those common factors, making the expression as simple as possible. This process not only simplifies the expression but also helps in understanding its fundamental structure and behavior, which is essential for further mathematical operations and analysis.

Solution:

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

Example 2: Simplifying ${\frac{12y^2}{6y}}$

Next up, we have ${\frac{12y^2}{6y}}$. This one looks a little more complex, but don't worry, we'll tackle it the same way. The key is to identify the common factors between the numerator and the denominator. Let's break down the terms: 12y² can be expressed as 2 * 6 * y * y (since y² means y multiplied by itself), and 6y is 6 * y. So, we can rewrite the expression as ${\frac{2 * 6 * y * y}{6 * y}}$. Now, let's hunt for those common factors! We see a 6 in both the numerator and the denominator, and we also have a y in both places. Just like in the previous example, we can cancel these common factors out. The 6 in the numerator and the 6 in the denominator cancel each other out, becoming 1. Similarly, one of the ys in the numerator cancels out with the y in the denominator. After canceling, we're left with 2 * y in the numerator. The denominator is essentially 1 (since everything canceled out). Thus, the simplified form of ${\frac{12y^2}{6y}}$ is 2y. Simplifying algebraic expressions involves systematically breaking down terms into their factors and identifying common elements that can be canceled out. This approach not only simplifies the expression but also reveals its underlying structure, making it easier to work with in more complex equations or mathematical problems.

Solution:

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

Example 3: Simplifying ${\frac{2x + 4}{x + 2}}$

Now we're moving onto something a bit trickier: ${\frac{2x + 4}{x + 2}}$. Notice that we have addition in both the numerator and the denominator. The crucial step here is factoring. Factoring is like the reverse of expanding – we're looking for common factors within the terms of the expression. In the numerator, 2x + 4, we can see that both terms have a common factor of 2. We can factor out the 2, which means dividing each term by 2 and writing it outside a set of parentheses. Factoring 2 out of 2x gives us x, and factoring 2 out of 4 gives us 2. So, the numerator becomes 2(x + 2). The denominator, x + 2, is already in its simplest form. Now, our expression looks like ${\frac{2(x + 2)}{x + 2}}$. Aha! We now have a common factor of (x + 2) in both the numerator and the denominator. Just like before, we can cancel these out because they're the same. Once we cancel (x + 2) from the top and the bottom, we're left with just 2 in the numerator. The denominator is effectively 1. Therefore, the simplified form of ${\frac{2x + 4}{x + 2}}$ is 2. Factoring is a fundamental technique in simplifying rational expressions, as it allows us to identify common terms that might not be immediately obvious. By factoring, we can rewrite the expression in a form that highlights the common factors, making it easier to simplify and work with in further calculations.

Solution:

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

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

Alright, let's tackle ${\frac{6x^2 - 3x}{3x}}$. This one involves subtraction in the numerator, so we'll use factoring again. Look at the numerator, 6x² - 3x. What common factors can we spot? Well, both terms have a factor of 3, and they both have at least one x. So, the greatest common factor (GCF) is 3x. Let's factor out 3x from the numerator. Dividing 6x² by 3x gives us 2x (because 6 divided by 3 is 2, and x² divided by x is x). Dividing -3x by 3x gives us -1. So, the numerator becomes 3x(2x - 1). Our expression now looks like ${\frac{3x(2x - 1)}{3x}}$. Do you see any common factors between the numerator and the denominator? You got it – we have 3x in both places! We can cancel these common factors out. The 3x in the numerator and the 3x in the denominator cancel each other out, leaving us with (2x - 1) in the numerator. The denominator is effectively 1. So, the simplified form of ${\frac{6x^2 - 3x}{3x}}$ is 2x - 1. Remember, guys, factoring out the greatest common factor is a super powerful technique for simplifying expressions. It helps us rewrite the expression in a way that makes the common factors more apparent, leading to easier simplification and a clearer understanding of the expression's structure.

Solution:

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

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

Last but not least, we have ${\frac{9m + 18}{3m + 6}}$. This one looks a bit like our earlier example with addition, so we'll use the same strategy: factoring. Let's start with the numerator, 9m + 18. What's the greatest common factor here? Both 9m and 18 are divisible by 9. Factoring out 9 from 9m gives us m, and factoring out 9 from 18 gives us 2. So, the numerator becomes 9(m + 2). Now let's tackle the denominator, 3m + 6. The greatest common factor here is 3. Factoring out 3 from 3m gives us m, and factoring out 3 from 6 gives us 2. So, the denominator becomes 3(m + 2). Our expression now looks like ${\frac{9(m + 2)}{3(m + 2)}}$. Any common factors jump out at you? We have (m + 2) in both the numerator and the denominator, so we can cancel those out. But wait, there's more! We also have 9 in the numerator and 3 in the denominator. These are both divisible by 3, so we can simplify further. 9 divided by 3 is 3, and 3 divided by 3 is 1. After all the canceling and simplifying, we're left with 3 in the numerator and 1 in the denominator (which we don't usually write). So, the simplified form of ${\frac{9m + 18}{3m + 6}}$ is 3. This example really showcases the power of factoring and simplifying in stages. By identifying and canceling common factors step-by-step, we can break down complex expressions into their simplest forms, making them much easier to understand and manipulate in further calculations.

Solution:

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

Conclusion

So, there you have it, guys! We've walked through simplifying five different rational algebraic expressions. Remember, the key is to look for common factors, whether they're simple numbers and variables or more complex expressions. Factoring is your best friend when you have addition or subtraction in the numerator or denominator. By breaking down each term and identifying the greatest common factor, you can rewrite the expression in a simpler form. And don't forget to cancel out those common factors from both the numerator and the denominator – that's where the magic happens! Simplifying rational expressions might seem challenging at first, but with practice, you'll become a pro at spotting those common factors and simplifying like a champ. Keep practicing, and you'll master this essential skill in no time. You've got this!