Simplifying Rational Expressions Dividing \(\frac{4}{21-9x}\) By \(\frac{8}{14-6x}\)

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Introduction

In the realm of mathematics, simplifying rational expressions is a fundamental skill, especially when dealing with algebraic fractions. This in-depth guide aims to dissect and demystify the process of dividing the rational expression ${\frac{4}{21-9x}}$ by ${\frac{8}{14-6x}}$. We will explore each step meticulously, ensuring a comprehensive understanding for students and enthusiasts alike. This article not only provides a step-by-step solution but also enriches your understanding of the underlying principles governing algebraic manipulations. Let's embark on this mathematical journey together, turning complexity into clarity.

Understanding Rational Expressions

Before we dive into the division problem, it’s crucial to grasp the essence of rational expressions. A rational expression is essentially a fraction where the numerator and the denominator are polynomials. These expressions can involve variables, constants, and various algebraic operations. To master rational expressions, one must be adept at factoring, simplifying, and performing operations such as addition, subtraction, multiplication, and, of course, division. These skills form the backbone of more advanced algebraic concepts, making this topic a cornerstone of mathematical literacy.

The significance of rational expressions extends beyond textbook problems. They appear frequently in real-world applications, such as physics, engineering, and economics, where mathematical models often involve ratios and proportions. Understanding how to manipulate these expressions, therefore, has practical implications in various fields. To fully appreciate the division of rational expressions, it’s beneficial to review the basic principles of polynomial manipulation and factoring techniques. This foundational knowledge will make the process not only easier but also more intuitive.

Problem Statement: Dividing Rational Expressions

Our main objective is to simplify the expression obtained when dividing ${\frac{4}{21-9x}}$ by ${\frac{8}{14-6x}}$. This problem is an excellent example of how algebraic fractions can be manipulated and simplified using basic algebraic principles. Division of rational expressions might initially seem daunting, but it can be broken down into manageable steps. By following these steps carefully, we can transform a seemingly complex problem into a straightforward exercise in algebraic simplification. The key lies in understanding the underlying principles and applying them systematically. We will start by recapping the fundamental rule for dividing fractions and then apply this rule to our specific problem.

Step 1: Reciprocal and Multiplication

The first critical step in dividing fractions is to understand that division is equivalent to multiplying by the reciprocal. This is a fundamental rule that applies not only to simple numerical fractions but also to algebraic fractions. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For instance, the reciprocal of ${\frac{a}{b}}$ is ${\frac{b}{a}}$. Applying this principle to our problem, we transform the division of ${\frac{4}{21-9x}}$ by ${\frac{8}{14-6x}}$ into the multiplication of ${\frac{4}{21-9x}}$ by the reciprocal of ${\frac{8}{14-6x}}$, which is ${\frac{14-6x}{8}}$. This transformation is crucial because multiplication is often easier to handle than division, especially when dealing with algebraic expressions.

Detailed Explanation

To reiterate, instead of dividing by ${\frac{8}{14-6x}}$, we multiply by ${\frac{14-6x}{8}}$. This can be written mathematically as:

${ \frac{4}{21-9x} \div \frac{8}{14-6x} = \frac{4}{21-9x} \times \frac{14-6x}{8} }$

This step is not just a mechanical transformation; it’s a reflection of a core mathematical principle. Understanding why this transformation works is vital for building a solid foundation in algebra. By changing division to multiplication by the reciprocal, we set the stage for further simplification and manipulation of the expression. This initial transformation is often the most crucial step in solving such problems, as it sets the direction for the subsequent algebraic operations.

Step 2: Factoring

Factoring is a cornerstone of simplifying rational expressions. It allows us to break down complex expressions into simpler components, revealing common factors that can be canceled out. In our problem, both the expressions ${21-9x}$ and ${14-6x}$ can be factored. The expression ${21-9x}$ has a common factor of 3, and ${14-6x}$ has a common factor of 2. Factoring these out simplifies the expressions significantly.

Detailed Factoring Process

Let’s factor each expression individually:

1. Factoring ${21-9x}$: We can factor out 3 from both terms:

   ${ 21 - 9x = 3(7 - 3x) }$

2. Factoring ${14-6x}$: Similarly, we can factor out 2 from both terms:

   ${ 14 - 6x = 2(7 - 3x) }$

Now, we substitute these factored expressions back into our equation. The ability to identify and extract common factors is a vital skill in algebra. It not only simplifies expressions but also makes it easier to spot potential cancellations in later steps. By factoring, we reduce the complexity of the problem, making it more manageable and less prone to errors. Factoring is a technique that is used extensively in algebra, so mastering it is crucial for success in this field.

Step 3: Substituting Factored Expressions

After factoring the expressions ${21-9x}$ and ${14-6x}$, the next step is to substitute these factored forms back into our equation. This substitution is a pivotal step as it sets the stage for simplifying the rational expression by canceling out common factors. Replacing the original expressions with their factored forms helps to reveal the underlying structure of the expression, making it easier to identify and eliminate redundant components. This step showcases the power of factoring in simplifying complex algebraic expressions.

