Fraction Simplification: Step-by-Step Guide

Hey guys! Let's dive into the world of fraction simplification. This guide will walk you through how to express complex expressions as fractions, step by step. We'll tackle some tricky problems, break them down, and make them super easy to understand. So, grab your pencils, and let's get started!

a) (3-2a)/(2a) + (1-a²)/(a²)

In this section, we'll focus on how to simplify the expression (3-2a)/(2a) + (1-a²)/(a²). This involves combining two fractions with different denominators. The key here is to find a common denominator, which will allow us to add the numerators together. Let's break it down step by step.

First, let's identify our main objective. We need to express the sum of these two fractions as a single fraction. To do this, we must first find a common denominator. Looking at the denominators 2a and a², we can see that the least common multiple (LCM) is 2a². This is because 2a² is the smallest expression that both 2a and a² can divide into evenly.

Now, we need to rewrite each fraction with the common denominator of 2a². For the first fraction, (3-2a)/(2a), we multiply both the numerator and the denominator by a. This gives us (a(3-2a))/(2a²), which simplifies to (3a - 2a²)/(2a²). For the second fraction, (1-a²)/(a²), we multiply both the numerator and the denominator by 2. This gives us (2(1-a²))/(2a²), which simplifies to (2 - 2a²)/(2a²).

With both fractions now having the same denominator, we can add them together. We add the numerators while keeping the denominator the same: (3a - 2a²)/(2a²) + (2 - 2a²)/(2a²) = (3a - 2a² + 2 - 2a²)/(2a²). Next, we combine like terms in the numerator. We have -2a² and -2a², which combine to -4a². So, the numerator becomes 3a - 4a² + 2. Our fraction now looks like this: (3a - 4a² + 2)/(2a²).

Finally, we check if the fraction can be simplified further. In this case, there are no common factors between the numerator and the denominator, so the fraction is in its simplest form. Thus, the simplified expression is (-4a² + 3a + 2)/(2a²). Remember, always look for opportunities to simplify after each step to make the problem more manageable. This step-by-step approach will help you tackle similar problems with confidence!

b) 1/1 + (4-3B)/(B²-2B) + B = 2

Let's break down how to simplify the expression 1/1 + (4-3B)/(B²-2B) + B = 2. This problem involves adding fractions and a whole number, and it requires careful attention to factoring and finding common denominators. We'll walk through each step to make it clear and easy to follow. Buckle up, guys!

First, let's clarify our goal. We need to simplify the left-hand side of the equation and express it as a single fraction. This will involve combining the terms and, if necessary, isolating B to solve for its value. The first term 1/1 is simply 1, so we can rewrite the expression as 1 + (4-3B)/(B²-2B) + B = 2. Now, let's focus on the fraction (4-3B)/(B²-2B). The denominator B²-2B can be factored by taking out a common factor of B. This gives us B(B-2). So, our fraction becomes (4-3B)/(B(B-2)).

Next, we need to find a common denominator for all terms on the left side. We have 1, (4-3B)/(B(B-2)), and B. To combine these, we need to express 1 and B with the denominator B(B-2). We can rewrite 1 as B(B-2)/(B(B-2)) and B as B²(B-2)/(B(B-2)). Now, our expression looks like this: B(B-2)/(B(B-2)) + (4-3B)/(B(B-2)) + (B²(B-2))/(B(B-2)) = 2.

With the common denominator, we can combine the numerators: (B(B-2) + (4-3B) + B(B-2))/(B(B-2)) = 2. Expanding the numerators, we get (B² - 2B + 4 - 3B + B²)/(B(B-2)) = 2. Combining like terms in the numerator, we have (2B² - 5B + 4)/(B(B-2)) = 2. Now, let's multiply both sides of the equation by B(B-2) to get rid of the fraction: 2B² - 5B + 4 = 2B(B-2). Expanding the right side, we have 2B² - 5B + 4 = 2B² - 4B.

Next, we simplify the equation by subtracting 2B² from both sides: -5B + 4 = -4B. Adding 5B to both sides gives us 4 = B. Thus, we have found that B = 4. Always remember to double-check your solution by plugging it back into the original equation to ensure it holds true. This careful, step-by-step method will help you solve even the trickiest equations with confidence!

c) 1/(3x+y) + 1/(3x-y)

Alright, guys! Let's tackle this fraction simplification problem: 1/(3x+y) + 1/(3x-y). This involves adding two fractions with different denominators. The key here is to find a common denominator, which will allow us to add the numerators together. We'll break it down step by step, so you can see exactly how it's done.

First, we need to identify our main goal: to express the sum of these two fractions as a single fraction. To do this, we need to find a common denominator. Looking at the denominators (3x+y) and (3x-y), we can see that the least common multiple (LCM) is simply the product of the two denominators, which is (3x+y)(3x-y). This is because these two expressions don't share any common factors, so their product is the smallest expression that both can divide into evenly.

Now, we need to rewrite each fraction with the common denominator of (3x+y)(3x-y). For the first fraction, 1/(3x+y), we multiply both the numerator and the denominator by (3x-y). This gives us (1(3x-y))/((3x+y)(3x-y)), which simplifies to (3x-y)/((3x+y)(3x-y)). For the second fraction, 1/(3x-y), we multiply both the numerator and the denominator by (3x+y). This gives us (1(3x+y))/((3x-y)(3x+y)), which simplifies to (3x+y)/((3x-y)(3x+y)).

With both fractions now having the same denominator, we can add them together. We add the numerators while keeping the denominator the same: (3x-y)/((3x+y)(3x-y)) + (3x+y)/((3x-y)(3x+y)) = (3x-y + 3x+y)/((3x+y)(3x-y)). Next, we combine like terms in the numerator. We have 3x + 3x, which is 6x, and -y + y, which cancels out to 0. So, the numerator becomes 6x. Our fraction now looks like this: (6x)/((3x+y)(3x-y)).

Finally, we check if the fraction can be simplified further. We can expand the denominator using the difference of squares formula, which states that (a+b)(a-b) = a² - b². In our case, a = 3x and b = y, so (3x+y)(3x-y) = (3x)² - y² = 9x² - y². Therefore, our fraction becomes (6x)/(9x² - y²). There are no common factors between the numerator and the denominator, so the fraction is in its simplest form. Thus, the simplified expression is (6x)/(9x² - y²). Remember, always simplify as much as possible to get to the final answer!

I hope this helps you better understand how to simplify these types of expressions! Keep practicing, and you'll become a fraction master in no time!