Simplify Algebraic Fractions: Step-by-Step

Hey guys! Today, we're diving deep into the sometimes tricky world of algebraic simplification, specifically tackling expressions involving fractions. You know, those problems that look a bit daunting at first glance with all the variables and denominators flying around? Well, fear not! By the end of this article, you'll be a pro at simplifying expressions like (3p+1)/(3p-1) - (3p-1)/(3p+1) divided by 36p²/(3p-1). We'll break it down piece by piece, making sure you understand every single step. So, grab your notebooks, and let's get this done!

Understanding the Problem: What Are We Simplifying?

Alright, let's get our heads around the expression we're working with: (3p+1)/(3p-1) - (3p-1)/(3p+1) : 36p²/(3p-1). It looks like a mouthful, right? But really, it's just a combination of a few algebraic concepts we're all familiar with: subtraction of fractions and division of fractions. The key here is to remember the order of operations (PEMDAS/BODMAS) – we handle the division after we've sorted out the subtraction within the parentheses. This is a super important rule in algebra, and messing it up can lead to totally wrong answers. So, the first step is always to look at the structure of the expression and figure out what needs to be done first. In our case, we have a subtraction happening inside parentheses, and that whole result is then divided by another fraction. This means we need to simplify the part inside the parentheses before we even think about the division. It's like unwrapping a present – you have to get through the outer layers first to reach the main event. So, focus on the subtraction first, guys. That's where the real work begins for this particular problem. Once that's squared away, the division part will be much easier to handle.

Step 1: Simplifying the Subtraction Within the Parentheses

Okay, team, let's tackle the first major part of our problem: the subtraction inside the parentheses. We have (3p+1)/(3p-1) - (3p-1)/(3p+1). To subtract fractions, whether they're numbers or algebraic expressions, we need a common denominator. This is probably the most crucial step in fraction arithmetic. Think about it: you can't just subtract apples from oranges, right? You need them to be in the same units to do the subtraction. The same applies here. Our denominators are (3p-1) and (3p+1). The easiest way to find a common denominator is usually to multiply the two denominators together. So, our common denominator will be (3p-1)(3p+1).

Now, we need to adjust each fraction so that it has this new, common denominator. For the first fraction, (3p+1)/(3p-1), we need to multiply its denominator by (3p+1) to get our common denominator. To keep the fraction equivalent, we must multiply the numerator by the same thing: (3p+1). So, the first fraction becomes [(3p+1)*(3p+1)] / [(3p-1)*(3p+1)], which simplifies to (3p+1)² / (3p-1)(3p+1).

For the second fraction, (3p-1)/(3p+1), we need to multiply its denominator by (3p-1). Again, we do the same to the numerator: (3p-1). So, this fraction becomes [(3p-1)*(3p-1)] / [(3p+1)*(3p-1)], which simplifies to (3p-1)² / (3p-1)(3p+1).

Now that both fractions have the same denominator, we can subtract their numerators: (3p+1)² - (3p-1)². Don't forget to keep that common denominator (3p-1)(3p+1) underneath!

Expanding the Numerators: A Little Algebra Magic

Before we subtract the numerators, we need to expand (3p+1)² and (3p-1)². Remember the perfect square formulas? (a+b)² = a² + 2ab + b² and (a-b)² = a² - 2ab + b².

For (3p+1)²: Here, a = 3p and b = 1. So, (3p)² + 2*(3p)*(1) + 1² = 9p² + 6p + 1. Easy peasy!

For (3p-1)²: Here, a = 3p and b = 1. So, (3p)² - 2*(3p)*(1) + 1² = 9p² - 6p + 1. Looking good!

Now, let's substitute these expanded forms back into our numerator subtraction: (9p² + 6p + 1) - (9p² - 6p + 1). Be super careful with the minus sign here, guys. It applies to everything inside the second set of parentheses. So, we distribute the minus sign: 9p² + 6p + 1 - 9p² + 6p - 1.

Let's combine like terms:

• 9p² - 9p² = 0 (These cancel out!)
• 6p + 6p = 12p
• 1 - 1 = 0 (These cancel out too!)