The Substitution Process

Recall that we transformed the division problem into the multiplication problem:

${ \frac{4}{21-9x} \times \frac{14-6x}{8} }$

We factored ${21-9x}$ as ${3(7-3x)}$ and ${14-6x}$ as ${2(7-3x)}$. Substituting these back into the equation, we get:

${ \frac{4}{3(7-3x)} \times \frac{2(7-3x)}{8} }$

This substitution might seem like a small step, but it’s a crucial link in the chain of simplification. By replacing the original expressions with their factored forms, we make the expression more amenable to further manipulation. The presence of common factors is now more apparent, paving the way for cancellation and simplification. This substitution is a clear example of how algebraic manipulation can transform a complex-looking expression into a simpler, more manageable form.

Step 4: Cancelling Common Factors

The magic of simplifying rational expressions often lies in the cancellation of common factors. After substituting the factored expressions, we arrive at:

${ \frac{4}{3(7-3x)} \times \frac{2(7-3x)}{8} }$

Here, we can observe the common factor ${(7-3x)}$ in both the numerator and the denominator. Canceling this common factor is a significant step in simplifying the expression. Additionally, we can simplify the numerical factors as well. This step not only reduces the complexity of the expression but also reveals its essential structure. The ability to identify and cancel common factors is a hallmark of algebraic proficiency.

Detailed Cancellation

Let's proceed with the cancellation step by step:

1. Cancel the common factor ${(7-3x)}$ from the numerator and the denominator:

   ${ \frac{4}{3\cancel{(7-3x)}} \times \frac{2\cancel{(7-3x)}}{8} = \frac{4}{3} \times \frac{2}{8} }$

2. Simplify the numerical factors. We can simplify ${\frac{2}{8}}$ to ${\frac{1}{4}}$:

   ${ \frac{4}{3} \times \frac{1}{4} }$

This cancellation process transforms the expression into a much simpler form. It’s a powerful technique that is widely used in algebra to reduce complex expressions to their simplest forms. By canceling common factors, we eliminate redundant components and reveal the underlying simplicity of the expression. This step highlights the elegance and efficiency of algebraic manipulation.

Step 5: Final Simplification

After canceling common factors, we have:

${ \frac{4}{3} \times \frac{1}{4} }$

The final step involves multiplying the remaining fractions and simplifying the result. This is a straightforward process, but it’s crucial to ensure accuracy in the arithmetic. Multiplying the numerators and the denominators gives us a single fraction, which can then be simplified to its lowest terms. This step completes the simplification process, leading us to the final, most concise form of the expression. The ability to perform this final simplification accurately is the culmination of all the previous steps, demonstrating a mastery of algebraic manipulation.

Detailed Simplification

Let’s multiply the fractions:

${ \frac{4 \times 1}{3 \times 4} = \frac{4}{12} }$

Now, simplify the fraction ${\frac{4}{12}}$ by dividing both the numerator and the denominator by their greatest common divisor, which is 4:

${ \frac{4 \div 4}{12 \div 4} = \frac{1}{3} }$

Thus, the final simplified form of the expression is ${\frac{1}{3}}$. This result showcases the power of algebraic simplification. What initially appeared to be a complex division problem involving rational expressions has been reduced to a simple fraction. This final step underscores the importance of careful arithmetic and attention to detail in mathematical problem-solving. The simplicity of the final result is a testament to the effectiveness of the techniques we have employed throughout the simplification process.

Conclusion

In conclusion, simplifying the rational expression ${\frac{4}{21-9x} \div \frac{8}{14-6x}}$ involves several key steps: converting division to multiplication by the reciprocal, factoring expressions, substituting factored forms, canceling common factors, and performing final simplification. Each step is crucial in reducing the expression to its simplest form, which in this case is ${\frac{1}{3}}$. This process not only demonstrates the mechanics of simplifying rational expressions but also reinforces the importance of understanding the underlying algebraic principles. The journey from the initial complex expression to the final simple fraction highlights the power of algebraic manipulation and the elegance of mathematical simplification.

Recap of the Process

Let’s briefly recap the steps we followed:

1. Division to Multiplication: We converted the division problem into a multiplication problem by using the reciprocal of the second fraction.
2. Factoring: We factored the expressions in the numerators and denominators to identify common factors.
3. Substitution: We substituted the factored expressions back into the equation.
4. Cancellation: We canceled the common factors present in both the numerator and the denominator.
5. Final Simplification: We performed the final multiplication and simplified the resulting fraction.

By following these steps meticulously, we were able to transform a complex expression into a simple fraction. This methodical approach is applicable to a wide range of algebraic problems and is a valuable skill for anyone studying mathematics. The final result, ${\frac{1}{3}}$, is not just an answer; it’s a testament to the power and beauty of algebraic simplification. Mastering these techniques is essential for success in algebra and beyond.