Simplifying (a*b) / (3*a - 6*b) * (4*a - 8*b) / (a^2 * B) A Step-by-Step Guide

In mathematics, simplifying expressions is a fundamental skill, especially when dealing with algebraic fractions. This article provides a detailed, step-by-step guide on how to simplify the expression ${\frac{a b}{3 a-6 b} \times \frac{4 a-8 b}{a^2 b}}$. We will explore the necessary techniques, including factoring, canceling common factors, and arriving at the simplest form of the expression. This guide is designed to help students and anyone interested in mathematics to enhance their understanding and skills in algebraic manipulation.

Understanding the Basics of Algebraic Fractions

Before we dive into the simplification process, it’s essential to understand the basics of algebraic fractions. An algebraic fraction is a fraction where the numerator and the denominator are algebraic expressions. Simplifying these fractions involves reducing them to their simplest form, much like reducing numerical fractions. The key techniques include factoring and canceling common factors.

In our expression, ${\frac{a b}{3 a-6 b} \times \frac{4 a-8 b}{a^2 b}}$, we have two algebraic fractions multiplied together. To simplify this, we will first factor the numerators and denominators, and then look for common factors that can be canceled out. This process makes the expression easier to manage and understand.

Factoring: The First Key Step

Factoring is the process of breaking down an algebraic expression into its constituent factors. This is a crucial step in simplifying algebraic fractions because it allows us to identify common factors in the numerator and the denominator. Let's start by factoring the denominators in our expression.

The first denominator is ${3a - 6b}$. We can factor out the common factor of 3 from both terms: ${ 3a - 6b = 3(a - 2b) }$

Next, we factor the second denominator, which is ${4a - 8b}$. Similarly, we can factor out the common factor of 4: ${ 4a - 8b = 4(a - 2b) }$

Now that we have factored the denominators, let’s rewrite the expression with the factored terms: ${ \frac{a b}{3(a-2 b)} \times \frac{4(a-8 b)}{a^2 b} }$

Canceling Common Factors: Simplifying the Expression

After factoring, the next step is to cancel common factors between the numerators and the denominators. This is based on the principle that any factor appearing in both the numerator and the denominator can be canceled out, as it is equivalent to dividing by 1. In our expression, we can identify several common factors.

Let's rewrite our expression: ${ \frac{a b}{3(a-2 b)} \times \frac{4(a-2 b)}{a^2 b} }$

First, notice the term ${(a - 2b)}$ appears in both the numerator and the denominator. We can cancel this factor out: ${ \frac{a b}{3 \cancel{(a-2 b)}} \times \frac{4\cancel{(a-2 b)}}{a^2 b} = \frac{a b}{3} \times \frac{4}{a^2 b} }$

Next, we can see that ${a}$ and ${b}$ are common factors. We have ${ab}$ in the first numerator and ${a^2b}$ in the second denominator. We can cancel out one ${a}$ and one ${b}$ from both terms:

${ \frac{\cancel{a b}}{3} \times \frac{4}{a^{\cancel{2}} \cancel{b}} = \frac{1}{3} \times \frac{4}{a} }$

Multiplying the Simplified Fractions

Now that we have canceled out all the common factors, we are left with a much simpler expression. The next step is to multiply the remaining fractions. To do this, we multiply the numerators together and the denominators together: ${ \frac{1}{3} \times \frac{4}{a} = \frac{1 \times 4}{3 \times a} }$

This simplifies to: ${ \frac{4}{3a} }$

So, the simplified form of the given expression is ${\frac{4}{3a}}$.

Step-by-Step Breakdown of the Simplification Process

To recap, let’s break down the entire process into clear, manageable steps. This will help reinforce your understanding and make it easier to apply these techniques to other problems.

1. Write down the original expression: ${ \frac{a b}{3 a-6 b} \times \frac{4 a-8 b}{a^2 b} }$

2. Factor the denominators:

   • Factor ${3a - 6b}$ as ${3(a - 2b)}$.
   • Factor ${4a - 8b}$ as ${4(a - 2b)}$.

3. Rewrite the expression with factored terms: ${ \frac{a b}{3(a-2 b)} \times \frac{4(a-2 b)}{a^2 b} }$

4. Cancel common factors:

   • Cancel the ${(a - 2b)}$ terms.
   • Cancel one ${a}$ and one ${b}$ term.

5. Rewrite the simplified expression: ${ \frac{1}{3} \times \frac{4}{a} }$

6. Multiply the remaining fractions: ${ \frac{1 \times 4}{3 \times a} }$

7. Simplify to the final form: ${ \frac{4}{3a} }$

By following these steps, you can systematically simplify complex algebraic expressions involving fractions. The key is to factorize and identify common factors that can be canceled out.

Common Mistakes to Avoid

When simplifying algebraic expressions, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct answer. Here are some common mistakes to watch out for:

1. Incorrect Factoring: Factoring is a critical step, and an error here can derail the entire simplification process. Always double-check your factored expressions to ensure they are correct. For instance, ensure you have factored out the greatest common factor and that the remaining expression is accurate.

