Simplifying Rational Expressions: A Step-by-Step Guide

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Hey guys! Let's dive into the world of simplifying rational expressions. This can sometimes seem a bit tricky, but trust me, with a clear understanding of the steps involved, you'll be acing these problems in no time. We're going to break down the given expression and find its equivalent form. So, let's get started!

Understanding the Problem: The Core Concepts

Our main goal here is to simplify a complex fraction. This means we have a fraction within a fraction. The key to tackling this is to remember how we divide fractions. When you divide by a fraction, it's the same as multiplying by its reciprocal. Also, we'll need to use our knowledge of factoring. Factoring will help us identify common factors in the numerator and denominator, which we can then cancel out. Keep in mind that we need to be careful about the values of m that would make the denominator zero, as division by zero is undefined. Let's make sure we're always aware of these excluded values.

Now, let's look at the expression we're given: $ rac{ rac{m+3}{m^2-16}}{ rac{m^2-9}{m+4}}$. It might look a little intimidating at first glance, but fear not! We'll simplify this step by step. Remember, the core idea is to change the division into multiplication using the reciprocal. We will rewrite the complex fraction as a multiplication problem. This makes the expression more manageable and easier to simplify. We need to remember the order of operations as we go through the simplification, as this is very important. This also means we'll apply our knowledge of factoring different types of expressions, such as difference of squares and quadratic expressions. This will allow us to see which factors cancel each other out, thus simplifying the initial expression.

Step-by-Step Simplification

Alright, let's start the simplification process, taking it one step at a time. This should clarify how to solve this kind of expression, and it helps to understand better how to break down complex expressions. We'll show you exactly how to transform the division problem into a multiplication problem, what to factor, and what cancels out. This means it can all be broken down into simpler terms.

First, we'll rewrite the division as a multiplication problem by taking the reciprocal of the denominator. So, the expression becomes: $ rac{m+3}{m^2-16} imes rac{m+4}{m^2-9}$. See, it's already looking less scary, right?

Next, we'll factor each of the expressions. Let's start with m² - 16. This is a difference of squares, and it factors into (m - 4)(m + 4). Next, we look at m² - 9, which is also a difference of squares and factors into (m - 3)(m + 3). So now, our expression looks like this: $ rac{m+3}{(m-4)(m+4)} imes rac{m+4}{(m-3)(m+3)}$. Look at how we can start to see things cancelling out already!

Now, we'll cancel out the common factors. We have (m + 3) in both the numerator and denominator, and we have (m + 4) in both the numerator and denominator. This leaves us with $ rac{1}{m-4} imes rac{1}{m-3}$. Finally, multiply these to have the final answer. The answer is $ rac{1}{(m-4)(m-3)}$.

Analyzing the Answer Choices

Now, let's look at the answer choices you provided and see which one matches our simplified expression. The question asks us to identify the expression equivalent to the given one, so the only one we need to match to is the final answer we came up with. We'll go through the given options and choose the correct one. This process of elimination is important to make sure we arrive at the correct final expression.

  • Option A: $ rac{1}{(m+4)(m+3)}$ This option is incorrect because the denominator has the wrong factors. It doesn't match our simplified expression, which is $ rac{1}{(m-4)(m-3)}$.
  • Option B: $ rac{1}{(m-4)(m-3)}$ This is the correct answer! It matches our simplified expression perfectly. We factored everything correctly and cancelled out the right terms, so we know it's the correct answer.
  • Option C: $ rac{m+3}{m+4}$ This option is incorrect because we cancelled out the factors of (m + 3) and (m + 4) during the simplification process. This does not match our simplified expression, which is $ rac{1}{(m-4)(m-3)}$.

Conclusion: Mastering Rational Expression Simplification

So there you have it, guys! We've successfully simplified the given rational expression. The answer is Option B: $ rac{1}{(m-4)(m-3)}$. By understanding the rules of dividing fractions, recognizing patterns in factoring, and carefully cancelling out common factors, you can conquer these types of problems. Remember to always double-check your work and pay attention to those excluded values. Practice makes perfect, so keep working through different examples, and you'll become a pro at simplifying rational expressions. Keep in mind that we're always looking to simplify the expressions by canceling out the same terms that appear in the numerator and the denominator.

Keep practicing, and you'll be acing these problems in no time! Keep in mind that these kinds of expressions are a fundamental aspect of algebra and are used in solving more complex mathematical problems. Mastering them is a key step towards higher-level math. Good luck, and keep up the great work!