Simplifying Rational Expressions Solving (4a + 4)/(2a) * A^2/(a + 1)

Alright, math enthusiasts! Let's dive into simplifying some rational expressions. This is a classic algebra problem that combines factoring, cancellation, and a bit of algebraic manipulation. We're going to break down the expression (4a + 4)/(2a) * a^2/(a + 1) step by step, so you'll not only get the answer but also understand the how and why behind it. Our mission is to figure out which of the provided options (A, B, C, or D) is equivalent to our given expression. So, buckle up and let's get started!

Before we even think about multiplying these fractions, let’s take a good look at each part and see if there’s anything we can simplify right off the bat. Factoring is your best friend in these situations. When you see a sum or difference in the numerator or denominator, ask yourself, “Can I factor anything out?” In our expression, 4a + 4 immediately screams, “Factor me!” There's a common factor of 4. So, let's rewrite 4a + 4 as 4(a + 1). Now, our expression looks like (4(a + 1))/(2a) * a^2/(a + 1). See how much cleaner it already looks? This is the magic of factoring, guys – it transforms complicated expressions into manageable bits. Factoring isn't just a trick; it's a fundamental tool in algebra. It helps us reveal hidden structures and simplifies calculations. By identifying and extracting common factors, we're essentially decluttering the expression, making it easier to spot opportunities for cancellation and further simplification. Remember, the goal is to break down the problem into its simplest components, and factoring is a powerful way to achieve this. So, whenever you encounter expressions like 4a + 4, make factoring your first instinct. It’s like having a secret weapon in your math arsenal. And trust me, this weapon will come in handy time and time again as you tackle more complex algebraic problems. The key takeaway here is that factoring is not just about finding common factors; it’s about transforming the expression into a form that reveals its underlying structure and makes it easier to manipulate. So, let’s keep this in mind as we move forward and continue simplifying our expression.

Step 1 Factoring and Simplifying

Now that we've factored 4a + 4 into 4(a + 1), let's rewrite the entire expression: (4(a + 1))/(2a) * a^2/(a + 1). The next thing to observe is whether any terms can be cancelled between the numerator and the denominator. Notice that we have an (a + 1) term in both the numerator and the denominator. This is fantastic news because it means we can cancel them out! When we cancel (a + 1) from both the numerator and the denominator, we're left with (4/2a) * a^2. Ah, much simpler, isn't it? But we're not done yet. There's still more simplification we can do. Always remember, simplifying algebraic expressions is like peeling an onion – you go layer by layer until you reach the core. In this case, we've peeled off the first layer by factoring and cancelling the (a + 1) terms. Now, let's look at what remains and see what other simplifications we can make. This step is crucial because it prevents us from getting bogged down in unnecessary complexity. By simplifying early, we keep the expression manageable and reduce the chances of making errors later on. Think of it like cleaning up your workspace before starting a new task – a clear workspace leads to clearer thinking and better results. So, let's continue our cleanup operation and see what other hidden gems we can uncover in this expression. Remember, the goal is to make the expression as simple and straightforward as possible, and we're well on our way to achieving that.

Step 2 Further Simplification

Looking at (4/2a) * a^2, we can simplify the numerical part of the fraction. Notice that 4 and 2 have a common factor. You guessed it! We can divide both 4 and 2 by 2. This gives us 2/a * a^2. Now, we have a single fraction multiplied by a term with a variable. What do we do next? Well, remember that a^2 is just a * a. So, we can rewrite the expression as (2/a) * (a * a). Do you see any more opportunities for cancellation? I sure do! We have an 'a' in the denominator of the fraction and an 'a' in the numerator (within the a * a term). We can cancel out one 'a' from each, leaving us with 2 * a. And there you have it! We've simplified the expression down to its core. This step is all about recognizing the relationships between terms and using the rules of algebra to our advantage. By breaking down a^2 into a * a, we made it easier to see the cancellation opportunity. This is a common strategy in simplifying expressions – look for ways to rewrite terms so that common factors become more apparent. It's like having a detective's eye, always searching for clues that can help you crack the case. And in this case, the clue was the hidden 'a' within a^2. So, remember, when you're simplifying expressions, don't be afraid to rewrite terms, break them down, and look for those hidden opportunities for cancellation. It's these small steps that lead to big simplifications.

Step 3 The Final Result

So, after all that simplifying, we've arrived at 2 * a, which is simply 2a. Now, let's circle back to our original question: Which expression is equivalent to (4a + 4)/(2a) * a^2/(a + 1)? We've successfully transformed the original expression into 2a. Looking at the options provided, we can see that option C, 2a, is the correct answer. Woo-hoo! We did it! But the real victory here isn't just getting the right answer; it's understanding the process. We took a seemingly complex expression, broke it down into smaller, manageable parts, and simplified it step by step. This is the essence of algebra – taking the complicated and making it simple. And this skill isn't just useful in math class; it's a valuable life skill. It teaches you to approach problems methodically, to look for patterns, and to break down challenges into smaller, more achievable steps. So, pat yourself on the back, not just for getting the answer right, but for mastering the process. You've added a powerful tool to your problem-solving toolkit. And remember, the more you practice these techniques, the more natural they'll become. So, keep simplifying, keep exploring, and keep conquering those algebraic expressions!

Answer

The equivalent expression is 2a, which corresponds to option C.

Final Answer: C