So, the simplified numerator is just 12p!

The Denominator: A Special Product

Now, let's look at our common denominator: (3p-1)(3p+1). This is another special product, known as the difference of squares: (a-b)(a+b) = a² - b².

In our case, a = 3p and b = 1. So, (3p-1)(3p+1) = (3p)² - 1² = 9p² - 1.

Putting it all together, the expression inside the parentheses simplifies to 12p / (9p² - 1).

Step 2: Performing the Division

Alright, we've conquered the subtraction part! Our expression now looks like this: [12p / (9p² - 1)] : [36p² / (3p - 1)]. Remember, the colon : means division. Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of a fraction is just the fraction flipped upside down. So, the reciprocal of 36p² / (3p - 1) is (3p - 1) / 36p².

Our division problem now becomes a multiplication problem: [12p / (9p² - 1)] * [(3p - 1) / 36p²].

When multiplying fractions, we multiply the numerators together and the denominators together: (12p * (3p - 1)) / ((9p² - 1) * 36p²).

Now, let's simplify this beast. We can often cancel out common factors between the numerator and the denominator before we multiply everything out. This makes things so much easier, guys!

Let's rewrite the denominator (9p² - 1) using the difference of squares we found earlier: (3p - 1)(3p + 1). So, our expression is: (12p * (3p - 1)) / ((3p - 1)(3p + 1) * 36p²).

Now, look for common factors:

• We have (3p - 1) in both the numerator and the denominator. They cancel out! Poof!
• We have 12p in the numerator and 36p² in the denominator. Both are divisible by 12p.
  • 12p / 12p = 1
  • 36p² / 12p = 3p

Let's substitute these cancellations back into our expression. Our numerator becomes 1 * 1 = 1. Our denominator becomes (3p + 1) * 3p.

So, the final simplified expression is 1 / (3p * (3p + 1)).

Step 3: Final Simplification and Checking Our Work

We have 1 / (3p * (3p + 1)). We could distribute the 3p in the denominator to get 1 / (9p² + 3p), but usually, leaving it in factored form is preferred in algebra, as it's often more useful for further calculations. So, 1 / (3p(3p + 1)) is our final answer!

Let's quickly recap the process:

1. Identify the operations: Subtraction inside parentheses, followed by division.
2. Simplify inside the parentheses: Find a common denominator for the subtraction, subtract the numerators, and expand/simplify.
3. Rewrite the denominator: Use the difference of squares formula.
4. Perform the division: Multiply by the reciprocal of the divisor.
5. Cancel common factors: Simplify the resulting multiplication.

This methodical approach is key to mastering algebraic simplification. Always take it one step at a time, be careful with signs, and remember your algebraic rules like common denominators, difference of squares, and multiplying by the reciprocal. Practice makes perfect, guys! The more you do these, the faster and more confident you'll become. Keep practicing, and you'll be simplifying expressions like this in your sleep!

Why This Matters: The Power of Algebraic Skills

So, why do we bother with all these steps, you ask? Well, simplifying algebraic expressions is a fundamental skill in mathematics. It's like learning your multiplication tables before tackling calculus. When you can simplify complex expressions, you make them easier to understand, analyze, and use in further calculations. This skill is crucial not just for passing your algebra exams, but also for fields like engineering, physics, computer science, economics, and even advanced statistics. Imagine trying to solve a complex physics problem with a giant, unsimplified equation – it would be a nightmare! By simplifying, you reveal the core relationships and make the problem tractable.

Furthermore, understanding how to manipulate algebraic fractions helps build a strong foundation for more advanced topics. Concepts like finding common denominators, factoring, and canceling terms are building blocks for solving equations, graphing functions, and understanding calculus. The process we followed – breaking down a complex problem into smaller, manageable steps – is a valuable problem-solving strategy that applies to many areas of life, not just math. So, while it might seem tedious at times, mastering these algebraic manipulations truly empowers you to tackle more complex challenges and unlock a deeper understanding of the mathematical world around us. Keep at it, guys; the effort is absolutely worth it!