2. Canceling Terms Instead of Factors: One of the most frequent errors is canceling terms that are not factors. Remember, you can only cancel factors that are multiplied, not terms that are added or subtracted. For example, in the expression ${\frac{a + 2}{a}}$, you cannot cancel the ${a}$ terms because ${2}$ is being added to ${a}$ in the numerator.

3. Forgetting to Multiply Remaining Terms: After canceling common factors, ensure you multiply the remaining terms correctly. It’s easy to overlook this step and end up with an incorrect simplification. Take your time and methodically multiply the numerators and denominators.

4. Not Simplifying Completely: Sometimes, students might cancel some factors but not simplify the expression to its fullest extent. Always look for additional common factors that can be canceled to reach the simplest form. This often involves reviewing your expression one last time after the initial simplification.

5. Sign Errors: Sign errors are common, especially when dealing with negative numbers. Pay close attention to signs when factoring and multiplying. A small sign mistake can change the entire outcome of the simplification.

By keeping these common mistakes in mind and practicing regularly, you can improve your accuracy and confidence in simplifying algebraic expressions.

Practice Problems

To reinforce your understanding and skills, let’s work through a few practice problems. These examples will help you apply the steps we’ve discussed and build your confidence in simplifying algebraic fractions.

Problem 1: Simplify the expression ${ \frac{2x^2}{5x} \times \frac{10}{4x} }$

Solution:

1. Write down the original expression: ${ \frac{2x^2}{5x} \times \frac{10}{4x} }$

2. Cancel common factors:

   • Cancel ${x}$ from ${x^2}$ and ${x}$ in the denominators.
   • Simplify ${\frac{10}{5}}$ to ${2}$ and ${\frac{2}{4}}$ to ${\frac{1}{2}}$.

3. Rewrite the simplified expression: ${ \frac{2x}{1} \times \frac{2}{4x} }$ ${ \frac{2x}{1} \times \frac{1}{2x} }$

4. Multiply the remaining fractions: ${ \frac{2x \times 1}{1 \times 2x} }$

5. Simplify to the final form: ${ \frac{2x}{2x} = 1 }$

So, the simplified form of the expression is ${1}$.

Problem 2: Simplify the expression ${ \frac{a^2 - b^2}{a + b} \times \frac{1}{a - b} }$

Solution:

1. Write down the original expression: ${ \frac{a^2 - b^2}{a + b} \times \frac{1}{a - b} }$

2. Factor the numerator:

   • Factor ${a^2 - b^2}$ as ${(a + b)(a - b)}$.

3. Rewrite the expression with factored terms: ${ \frac{(a + b)(a - b)}{a + b} \times \frac{1}{a - b} }$

4. Cancel common factors:

   • Cancel the ${(a + b)}$ terms.
   • Cancel the ${(a - b)}$ terms.

5. Rewrite the simplified expression: ${ \frac{\cancel{(a + b)}\cancel{(a - b)}}{\cancel{a + b}} \times \frac{1}{\cancel{a - b}} }$

6. Simplify to the final form: ${ 1 }$

So, the simplified form of the expression is ${1}$.

Problem 3: Simplify the expression ${ \frac{x^2 + 4x + 4}{x + 2} \times \frac{1}{x + 2} }$

Solution:

1. Write down the original expression: ${ \frac{x^2 + 4x + 4}{x + 2} \times \frac{1}{x + 2} }$

2. Factor the numerator:

   • Factor ${x^2 + 4x + 4}$ as ${(x + 2)(x + 2)}$.

3. Rewrite the expression with factored terms: ${ \frac{(x + 2)(x + 2)}{x + 2} \times \frac{1}{x + 2} }$

4. Cancel common factors:

   • Cancel one ${(x + 2)}$ term.

5. Rewrite the simplified expression: ${ \frac{\cancel{(x + 2)}(x + 2)}{\cancel{x + 2}} \times \frac{1}{x + 2} }$

6. Simplify to the final form: ${ 1 }$

So, the simplified form of the expression is ${1}$.

These practice problems illustrate how to apply the step-by-step method to various algebraic expressions. Remember to always factorize first, cancel common factors, and then simplify the remaining terms.

Conclusion: Mastering Simplification Techniques

In conclusion, simplifying algebraic expressions involving fractions is a crucial skill in mathematics. By following a systematic approach—factoring, canceling common factors, and multiplying—you can reduce complex expressions to their simplest forms. This article has provided a comprehensive guide, complete with step-by-step instructions, common mistakes to avoid, and practice problems to enhance your understanding.

Mastering these simplification techniques not only improves your problem-solving abilities but also lays a solid foundation for more advanced mathematical concepts. Consistent practice and attention to detail are key to becoming proficient in simplifying algebraic fractions. Whether you are a student learning these concepts for the first time or someone looking to refresh your skills, this guide offers the tools and knowledge you need to succeed.

By understanding the importance of factoring, identifying common factors, and avoiding common mistakes, you can confidently tackle a wide range of simplification problems. Keep practicing, and you’ll find that simplifying algebraic expressions becomes second nature. Remember, mathematics is a skill that improves with practice, and with the right approach, you can master even the most challenging concepts.