Simplifying $\frac{4a^2-36a}{2a^4-24a^3+54a^2}$

Hey math enthusiasts! Let's dive into the world of algebraic fractions and learn how to simplify expressions like the one we've got: $\frac{4a^2-36a}{2a^4-24a^3+54a^2}$. Don't worry, it might look a bit intimidating at first, but trust me, it's all about breaking it down into manageable steps. By the end of this, you'll be simplifying these fractions like a pro! We'll go through the process step-by-step, making sure we cover all the bases. So, grab your pencils, and let's get started. Simplifying algebraic fractions is a fundamental skill in algebra, and it's essential for solving more complex equations and problems. The key is to factorize both the numerator and the denominator and then cancel out any common factors. This makes the expression much easier to work with, and it also reveals the underlying structure of the expression. Let's learn how to simplify the given fraction.

Step 1: Factor Out the Greatest Common Factor (GCF) from the Numerator

Alright, first things first, we've got to look at the numerator, which is $4a^2 - 36a$. Our goal here is to find the greatest common factor (GCF) of the terms. In this case, both terms, $4a^2$ and $-36a$, share a common factor of $4a$. Let's factor that out. When we do that, we get:

$4a^2 - 36a = 4a(a - 9)$.

So, we've essentially rewritten the numerator in a factored form. This is a crucial step because it helps us identify potential cancellations later on. Remember, finding the GCF is all about finding the largest expression that divides evenly into all terms. This step simplifies the process significantly and gives us a clearer picture of the expression. Always start here, and you'll be well on your way to simplifying algebraic fractions efficiently. This ensures that we are working with the simplest form of the numerator. Pay close attention to the signs while doing this, because a sign error can change the entire result.

Now, let's move on to the next step where we handle the denominator. Getting the GCF right is like building a solid foundation for a house – it supports everything else that comes after!

Step 2: Factor Out the Greatest Common Factor (GCF) from the Denominator

Now, let's shift our focus to the denominator, which is $2a^4 - 24a^3 + 54a^2$. Just like we did with the numerator, we need to find the GCF. Here, the GCF is $2a^2$. Factoring this out, we get:

$2a^4 - 24a^3 + 54a^2 = 2a^2(a^2 - 12a + 27)$.

We've successfully factored the GCF from the denominator, making it easier to see if there are any common factors that we can cancel out. Now, we're not done yet, because the quadratic expression inside the parentheses, $a^2 - 12a + 27$, can also be factored further. This is where things get a bit more interesting, but don't worry, we'll break it down.

Finding the GCF in the denominator may seem complex, but it simplifies the overall process greatly. Keep in mind that a well-factored denominator makes the whole process smoother. Make sure to identify and factor out the greatest common factor correctly; this is a critical skill in algebra. When we have the GCF, we prepare to further factorize the expression within the parentheses, we are preparing the expression for a potential cancellation.

Step 3: Factor the Quadratic Expression in the Denominator

Alright, let's zoom in on the quadratic expression $a^2 - 12a + 27$. We need to factor this into two binomials. To do this, we're looking for two numbers that multiply to give us $27$ (the constant term) and add up to $-12$ (the coefficient of the $a$ term). After some thinking, you'll realize that $-3$ and $-9$ fit the bill because $(-3) * (-9) = 27$ and $(-3) + (-9) = -12$. Thus, we can rewrite the quadratic as:

$a^2 - 12a + 27 = (a - 3)(a - 9)$.

Now, our denominator becomes $2a^2(a - 3)(a - 9)$. We've completely factored the denominator, which is a big win! This step is critical because it reveals potential common factors between the numerator and denominator, which we'll use to simplify the fraction. Factoring quadratics can take a bit of practice, but with enough exercises, you will master it.

Also, remember, the goal here is to make the expression as simple as possible. This step prepares us for the final stage of simplification, where we will eliminate common factors. Make sure to understand this step since it is necessary to solve more complex problems in the future. Now, we are ready to move to the next stage where we are going to look for common factors between the numerator and the denominator, and then simplify.

Step 4: Rewrite the Entire Expression with the Factored Numerator and Denominator

Now that we've factored both the numerator and the denominator, let's put it all back together. Our original expression $\frac{4a^2 - 36a}{2a^4 - 24a^3 + 54a^2}$ becomes:

$\frac{4a(a - 9)}{2a^2(a - 3)(a - 9)}$.

This step is all about organizing our work. It makes it easier to spot the common factors that we can cancel out. We are now one step away from simplifying the whole expression. You can visualize this step as assembling all the pieces of a puzzle. Ensure that you have factored everything correctly. Be very careful with the signs and also the numerical factors. We have the numerator and denominator in the factored form, and the common factors will become more apparent.

Carefully rewriting the entire expression will prevent us from making mistakes. You can recheck if you have factored all the terms correctly to prevent any further errors. The proper arrangement prepares us for the last, crucial step: simplification by cancellation. Now we are ready to find the common factors between the numerator and denominator and then eliminate them, which will help us to simplify the algebraic fraction.

Step 5: Cancel Out Common Factors

This is where the magic happens! Look closely at the expression $\frac{4a(a - 9)}{2a^2(a - 3)(a - 9)}$. We can see that $(a - 9)$ is a common factor in both the numerator and the denominator. We can cancel these out. Also, we can simplify the numbers: $4$ in the numerator and $2$ in the denominator. $4$ divided by $2$ equals $2$. Also, we can cancel out one $a$ from the numerator and denominator. This leaves us with:

$\frac{2}{a(a - 3)}$.

And that's it! We have successfully simplified the algebraic fraction. The original expression has been transformed into a much simpler form, making it easier to work with. Canceling out common factors is the cornerstone of simplifying algebraic fractions, and it is a key skill. Always make sure to check for common factors before you declare the fraction simplified. After this step, the given fraction is in its simplest form. We are done!

This final step is the culmination of all the previous steps. With practice, you'll become adept at identifying and canceling common factors. It's like finding the hidden treasure in the expression! We have successfully simplified it to its simplest form. Now, the expression is much easier to work with and understand. We have reduced the expression to its simplest form. Make sure that you understand all the steps before moving on to other more complex problems. Excellent work! You have simplified the given expression. Feel proud of yourself.